Useful stuff

Here I'm putting useful commands, dotfiles and links to remember.

Links

I love:

• Owls in Towels
• Electron Band Structure In Germanium, My Ass
• Max Carr's PCT blog

Commands

To convert a bunch of images from one format to another via ImageMagick:

mogrify -format jpg *.heic

To enable scientific notation in qalc

set exp # set exp 0 to disable

Dotfiles

[env]
TERM = "xterm-256color"

[window]
opacity = 0.8
blur = true

[window.dimensions]
columns=100
lines=30

[font]
normal.family = "MesloLGS NF"
size = 12

Using the PIC16F88 as an I2C device

Final project for PHYS 335A: Digital Electronics, taught Spring of 2024 by Dr. David Pengra at the University of Washington.

The PIC16F88 is a 16-bit, lightweight microcontroller from Microchip devices which is useful for a variety of low-cost and low-power, high speed embedded systems operations.

Here I describe some details of how I try to set up an I${}^2$C implementation on the PIC16F88, and design a program using MPASM Assembly language to operate it as an I2C target device (slave), to be controlled by higher-level controller (master) devices such as Arduinos or Raspberry Pis.

Note: MPASMx is no longer supported as of MPLAB X v5.40 - use >v5.35 if you're intending to use it here. See the footnote¹ for more details on this & using OSx.

Introduction to I2C

I${}^2$C (or inter-integrated circuit) is, like so many things in electrical engineering, a complicated-looking protocol that's actually pretty simple in a clever sort of way.

When compared to its closest serial communications cousin SPI (or serial peripheral interface), I${}^2$C only requires 2 datalines (SDA and SCL) compared to SPI's 4 (MISO, MOSI, SCLK and CS), and allows for far more devices to be connected due to the use of peripheral addressing rather than SPI's chip select.

The master device / controller, in my case an Arduino Mega 2560, specifies some baudrate to operate the clock frequency on. When no data transmission is happening, both SDA and SCL are pulled to Vdd through the pullup resistors ($R$ above - typical choices range from 4.7k$\Omega$ to 10k$\Omega$).

Since this implementation of I${}^2$C is open-drain, the controller and target oscillate lines by just directly grounding them - the pullup resistors avoid shorting the circuit out this way. Note this means that the longer SCL=0 or SDA=0, the more power the circuit will use by extension.

Pullup resistor choice is important if you're trying to optimize - small PU resistors have high power consumption but also high speeds, while large PU resistors have low power consumption but also create capacitance delays. Texas Instruments has a good description of this.

By my understanding, I${}^2$C (in 7-bit addressing space²) generally operates like this:

1. Controller device sends a START bit by holding SDA low with SCL high.
2. Controller sends a 7-bit address, one bit per clock cycle.
3. At the 8th bit (8th time SDA goes HIGH), target devices compare the sent address with their own. If the two match, continue. Otherwise, ignore - this stream isn't meant for that device.
4. Controller and specified target communicate.
5. END bit set by either to indicate the conversation ifs over.

Each 'communication sequence' happens in 8-bit intervals - 7 bits for data, the 8th allowing time for each device to tell the other what to expect next. The PIC16F88 SSP section of the datasheet has a handy visualization of the target transmitting data back to the host - what we're planning on doing here.

So - let's talk about the project.

I2C Rangefinder Project

We'll use the PIC16F88 in conjunction with the HC-SR04 ultrasonic rangefinder module to create a rangefinding device, retrieving the rangefinder values from the PIC by communicating over I${}^2$C.

For our controller, we'll use an Arduino Mega 2560 - chosen because it allows serial console monitoring & has 5V I${}^2$C logic by default (the PIC16F88 uses 5V, and thus has 5V I${}^2$C outputs), and has a number of high-level logic libraries for interfacing with I${}^2$C (see the Wire library for more information).

It is possible to use an I${}^2$C controller with 3.3V logic (such as a Raspberry Pi) with 5V target devices or visa-versa, however you'll need to convert logic levels between the two, either with a dedicated logic-level conversion device or by using N-channel MOSFETs and pull-up resistors. Refer to NXP note AN10441 for more information.

Hardware

Making a circuit diagram for this,

For my pullup resistors I decided to use 4.7k$\Omega$, as I'm not particularly worried about power consumption on a proof-of-concept. Note that RA<x> refers to pins PORTA<x>, and similarly RB<x> refer to pins PORTB<x>.

Note: it's quite important that the controller and target devices share a common ground - otherwise, the "reference ground voltage" for I${}^2$C pulses may not be the same and a circuit may not function as hoped!

Here's the physical implementation of this circuit.

Software

First, I made sure code for the rangefinder was working before fiddling with I${}^2$C. I made use of some of the code provided in Lab 8 - Sonic Ranger with Interrupts & Hardware Control, ensuring the rangefinder worked by tinkering with the debugger and observing variable changes in response to changed echo distances from the rangefinder. This could have also been done by reusing our Lab 7 code, but I thought the Lab 8 code looked much neater.

Now - onto the software side of I${}^2$C. The SSP section on I${}^2$C in the PIC16F88 datasheet is unfortunately quite lacking with regards to exact implementation methods, so I was forced to scrounge around a bit through sources official and otherwise for information.

In brief, the PIC16F88 doesn't easily support a controller-mode implementation of I${}^2$C out of the box, lacking the MSSP register found in other PICs that would allow easier implementation.

Instead, the 'F88 supports a variety of both SPI and I${}^2$C target device implementations that may be used instead - refer to Register 10.2 in the datasheet SSP section. If you're looking to implement master-mode on a PIC, look to other devices such as the PIC16F1508 which have MSSP registers.

Using the PIC16F88 datasheet section on SSP (paying particular attention to section 10.3.1) as well as the section on interrupts and the section on configuring PORTB, here's the process I followed to implement "I${}^2$C Slave mode, 7-bit address with Start and Stop bit interrupts enabled".

1. Clear ANSEL to enable digital (instead of analog) inputs.
2. Set TRISB<1> (SDA) and TRISB<4> (SCL) pins, turning them into inputs.
   • "But wait! I${}^2$C is bidirectional!" you might argue - this is true. The SSP (synchronous serial port) module in the PIC16F88 will automatically set and clear TRISB<1>/<4> in response to SDA and SCL events.
3. Set bits in SSPCON (SSP control register):
   • SSPCON<5> - SSPEN, enables SSP
   • SSPCON<4> - CKP, clock polarity, allowing controller to oscillate SCL
   • SSPCON<3:0>=1110 - SSPM<3:0>, "SSP mode select". See Register 10.2 in the SSP section of the datasheet for other options - by using 1110, we're setting our SSP mode to I${}^2$C target mode with 7 bit addressing, start/stop interrupts enabled.
4. Choose an address for your device. It must not be 0x00 or 0x01, other values allowed up to 0xFF. I chose 0x77 arbitarily. Write it into SSPADD.
   • Since only bits SSPSR<7:1> are compared to SSPADD, run an RLF instruction (rotate-left-through-carry) on SSPADD to make sure your address matches the checked one.
5. Enable interrupts. Set bits:
   • INTCON<7> - GIE, enable global interrupts
   • INTCON<6> - PEIE, enable peripheral interrupts
   • PIE1<3> - SSPIE, enable SSP interrupts
6. Create interrupt functions in your ISR space
   • Check if PIR<3> (SSPIF) is set to make sure the interrupt was caused by I${}^2$C, if there are multiple possible interrupts in your code.
   • Function to Save STATUS and W as you normally would.
   • WriteData - to send data, write (less than 8 bits total) to SSPBUF, then set SSPCON<4> to indicate to the controller that you (the target) are ready to transmit. SSPBUF will now start automatically sending.
   • Function to Load as you normally would
   • Clear PIR1<3> (SSPIF) to reset the SSP ...
7. ... and RETFIE.

My ISR implementation following 6-7 above looks like this:

; Interrupt Service Routine ----------------------------------------------------

	ORG 0x0004 ; ISR beginning

SaveState
	MOVWF SAVE_W ; Save W register
	SWAPF STATUS, W ; Save STATUS reg
	MOVWF SAVE_STAT ; ... into temporary reg

LoadAndSend
	MOVFW TimerCounts ; Load last pulse period into W
	MOVWF SSPBUF ; Load SSPBUF with W (will be sent)
	BSF SSPCON,CKP ; Set CKP bit to indicate our buffer is ready

LoadState
	SWAPF SAVE_STAT,W ; Load STATUS
	MOVWF STATUS
	SWAPF SAVE_W, F ; Load W
	SWAPF SAVE_W, W ; Load W into W
	
	BCF PIR1,SSPIF ; Clear serial interrupt flag
	RETFIE ; ... and return to program execution
	
; End ISR ----------------------------------------------------------------------

and relevant I${}^2$C code blocks like this:

; I2C initialization subroutine ------------------------------------------------

SetI2C
    BANKSEL	ANSEL           ; Bank 1
    CLRF	ANSEL           ; Set to all-digital inputs
    BSF		TRISB, TRISB1	; Set PORTB<1> (SDA) as an input
    BSF		TRISB, TRISB4	; Set PORTB<4> (SCL) as an input
    CLRF	SSPSTAT		; Reset SSPSTAT
    
    BANKSEL	SSPCON		; Bank 0
    BSF		SSPCON, SSPEN	; Turn on SSP
    BSF		SSPCON, CKP	; Enable clock (if 0, holds clock low)
    BSF		SSPCON, SSPM3	; I2C 7-bit slave-mode w/ int is SSPM<3:0>=1110
    BSF		SSPCON, SSPM2
    BSF		SSPCON, SSPM1
    BCF		SSPCON, SSPM0
    
    BANKSEL	SSPADD		; Bank 1
    MOVLW	I2C_ADDR	; Load I2C address into W
    MOVWF	SSPADD		; Load SSPADD with I2C_ADDR (0xIC random)
    RLF		SSPADD,F	; Left-shift since SSPSR<7:1> compared
    
    ; Interrupt configuration
    BANKSEL	INTCON		; Bank 0
    BSF		INTCON, GIE	; Enable global interrupts - SSP interrupt on START
    BSF		INTCON, PEIE	; Enable peripheral interrupts
    BANKSEL	PIE1		; Select PIE register
    BSF		PIE1, SSPIE	; Enable SSP interrupts in peripheral interrupt reg
	
; End subroutine ---------------------------------------------------------------

Results

Using the Arduino IDE with the Wire library, I set up a basic script that, after initializing I${}^2$C and the serial monitor both at a baudrate of 9600, would send an I${}^2$C "read" query down SDA to address 0x77 (our PIC) to try and retrieve our TimerCounts value from our interrupt function LoadAndSend shown above.

#include <Wire.h>

// Notes: D21 is SCL, D20 is SDA for MEGA 2560

void setup() {
  // Initialize I2C
  Wire.begin();
  
  // Start serial output for console monitoring
  Serial.begin(9600);  
  Serial.println("Setup complete. Starting loop ...");
}

// Poll 0x77 address - then sleep for two seconds - then poll.
void loop() {
  // Request 2 bits from 0x77
  Wire.requestFrom(0x77, 1);

  // Send results (if any) to serial monitor
  while (Wire.available()) {
    char c = Wire.read();
    Serial.print("Distance: ");
    Serial.print(c, DEC);
    Serial.println(" cm");
  }

  delay(2000);
}

And, after some tinkering, checking the serial monitor revealed ...

Voilà!!

Notes

I definitely didn't get this first try - I wrote the Arduino script pretty early on to monitor the PIC16F88 while I figured out the maze of interrupt flags necessary to work with I${}^2$C.

Something I should note - I've wasted at least a few hours during this project because I wasn't operating on the correct bank while trying to operate on some register. Such problems won't often be immediately apparent, only becoming clear when you step through your code with a debugger and realize a value isn't changing when it should. BANKSEL is a friend!

Code

The full program I used for this is written below.

;*******************************************************************************
;                                                                             
;    Filename: Final Project I2C Ranger                                       
;    Date: 5/31/2024                                                           
;    File Version: 1.0                                                            
;    Author: Parker Lamb                                                                   
;    Description: Sets PIC16F88 up as I2C-ready ultrasonic rangefinding device.
;                                                                              
;*******************************************************************************

;*******************************************************************************
;
;                Procesor initial setup
;
;*******************************************************************************
    
    list       F=inhx8m, P=16F88, R=hex, N=0 ; File format, chip, and default radix

#include p16f88.inc ;   PIC 16f88 specific register definitions

    __config _CONFIG1, _MCLR_ON & _FOSC_INTOSCCLK & _WDT_OFF & _LVP_OFF & _PWRTE_OFF & _BODEN_ON & _LVP_OFF & _CPD_OFF & _WRT_PROTECT_OFF & _CCP1_RB0 & _CP_OFF
    __config _CONFIG2 , _IESO_OFF & _FCMEN_OFF

    Errorlevel -302 ; switches off msg [302]: Register in operand not in bank 0.

;*******************************************************************************
;
;                Constants and variables
;
;*******************************************************************************

; Program vars
TimerCounts EQU     h'20'	; Saving timer counts
 
;  vars
SAVE_W	    EQU	    h'21'	; Interrupt temporary W storage
SAVE_STAT   EQU	    h'22'	; Interrupt temporary STATUS FSR storage
   
; I2C status
I2C_STAT    EQU	    h'23'	; Check if I2C is connected
I2C_ADDR    EQU	    0x77	; I2C address (const)
	    
; Delay count registers
DInd1	    EQU	    h'24'
DInd2	    EQU	    h'25'
	    
; Delay times
DTime1	    EQU	    .199	; 60 ms delay - outer loop
DTime2	    EQU	    .60		; Nested loop runs for 59,941 cycles
	    
; SCL and SDA locations
#define	    _SDA    PORTB,RB1	; Make _SDA easier to check
#define	    _SCL    PORTB,RB4	; Make _SCL easier to check
	    
;*******************************************************************************
;
;                Memory init & interrupts
;
;*******************************************************************************
 
	    ORG	    0x00
	    GOTO    Init
	    
; Interrupt Service Routine ----------------------------------------------------
            ORG     0x0004               ; ISR beginning
		    
SaveState
    MOVWF	SAVE_W		; Save W register
    SWAPF	STATUS, W	; Save STATUS reg
    MOVWF	SAVE_STAT	; ... into temporary reg
    
LoadAndSend
    MOVFW	TimerCounts	; Load last pulse period into W
    MOVWF	SSPBUF		; Load SSPBUF with W (will be sent)
    BSF		SSPCON,CKP	; Set CKP bit to indicate our buffer is ready
    
LoadState
    SWAPF	SAVE_STAT,W	; Load STATUS
    MOVWF	STATUS
    SWAPF	SAVE_W, F	; Load W
    SWAPF	SAVE_W, W	; Load W into W
    
    BCF		PIR1,SSPIF
    RETFIE			; ... and return to program execution

;*******************************************************************************
;
;                Program start
;
;*******************************************************************************
Init	    ORG	    0x0020
	    
; Set up oscillator to 4 MHz
SetOsc_4MHz 
    BANKSEL	OSCCON
    CLRF	OSCCON
    BSF		OSCCON, IRCF1   ; Bit 5
    BSF		OSCCON, IRCF2   ; Bit 6

    ; Tuned value from Lab 7 - adjust as needed
    MOVLW	0x16
    MOVWF	OSCTUNE

; Reset IO to all digital, all outputs, clear latches
ResetIO
    BANKSEL	PORTA   ; Clear data latch registers
    CLRF	PORTA
    CLRF	PORTB
    BANKSEL	TRISA   ; Data direction on PORTA / PORTB
    CLRF	TRISA   ; Set PORTA to all output
    CLRF	TRISB   ; Set PORTB to all output
    BANKSEL	ANSEL   ; Select digital / analogue register
    CLRF	ANSEL   ; And set to all-digital inputs
	  
; ... and set what we need to
SetIO
    BANKSEL	TRISA
    MOVLW	0xFF
    MOVWF	TRISA		; Set PORTA direction to all inputs
    BCF		TRISA, RA0	; ... except PORTA<0>, an output
 
SetTMR0
    ; TMR0 duration between pulses indicates our radar distance
    BANKSEL	OPTION_REG	; TMR0 operation controlled via OPTION_REG
    CLRF	OPTION_REG	; Clear it - sets PSA to TMR0, low-hi edge
    BSF		OPTION_REG, 0x0
    BSF		OPTION_REG, 0x2	; Set prescaler rate to 1:64 w.r.t oscillator (101)
    BANKSEL	INTCON		; Select interrupt control register
    BCF		INTCON,TMR0IE	; Disable TMR0 OF interrupt (not using here)
    BCF		INTCON,TMR0IF	; Clear flag (maybe unnecessary - do anyway)
    
;*******************************************************************************
;
;                I2C configuration
;   We're using this device in 7-bit I2C slave-mode with stop + start
;   interrupts enabled. 
;
;*******************************************************************************
    
SetI2C
    BANKSEL	TRISB		; Bank 1
    BSF		TRISB, TRISB1	; Set PORTB<1> (SDA) as an input
    BSF		TRISB, TRISB4	; Set PORTB<4> (SCL) as an input
    
;    BANKSEL	SSPSTAT		; Bank 1
    CLRF	SSPSTAT		; Reset SSPSTAT
    
    BANKSEL	SSPCON		; Bank 0
    BSF		SSPCON, SSPEN	; Turn on SSP
    BSF		SSPCON, CKP	    ; Enable clock (if 0, holds clock low)
    BSF		SSPCON, SSPM3	; I2C 7-bit slave-mode w/ int is SSPM<3:0>=1110
    BSF		SSPCON, SSPM2
    BSF		SSPCON, SSPM1
    BCF		SSPCON, SSPM0
    
    BANKSEL	SSPADD		; Bank 1
    MOVLW	I2C_ADDR	; Load I2C address into W
    MOVWF	SSPADD		; Load SSPADD with I2C_ADDR (0xIC random)
    RLF		SSPADD,F	; Left-shift since SSPSR<7:1> compared
    
    ; Interrupt configuration
    BANKSEL	INTCON		; Bank 0
    BSF		INTCON, GIE	; Enable global interrupts - SSP interrupt on START
    BSF		INTCON, PEIE	; Enable peripheral interrupts
    BANKSEL	PIE1		; Select PIE register
    BSF		PIE1, SSPIE	; Enable SSP interrupts in peripheral interrupt reg

;*******************************************************************************
;
;                Main program loop
;   Cause the sonic module to pulse every 10 microseconds. Wait for a response,
;   then save time it took into a register. Repeatedly do this.
;
;   Once I2C interrupt is detected, send information via I2C. 
;
;   TODO only start this loop if START condition detected, stop with I2C STOP.
;
;*******************************************************************************
    ; Reset to bank 0
    BCF	    STATUS, RP0
    BCF	    STATUS, RP1	    ; All of the action is in Bank 0 now
	    
MainLoop    
; Send sonic device a 10-microsecond pulse
Pulse
    BSF		PORTA, RA0	; Set PORTA<0> output to high
    NOP
    NOP
    NOP
    NOP
    NOP
    NOP
    NOP
    NOP
    NOP
    BCF		PORTA, RA0	; Set PORTA<0> output low

; Loop until PORTA<1> goes HI, indicating a response
WaitForResp
    BTFSS	PORTA, RA1	; Check PORTA<1>
    GOTO	WaitForResp	; ... and loop if it's still zero

    ; Start of our response
    CLRF	TMR0		; Start timer from now
WaitUntilLow
    BTFSC	PORTA, RA1	; Check PORTA<1> to see if response finished
    GOTO	WaitUntilLow	; ... loop if still not LOW
    
    ; We have a response! Store timer value in a variable
    MOVFW	TMR0
    MOVWF	TimerCounts
    
    ; Delay for a bit so we aren't constantly polling
    CALL	Delay
    
    ; Return to MainLoop
    GOTO	MainLoop

;*******************************************************************************
;
;                Subroutines
;
;*******************************************************************************
; Loop from Template_for_Ranger_with_interrupts.asm, course lab template
Delay
		MOVLW   DTime2
		MOVWF   DInd2
Loop1		MOVLW   DTime1
		MOVWF   DInd1
Subloop1	NOP
		NOP
		DECFSZ  DInd1,F
		GOTO    Subloop1
		DECFSZ  DInd2,F
		GOTO    Loop1
	RETURN
    
Finish
    END

The code for the Mega 2560 is written in Results.

1. With version v5.40 of MPLAB X, Microchip's dedicated compiler/IDE, Microchip upgraded all their binaries to 64-bit. However, since mpasmx was not updated from 32 bits, and Mac OSx does not support 32-bit applications since Mojave, Mac users will need to either use Windows or a VM. ↩

2. See Wikipedia's page for addressing structure for more details. ↩

Building an ASCOM frontend for the MRO

As part of one of my final projects with the UW AUEG, this post covers the construction of a custom frontend interface to interact with the new Sidereal Technology ForceOne (F1) telescope controller we're installing April-June 2025.

Many of the telescope systems within the UW Manastash Ridge Observatory haven't been updated in a while, with 'a while' meaning (in the 21st century) to (ever). Given the telescope was originally built in 1975, that means many of the systems and electronics are ancient, creaky and sometimes just plain don't work.

One of those systems in the 'ever' category is the old telescope control system, which controls the telescope mount and helps tell it where to point - quite a critical system. Alas, like a cranky child, it's one that is prone to issues, needs constant monitoring and wire-jiggling, and just generally can be a headache sometimes.

So, after summoning our inner crusty old engineers and putting on some gloves, we've decided to rip out everything and replace it with ✨ better stuff ✨.

The Promised Land a.k.a. the ForceOne Brushless Controller uses a protocol called ASCOM ("Astronomy Common Object Model") which is a standardized set of instructions that we can use to communicate with motorized telescope mounts.

ASCOM¹ is really just a set of standards for communicating with a telescope - it's like the metric measuring system, or Python's PEP8 style guide. It covers most of the use cases for interacting with a telescope, including focusing, GOTO, dome controls, filter wheels, even safety monitoring.

We're only focusing on one subset of these commands: those that can be used to control the telescope mount.

There are some pre-existing GUIs and frontends for working with ASCOM, but I didn't like any of them and some of them were expensive so we decided to make our own.

In all seriousness, most existing GUIs are designed around general use cases for ASCOM, and either have too many options for our limited use, or too few. By making our own user interface, we can make something for our own use case, and just for it.

Design and implementation

Generating a mockup for such a frontend in Figma as a way to guide development:

1. ASCOM is cool. If you're a nerd like me and love cool APIs, see the ASCOM API which is a comprehensive set of HTTP API endpoints that are broadcast by the controller (the F1 in our case) and are how we communicate with the telescope base. ↩

Chapter 1 - Relevant Mathematics

Reference "Introduction to Electrodynamics" by David Griffiths.

Vectors

We start with a brief review of vector algebra - I'll skip most of it but keep the key things.

Dot product

The dot product is a measure of how "parallel" two vectors are, maximized when parallel, minimized (0) when perpendicular.

$$\begin{array}{rl}\vec{a}\cdot \vec{b}& =a_1{b}_1+a_2{b}_2+a_3{b}_3\\ & =|\vec{a}||\vec{b}|\cos \theta \end{array}$$

It is commutative ($\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$) and distributive ($\vec{a}\cdot (\vec{b}+\vec{c})=\vec{a}\cdot \vec{b}+\vec{a}\cdot \vec{c}$). The result of the dot product is a scalar.

Cross product

The cross product yields a third vector orthogonal to both $\vec{a}$ and $\vec{b}$, maximized in magnitude when $\vec{a}$ and $\vec{b}$ are themselves orthogonal to one another.

The cross product is distributive, but not exactly commutative - instead, $$(\vec{b}\times \vec{a})=-(\vec{a}\times \vec{b})$$ Some other properties:

• The result of the cross product is a vector, not a scalar.
• The magnitude of the cross product is the area of the parallelogram generated by $\vec{a}$ and $\vec{b}$.
• The cross product of a vector with itself is zero.

We can calculate a cross product like so:

$$\begin{array}{rl}\vec{a}\times \vec{b}& =\left|\begin{array}{ccc}\hat{x}& \hat{y}& \hat{z}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|\\ & =|\vec{a}||\vec{b}|\sin \theta \hat{n}\end{array}$$ where $\hat{n}$ is the normal to the plane formed by $\vec{a}$ and $\vec{b}$.

Vector triple product

Triple products are combinations of cross and dot products.

1. Scalar triple products: the magnitude of this is the volume of the parallelepiped (3D parallelogram) generated by $\vec{a}$, $\vec{b}$ and $\vec{c}$. $$\begin{array}{rl}\vec{a}\cdot (\vec{b}\times \vec{c})& =\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|\end{array}$$

2. Vector triple products: there's no easy geometric interpretation of this, but is useful to reduce complex cross product calculations. It can be memorized by the mnemonic BAC-CAB. $$\vec{a}\times (\vec{b}\times \vec{c})=\vec{b}(\vec{a}\cdot \vec{c})-\vec{c}(\vec{a}\cdot \vec{b})$$

Spatial vectors

The position vector indicates the position of a point relative to the origin.

In electrodynamics, usually we'll have two points: one for the source charge and one for the test charge. The vector $\vec{\mathcal{R}}$ from one to the other is the useful quantity then.

The Del Operator

Del ($\nabla$) is a vector operator that acts upon functions, and is defined as $$\nabla =\frac{d}{dx}\hat{x}+\frac{d}{dy}\hat{y}+\frac{d}{dz}\hat{z}$$ $\nabla$ really only is meaningful when applied to functions, and has three (usual) ways of being applied:

1. To a scalar function ($\nabla f$) to get the gradient of that function.
2. To a vector function via the dot product ($\nabla \cdot \vec{a}$) to get that function's divergence.
3. To a vector function via the cross product ($\nabla \times \vec{a}$) to get that function's curl.

Gradients

The gradient of some scalar function $f$ gives a vector result that will point in the direction of the max rate of increase of that function.

$$\nabla f=\frac{df}{dx}\hat{x}+\frac{df}{dy}\hat{y}+\frac{df}{dz}\hat{z}$$ and results in a vector with the derivative of the original function's units.

Fundamental theorem for gradients

Important: The integral of a derivative (here, the gradient) is given by the value of the function at the boundaries $\vec{a}$ and $\vec{b}$. $${\int }_{\vec{a}}^{\vec{b}}(\nabla T)\cdot d\vec{l}=T(\vec{b})-T(\vec{a})$$

Divergence

The divergence of some vector function $\vec{a}$ represents how much the function "spreads out" or diverges from a given point. $$\nabla \cdot \vec{a}=\frac{d{a}_x}{dx}+\frac{d{a}_y}{dy}+\frac{d{a}_z}{dz}$$ and results in a dimensionless scalar value indicating the rate of divergence from a point.

Green's theorem

Green's theorem is a special application of Stoke's theorem (see below). Geometrically, it relates the sources contained in some surface to the flux emitted by those sources around the boundary of a surface. $${\int }_V(\nabla \cdot \vec{v})d\tau ={\oint }_S\vec{v}\cdot d\vec{a}$$ In the case of electrostatics, this might be the flux field generated by some charges in a volume to the total, superposition flux coming out of that entire region - turning one "field of charges" into a single charge.

The integral of a derivative over some region will always be equal to the value of the function at the boundary ($P$) - in this case, the boundary term is an integral (a simple line may just have two endpoints, but a line bounding a volume forms a closed sufrface).

Curl

The curl of some vector function $\vec{a}$ represents how much the function "swirls around" some given point. Positive curl is given by the right-hand rule (normally counterclockwise). $$\nabla \times \vec{a}=\hat{x}\left(\frac{d{a}_z}{dy}-\frac{d{a}_y}{dz}\right)+\hat{y}\left(\frac{d{a}_x}{dz}-\frac{d{a}_z}{dx}\right)+\hat{z}\left(\frac{d{a}_y}{dx}-\frac{d{a}_x}{dy}\right)$$

Stokes' Theorem

The integral of the curl over some surface is the "total amount of swirl" on that surface - and mathematically can be distilled into just finding how much the flow is following the boundary of that surface. $${\int }_S(\nabla \times \vec{v})\cdot d\vec{a}={\oint }_P\vec{v}\cdot d\vec{l}$$ This last quantity, ${\oint }_P\vec{v}\cdot d\vec{l}$ is sometimes called the "circulation" of some vector field $\vec{v}$.

A Note on Del Squared (${\nabla }^2$)

Enumerating over the ways we can apply $\nabla$ twice:

1. Laplacian: the cross product of $\nabla$ with the gradient of a function results in the Laplacian, such that $$\nabla \cdot (\nabla f)={\nabla }^{2}f=\frac{d^{2}f}{d{x}^2}+\frac{d^{2}f}{d{y}^2}+\frac{d^{2}z}{d{z}^2}$$
2. Curl of gradient: always zero. $$\nabla \times (\nabla f)=0$$
3. Gradient of divergence: not a lot of physical applications, calculable though. $$\nabla (\nabla \cdot \vec{a})$$
4. Divergence of curl: always zero. $$\nabla \cdot (\nabla \times \vec{v})=0$$
5. Curl of curl: most easily defined as the acceleration of the swirl, like a hurricane speeding up. $$\nabla \times (\nabla \times \vec{a})=\nabla (\nabla \cdot \vec{a})-{\nabla }^2\vec{a}$$

Integrals

In electrodynamics, we have line (path) integrals, surface integrals (or flux) and volume integrals.

Line integrals

Iterate over infinitesimally small displacement vectors $d\vec{l}$. $${\int }_a^b\vec{a}\cdot d\vec{l}$$ If $a=b$, the line integral iterates over a closed loop and we can write $$\oint \vec{a}\cdot d\vec{l}$$

In elementary physics, a common example is work done by a force. $$W=\int \vec{F}\cdot d\vec{l}$$

While path taken normally matters (distance), there are some vectors where only displacement matters (for forces, these are called conservative forces, like gravity).

Wikipedia has a really cool animation of line integrals:

Surface integrals

Surface integrals iterate over infinitesimally small areas $d\vec{a}$, with norms perpendicular to the surface. $${\int }_{\mathcal{S}}\vec{v}\cdot d\vec{a}$$ In this case, if the surface is closed (i.e. like a sphere, rather than a hill), then we can write it in closed loop form. $$\oint \vec{v}\cdot d\vec{a}$$

Surface integrals are really useful for things like flux where something is moving through some area - i.e. imagine radio waves passing through a curved gas cloud or radiation falling "through" a planet.

Volume integrals

Volume integrals integrate over infinitesimally small volumes $d\tau =dxdydz$.

$$\int \vec{a}d\tau =\int (a_x\hat{x}+a_y\hat{y}+a_z\hat{z})d\tau$$

If some function $f(x,y,z)$ represented density of a substance, then the volume integral over it would be total mass.

Integration by Parts

Also known as the "inverse product rule". $$\int udv=uv-\int vdu$$

Curvilinear Coordinates

Taking another look at spherical and cylindrical coordinates, as well as how to convert between them.

Spherical coordinates

Three terms:

• $r$: the distance from the origin (range 0 to $\infty$)
• $\theta$: the polar angle from the $z$-axis (range 0 to $\pi$)
• $\varphi$: the azimuthal angle along the $xy$-axis (range 0 to $2\pi$)

$$\begin{array}{rl}x& =r\sin \theta \cos \varphi \\ y& =r\sin \varphi \sin \theta \\ z& =r\cos \theta \end{array}$$ Alternatively in terms of the unit vectors:

$$\begin{array}{r}\left[\begin{array}{c}\hat{x}\\ \hat{y}\\ \hat{z}\end{array}\right]=\left[\begin{array}{ccc}\sin \theta \cos \varphi & \cos \theta \cos \varphi & -\sin \theta \\ \sin \theta \sin \varphi & \cos \theta \sin \varphi & \cos \theta \\ \cos \theta & -\sin \varphi & 0\end{array}\right]\left[\begin{array}{c}\hat{r}\\ \hat{\theta }\\ \hat{\varphi }\end{array}\right]\end{array}$$

The interesting thing about the matrix form is that it is orthogonal - according to the invertible matrix theorem, $Q^{-1}=Q^T$ for orthogonal matrices.

We can use this to solve for $\hat{r}$, $\hat{\theta }$ and $\hat{\varphi }$ easily then by just taking the transpose of the above matrix.

$$\begin{array}{r}\left[\begin{array}{c}\hat{r}\\ \hat{\theta }\\ \hat{\varphi }\end{array}\right]=\left[\begin{array}{ccc}\sin \theta \cos \varphi & \sin \theta \sin \varphi & \cos \theta \\ \cos \theta \cos \varphi & \cos \theta \sin \varphi & -\sin \theta \\ -\sin \theta & \cos \varphi & 0\end{array}\right]\left[\begin{array}{c}\hat{x}\\ \hat{y}\\ \hat{z}\end{array}\right]\end{array}$$ Our line element in spherical coordinates is

$$d\vec{l}=dr\hat{r}+rd\theta \hat{\theta }+r\sin \theta d\varphi \hat{\varphi }$$

In terms of triple integrals, the textbook has a good graphic to represent the three displacements $d{l}_r,d{l}_{\theta },d{l}_{\varphi }$.

And a infinitesimal volume element is the product of these three:

$$d\tau =d{l}_{r}d{l}_{\theta }d{l}_{\varphi }=r^2\sin \theta drd\theta d\varphi$$

$r$ has a possible range from 0 to $\infty$, $\theta$ from 0 to $\pi$ and $\varphi$ from 0 to $2\pi$.

The textbook represents the spherical representations of the gradient, divergence, curl and Laplacian in eqs. 1.70-1.73.

Cylindrical coordinates

Three terms:

• $s$: distance from $z$-axis
• $\varphi$: azimuthal angle along $xy$-axis
• $z$: height along z-axis

$$\begin{array}{rl}x& =s\cos \varphi \\ y& =s\sin \varphi \\ z& =z\end{array}$$ with unit vectors $$\left[\begin{array}{c}\hat{x}\\ \hat{y}\\ \hat{z}\end{array}\right]=\left[\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}\hat{s}\\ \hat{\varphi }\\ \hat{z}\end{array}\right]\left[\begin{array}{c}\hat{s}\\ \hat{\varphi }\\ \hat{z}\end{array}\right]=\left[\begin{array}{ccc}\cos \varphi & \sin \varphi & 0\\ -\sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}\hat{x}\\ \hat{y}\\ \hat{z}\end{array}\right]$$

For triple integrals, the infinitesimal displacements are $$d{l}_s=ds,d{l}_{\varphi }=sd\varphi ,d{l}_z=dz$$ our volume element is $$d\tau =sdsd\varphi dz$$ and line displacement is $$d\vec{l}=ds\hat{s}+sd\varphi \hat{\varphi }+dz\hat{z}$$

Dirac Delta Function

Imagine we have an infinite point mass at the origin, and it is the only thing in the universe.

The total mass of the universe must be just the mass of the point charge; but the point charge exists only at $(0,0,0)$, and is a point charge - it has no spatial dimension. If we were to integrate over the entire universe, we'd find we have just the mass of the point charge - and can mathematically represent that with the Dirac delta function.

$$\delta (x)=\{\begin{array}{ll}0& \text{if }x\ne 0\\ \infty & \text{if }x=0\end{array}$$ and $${\int }_{-\infty }^{\infty }\delta (x)dx=1$$ Therefore, if we have some continuous function $f(x)$ (could be a constant or a function, anything attached to $\delta (x)$). $$f(x)\delta (x)=\{\begin{array}{ll}0& \text{if }x\ne 0\\ f(0)& \text{if }x=0\end{array}$$ which brings us to the neat and nice identity: $${\int }_{-\infty }^{\infty }f(x)\delta (x-a)dx=f(a)$$

Dirac Delta in 2D, 3D

$${\delta }^3(\vec{r})=\delta (x)\delta (y)\delta (z)$$

And, unsurprisingly, $${\int }_{\text{all space}}f(\vec{r}){\delta }^3(\vec{r}-\vec{a})d\tau =f(\vec{a})$$

Note on $\mathcal{R}$: with $\mathcal{R}=\vec{r}-\overrightarrow{r^{\prime }}$ (the separation vector), $$\nabla \cdot \left(\frac{\vec{\mathcal{R}}}{{\mathcal{R}}^2}\right)=4\pi {\delta }^3(\vec{\mathcal{R}})$$ and $${\nabla }^2\frac{1}{\mathcal{R}}=-4\pi {\delta }^3(\vec{\mathcal{R}})$$ Reference the textbook equations 1.100-1.102 for derivation.

Chapter 2 - Electrostatics

Reference "Introduction to Electrodynamics" by David Griffiths.

Fundamentally, electrodynamics seeks to model the interaction of some set of charges $q_1,q_2,q_3$ on some other charge, $Q$. We can do this via superposition, which states the interaction between two charges is unaffected by the presence of others - i.e. $\overrightarrow{F_1}$, the force between $q_1$ and $Q$, is independent of the presence of $q_2$ and $q_3$. We can then sum up the forces to get our net force on $Q$: $${\vec{F}}_{\text{net}}=\overrightarrow{F_1}+\overrightarrow{F_2}+\overrightarrow{F_3}$$ In an electrostatic system (no moving charges as opposed to an electrodynamic system), the forces can be calculated via Coulomb's law.

Coulomb's Law

Electrostatics simply takes into account distance and charge strength. Coulomb's law is

$$\vec{F}=\frac{1}{4\pi {ϵ}_0}\frac{qQ}{{\mathcal{R}}^2}\hat{\mathcal{R}}\equiv \frac{1}{4\pi {ϵ}_0}\frac{qQ}{{\mathcal{R}}^3}\vec{\mathcal{R}}$$

${ϵ}_0$ is the permittivity of free space (${ϵ}_0=8.85\times 10^{-12}$ ${\text{C}}^2/\text{N}\cdot {\text{m}}^2)$ and $\mathcal{R}=\vec{r}-{\vec{r}}^{\prime }$ (Griffith's script-r).

in electrostatics, force increases with higher-magnitude charges, and decreases with distance.

If we have several charges, we just sum their individual Coulomb forces as calculated above:

$$\begin{array}{rl}\vec{F}& =\overrightarrow{F_1}+\overrightarrow{F_2}+\cdots \\ & =\frac{Q}{4\pi {ϵ}_0}\left(\frac{q_1}{{{\mathcal{R}}_1}^2}\widehat{{\mathcal{R}}_1}+\frac{q_2}{{{\mathcal{R}}_2}^2}\widehat{{\mathcal{R}}_2}+\cdots \right)\end{array}$$ In shorthand, $$\vec{F}=Q\vec{E}$$ where $\vec{E}$, the electric field, is $$\vec{E}=\frac{1}{4\pi {ϵ}_0}\sum _{i=1}^n\frac{q_i}{{{\mathcal{R}}_i}^2}\widehat{{\mathcal{R}}_i}$$ where ${ϵ}_0=8.85\times 10^{-12}\frac{C^2}{N\cdot m}$.

The electric field is the force per unit charge - this force can only be imparted by the presence of other charges.

Continuous Charge Distributions

If we have a set of charges $q_i$, then our electric field is generated from a discrete charge distribution. If, however, the charge is distributed continuously over some region, then we have a continuous charge distribution generating the $\vec{E}$-field.

The electric field calculated as an integral over that continuous distribution: $$\vec{E}(\vec{r})=\frac{1}{4\pi {ϵ}_0}\int \frac{1}{{\mathcal{R}}^2}\hat{\mathcal{R}}dq$$ where $dq$ varies for our continuous distribution type (1D, 2D or 3D), such that:

• Line: $dq=\lambda ({\vec{r}}^{\prime })d{l}^{\prime }$
• Area: $dq=\sigma ({\vec{r}}^{\prime })d{a}^{\prime }$
• Volume: $dq=\rho ({\vec{r}}^{\prime })d{\tau }^{\prime }$

Regarding the vectors,

• $\vec{r}$ is from the origin to test point; above a disk, this might be be $z\hat{z}$.
• ${\vec{r}}^{\prime }$ is from the origin to infinitesimal elements; often $r\hat{r}$ or $x\hat{x}$.
• $\vec{\mathcal{R}}=\vec{r}-{\vec{r}}^{\prime }$, ${\mathcal{R}}^2=|\vec{\mathcal{R}}{|}^2$.

The $\hat{R}$ will usually yield two unit vectors (i.e. $z\hat{z}-x\hat{x}$). Use two separate integrals with the same $dq$ element to handle.

Divergence and Curl of Electrostatic Fields

Griffiths 2.2.

Field lines are a way to visualize electric fields on a 2D or 3D medium.

Source.

where the density of field lines indicates the strength of the E-field at that location (strongest near the center in the above image). The flux of $\vec{E}$ through some surface $S$ is $${\varphi }_E={\int }_S\vec{E}\cdot d\vec{a}$$ and is a way to measure the total charge contained within a closed surface, while those outside of the surface don't affect it.

This mathematically represented by Gauss's law, which states $$\oint \vec{E}\cdot d\vec{a}=\frac{1}{{ϵ}_0}{Q}_{\text{enc}}$$ There's also a differential version of Gauss's law (found by applying the Divergence Theorem) where $\rho$ is charge density: $$\vec{\nabla }\cdot \vec{E}=\frac{1}{{ϵ}_0}\rho$$

For example: in 3D, if a point charge $q$ is at the origin then the flux of $\vec{E}$ through a sphere around it is $$\oint \vec{E}\cdot d\vec{a}=\int \frac{1}{4\pi {ϵ}_0}\left(\frac{1}{r^2}\hat{r}\right)\cdot (r^2\sin \theta d\theta d\varphi \hat{r})=\frac{1}{{ϵ}_0}q$$

We're only evaluating at the surface of the sphere, so $r$ is constant.

Note that Gauss's law is only useful with objects where the E-field is pointing in the same direction as elements $d\vec{a}$. This requires both a symmetrical charge distribution within the object, and for the Gaussian surface to be symmetric according to one of the following:

1. Spherically symmetric (concentric spheres)
2. Cylindrically symmetric (coaxial cylinders)
3. Plane symmetric (like a "box" that is cut in half by the plane)

The associated coordinate system is used for each.

Note: flux from external charges is ignored, since the flux entering a Gaussian surface from an external charge will be equal to the flux leaving that surface from the other side - hence, net flux from those external charges is zero.

If the object we're evaluating doesn't have perpendicular surface elements (i.e. a charge at the north pole of a sphere instead of the center), then some problems can be approached by creating another Gaussian element centered over the charge which subtends the same angle, though this can get geometrically complicated.

Divergence of $\vec{E}$

$$\begin{array}{rl}\vec{\nabla }\cdot \vec{E}& =\frac{1}{4\pi {ϵ}_0}\int \vec{\nabla }\cdot \left(\frac{\hat{\mathcal{R}}}{{\mathcal{R}}^2}\right)\rho ({\vec{r}}^{\prime })d{\tau }^{\prime }\end{array}$$ Since $\vec{\nabla }\cdot \left(\frac{\hat{\mathcal{R}}}{{\mathcal{R}}^2}\right)=4\pi {\delta }^3(\vec{\mathcal{R}})$, then this just becomes $$\vec{\nabla }\cdot \vec{E}=\frac{1}{4\pi {ϵ}_0}\int 4\pi {\delta }^3(\vec{r}-{\vec{r}}^{\prime })\rho (\overrightarrow{r^{\prime }})d{\tau }^{\prime }=\frac{1}{{ϵ}_0}\rho (\vec{r})$$ Or, in shortform, $$\vec{\nabla }\cdot \vec{E}=\frac{1}{{ϵ}_0}\rho (\vec{r})$$

Curl of $\vec{E}$

For any electrostatic charge distribution, the curl of the field is always zero. $$\vec{\nabla }\times \vec{E}=\vec{0}$$

Electric Potential $V$

Griffiths 2.3.

Electric potential $V$ is a good indicator of the "strength" of an electric field between two points, and is defined as $$V(\vec{r})=-{\int }_{\mathcal{O}}^{\vec{r}}\vec{E}\cdot d\vec{l}$$ with units $J/C$.

A "natural" origin $\mathcal{O}$ is a point infinitely far from the charge, though only when the charge distribution itself doesn't also extend to infinity.

in differential form, it's the reverse: $$\vec{E}=-\vec{\nabla }V$$ Voltage is path-independent and only displacements matter - hence the electric potential between points $\vec{a}$ and $\vec{b}$ is $$V(\vec{b})-V(\vec{a})=-{\int }_{\vec{a}}^{\vec{b}}\vec{E}\cdot d\vec{l}$$

Poisson & Laplace Equations

Poisson's equation says $${\nabla }^{2}V=-\frac{\rho }{{ϵ}_0}$$ If there is no charge ($\rho =0$), Poisson's equation turns into Laplace's equation: $${\nabla }^{2}V=0$$

Voltage equations

For a point charge $q$ $$V(\vec{r})=\frac{1}{4\pi {ϵ}_0}\frac{q}{\mathcal{R}}$$

$\vec{r}$ is the reference point from origin, $\mathcal{R}$ from the reference point $\vec{r}$ to the infinitesimal charge ${\vec{r}}^{\prime }$ (or reference point to point charge).

For a set of point charges, we can use the principle of superposition: $$V(x,y,z)=\frac{1}{4\pi {ϵ}_0}\left[\frac{q}{\sqrt{x^2+y^2+(z-d{)}^2}}-\frac{q}{\sqrt{x^2+y^2+(z+d{)}^2}}\right]=\frac{1}{4\pi {ϵ}_0}\sum _{i=1}^n\frac{q_i}{{\mathcal{R}}_i}$$

For a continuous distribution, $$V(\vec{r})=\frac{1}{4\pi {ϵ}_0}\int \frac{1}{\mathcal{R}}dq$$ and for a volume (to compute $V$ when we know $\rho$): $$V(\vec{r})=\frac{1}{4\pi {ϵ}_0}\int \frac{\rho ({\vec{r}}^{\prime })}{\mathcal{R}}d{\tau }^{\prime }$$

Note: this is similar to the formula for electric field, but is missing $\hat{\mathcal{R}}$. $V$ is a scalar quantity, directionless.

Note on continuity

For some surface,

• The normal component of ${\vec{E}}_{\perp }$ is discontinuous by some amount $\sigma /{ϵ}_0$ at a boundary.
  • So, if the surface is uncharged ($\sigma =0$), $\vec{E}$ is continuous across the boundary.
• The parallel component of ${\vec{E}}_{||}$ is continuous (${\vec{E}}_{\text{above}}^{||}={\vec{E}}_{\text{below}}^{||}$). Representing both, $${\vec{E}}_{\text{above}}-{\vec{E}}_{\text{below}}=\frac{\sigma }{{ϵ}_0}\hat{n}$$ The electric potential $V$ is always continuous across any boundary (however the gradient of $V$ inherits the discontinuity in $\vec{E}$, since $\vec{E}=-\vec{\nabla }V$).

Summary

The three "fundamentals" of electrostatics are $\rho$, $\vec{E}$ and $V$, and are related by the following:

Griffith's Fig. 2.35.

Work and Energy

Griffith's 2.4.

For a conservative force, work is force times displacement. In electrostatics, our equivalent version is $$\Delta V=\frac{W}{Q}W=Q\Delta V$$

More extended, $$W={\int }_{\vec{a}}^{\vec{b}}\vec{F}\cdot d\vec{l}=-Q{\int }_{\vec{a}}^{\vec{b}}\vec{E}\cdot d\vec{l}=Q[V(\vec{b})-V(\vec{a})]$$

where work here is the work it takes, per unit charge, to carry a charge from $\vec{a}$ to $\vec{b}$. If our reference point $\vec{a}=\infty$ and we let $\vec{b}=\vec{r}$, then $$W=Q[V(\vec{r})-V(\infty )]\equiv QV(\vec{r})$$

Work in a Point Charge Distribution

Imagine we had some charge $q_1$ - then brought in a new charge $q_2$ to join $q_1$. Our work done would be

It would cost $$W_2=q_2{V}_1(\overrightarrow{r_2})=\frac{1}{4\pi {ϵ}_0}{q}_2\left(\frac{q_1}{r_{12}}\right)$$ Then, to bring in a third charge $q_3$ would act against the superposition of $q_1$ and $q_2$: $$W_3=\frac{1}{4\pi {ϵ}_0}{q}_3\left(\frac{q_1}{r_{13}}+\frac{q_2}{r_{23}}\right)$$ ... etc. To assemble $n$ charges, we'd need a work of

$$W=\frac{1}{2}\sum _{i=1}^n{q}_i\left(\sum _{j\ne i}^n\frac{1}{4\pi {ϵ}_0}\frac{q_j}{r_{ij}}\right)=\frac{1}{2}\sum _{i=1}^n{q}_{i}V(\overrightarrow{r_i})$$

Work in a Continuous Charge Distribution

For a continuous charge distribution, we use our last equation for our point charge distribution system and replace the sum $\sum$ with an integral $\int$. $$W=\frac{1}{2}\int \rho Vd\tau$$

Note: $V$ is not volume, but potential. This can be done via Gauss's law as well, see this hyperphysics article.

With a bit of mathematics, this is equivalent to $$W=\frac{{ϵ}_0}{2}\int E^{2}d\tau$$

Conductors

Griffith's 2.5.

Conductors are materials in which electrons are free to roam (like cows, on an open ranch 👍). In contrast, insulators have electrons pretty much immobile and packed-together (like Monsanto farms 👎).

We can approximate metals as ideal-case conductors (though perfect conductors don't yet exist, though we're gradually coming closer) with the following attributes:

1. $\vec{E}=\vec{0}$ inside a conductor - or, the induced field will cancel the external field. If an external field $\overrightarrow{E_0}$ is applied to a conductor, free electrons will move toward the E-field until they all sit on the surface, creating a deficit of charges on the opposite surface and positively charging it. Crucially, the net E-field is zero, so the fields must cancel inside: $$\overrightarrow{E_1}+\overrightarrow{E_0}=0\to \overrightarrow{E_1}=-\overrightarrow{E_0}$$ Charge will continue to flow until the cancellation is complete. Outside the conductor, $\vec{E}\ne 0$, since the two fields don't tend to cancel.
2. Charge density is zero $(\rho =0)$: Since Gauss's law says $\vec{\nabla }\cdot \vec{E}=\rho /{ϵ}_0$, zero E-field means zero charge density in the conductor.
3. Net charge resides on surface: positive and negative charges will only sit on the surface after enough time passes.
4. $\vec{E}$ is perpendicular to the surface: somewhat obvious, but bears mentioning.

Any dynamical system will try to minimize potential energy - the charges residing on the surface are an extension of this. It might take some time, but will eventually happen.

Induced Charges

If we hold a charge $+q$ near a conductor, the conductor will move toward the charge - this is because negative charges will accumulate closer to $+q$ than the "effective" $+$ charges on the far side. Force falls off by $r^2$, so the conductor will be attracted to the $+q$ charge.

Cavities

Let's say we have a cavity inside our conductive surface. There are two scenarios here:

Empty cavity: If the cavity has no charge, the field within the cavity is zero, regardless of the external fields applied. This is the principle of the Faraday cage.

Non-empty cavity: The charge contained by the cavity will induce an opposite charge $-q$ uniformly distributed on the walls of the cavity - the only information transmitted to an external observer is the distribution of charge on the exterior wall (i.e. the magnitude of the internal E-field, or the amount of net charge contained).

Surface Charge and Force on a Conductor

Through the field inside a conductor is zero, the field immediately outside is $$\vec{E}=-\frac{\sigma }{{ϵ}_0}\hat{n}\text{where}\sigma =-{ϵ}_0\frac{dV}{dn}$$

If we only care about the magnitude of the E-field, then $E=-1/{ϵ}_0\sigma$.

However, the electric field is discontinuous at a surface charge, so when calculating the force per unit area (or pressure) of an E-field at the surface of a conductor, we average the E-fields above and below the surface, such that $$\frac{\vec{F}}{A}=\frac{1}{2}\sigma ({\vec{E}}_{\text{above}}+{\vec{E}}_{\text{below}})=\frac{1}{2{ϵ}_0}{\sigma }^2\hat{n}$$ This is the outward electrostatic pressure on the surface, tending to draw the conductor into a given field, regardless of the sign of $\sigma$ (squared away).

The pressure at this point, expressed in terms of the field just outside the surface, is $$P=\frac{{ϵ}_0}{2}{E}^2$$

Capacitors

Put two conductors beside one another, with equal and opposite uniform net charges $+Q$ on one and $-Q$ on the other:

Since the charge density is uniform on each surface, $\sigma =Q/A$ (on the left plate) and the field between the two is $$E=\frac{\sigma }{{ϵ}_0}=\frac{1}{{ϵ}_0}\frac{Q}{A}$$ with a voltage of $$\begin{array}{rl}V& =\int Edl\\ & =Ed\\ & =\frac{\sigma }{{ϵ}_0}d\\ & =\frac{1}{{ϵ}_0}\frac{Q}{A}d\end{array}$$

$V=Ed$ in a uniform E-field.

We'll define a new term to represent the proportionality of the arrangement, capacitance: $$C=\frac{Q}{V}\equiv \frac{Q}{Ed}\equiv {ϵ}_0\frac{A}{d}$$ with units of Farads $(\text{F}=\frac{\text{C}}{\text{V}})$, usually expressed in $\mu \text{F}(10^{-6})$ or $p\text{F}(10^{-12})$. The work to go from one side to the other is $$\begin{array}{rl}W& =\frac{1}{2}C{V}^2\\ & =\frac{1}{2}\frac{Q^{2}d}{{ϵ}_{0}A}\end{array}$$

Chapter 3 - Potentials

Reference "Introduction to Electrodynamics" by David Griffiths.

The goal of electrostatics is to find the electric field given some stationary, immobile charge distribution.

Coulomb's Law is the way we do this for simple charge configurations, but for more complex charge configurations it's often easier to work with potential $V$. For areas of nonzero charge density $\rho \ne 0$ such as point, surface or volume charges, we use Poisson's equation: $${\nabla }^{2}V=-\frac{1}{{ϵ}_0}\rho$$ Outside of these charge regons (such as in regular space), this reduces to Laplace's equation: $${\nabla }^{2}V=0$$ $$\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}=0$$ with solutions called harmonic functions.

Laplace's equation is fundamental to the study of electrostatics according to Griffiths.

Laplace's Equation

In three dimensions, visualization is challenging - but the same two properties apply, with the first this time being

1. The value of $V(x,y,z)$ is the average $V$ over a spherical surface of radius $R$ centered at that point, with $$V(x,y,z)=\frac{1}{4\pi R^2}{\oint }_{\text{sphere}}Vda$$
2. $V$ can have no local maxima or minima, with extreme values only permitted at the boundaries (i.e. surface of the sphere).

Point 2 is particularly relevant in each of these circumstances - the value of $V$ at some point is the average of the surrounding $V$ on some surrounding boundary.

Uniqueness Theorems

The proof that a proposed set of boundary conditions will suffice takes the form of a uniqueness theorem (alternatively, a criterion to determine whether a solution to the Laplace or Poisson equations is unique).

Uniqueness Theorem 1: the solution to Laplace's equation ${\nabla }^{2}V=0$ in a volume $\mathcal{V}$ is unique if $V$ is specified on a boundary surface $\mathcal{S}$ enclosing the volume $\mathcal{V}$.

The surface does not have to be Gaussian - it can look totally eldritch and crazy and the theorem would hold.

Uniqueness Theorem 2: given a volume $\mathcal{V}$ surrounded by conductors and containing some charge density $\rho$, the electric field is uniquely determined if the total charge on each conductor is given.

Method of Images

Griffiths 3.2.

Say we have some charge $q$ a distance $d$ above an infinite grounded plane; what is the potential $V$ in the region above the plane? Our boundary conditions state

$$\{\begin{array}{ll}V=0& z=0\\ V\to 0& \text{far from the charge}\end{array}$$ We could solve Poisson's equation for this region, but a much easier technique is to use the Method of Images. Wikipedia says on the subject:

The method of image charges is used in electrostatics to simply calculate or visualize the distribution of the electric field of a charge in the vicinity of a conducting surface.

It is based on the fact that the tangential component of the electrical field on the surface of a conductor is zero, and that an electric field E in some region is uniquely defined by its normal component over the surface that confines this region (the uniqueness theorem).

Start by removing the conductor, and placing an opposite charge $-q$ at $-d$:

Then, the potential is easy to calculate: $$V(x,y,z)=\frac{1}{4\pi {ϵ}_0}\left[\frac{q}{\sqrt{x^2+y^2+(z-d{)}^2}}-\frac{q}{\sqrt{x^2+y^2+(z+d{)}^2}}\right]$$ which obeys both of the boundary conditions of the original problem:

1. $V=0$ when $z=0$
2. $V\to 0$ for $x^2+y^2+z^2\gg d$

Thus by uniqueness theorem 1, is the solution to our original problem.

Uniqueness theorem 1 means that if a solution satisfies Poisson's equation in the region of interest and assumes the correct value at the boundaries, it must be right.

The Method of Images can be used in any scenario where we have a stationary charge distribution near a grounded conducting plane.

Induced Surface Charges

The surface charge density induced on a conductor is $$\sigma =-{ϵ}_0\frac{\partial V}{\partial n}$$ where $\partial V/\partial n$ is the normal derivative of $V$ at the surface. In the above case, this is is in the $z$ direction - if we take this partial derivative of our above calculated voltage, then $$\sigma (x,y)=\frac{-qd}{2\pi (x^2+y^2+d^2{)}^{3/2}}$$

As expected from a positive charge, the induced surface charge is negative and greatest at $x=y=0$.

The total induced charge is $$\begin{array}{rl}Q=\int \sigma da& ={\int }_0^{2\pi }{\int }_0^{\infty }-\frac{qd}{2\pi (r^2+d^2{)}^{3/2}}rdrd\varphi \\ & =\frac{qd}{\sqrt{r^2+d^2}}{|}_0^{\infty }\\ & =-q\end{array}$$ Yes!

Force & Energy

Since the induced charge on the conductor is $-q$ and our charge is $+q$, it is attracted to the plane with a force given by Coulomb's Law: $$\vec{F}=-\frac{1}{4\pi {ϵ}_0}\frac{q^2}{(2d{)}^2}\hat{z}$$ While force is the same in our mirror problem, energy is not. For two point charges, $W=\frac{1}{2}{\sum }_{i=1}^n{q}_{i}V(\overrightarrow{r_i})$, such that $$W=-\frac{1}{4\pi {ϵ}_0}\frac{q^2}{2d}$$ But for a single charge and conducting plane (continuous charge distribution), energy is half of this. $$W=-\frac{1}{4\pi {ϵ}_0}\frac{q^2}{4d}$$

The work to bring two point charges towards one another does work on both of them, while to bring a point charge toward a grounded conductor has us only doing work on one charge - only half the work is necessary.

Separation of Variables

Griffiths' 3.3

Separation of variables is a way to solve ODEs and PDEs by rewriting equations such that each of the two variables occur on different sides of the equation.

Separable equations must be able to be written in the form $$\frac{dx}{dy}=g(x)h(y)$$ We can rearrange the terms to get $$dy\frac{1}{h(y)}=g(x)dx$$ integrate, $$\int \frac{1}{h(y)}dy=\int g(x)dx$$ and add some constant term to one side to represent all our constants of integration. $$H(y)=G(x)+C$$

In the context of electrostatics, separation of variables is very useful when solving 2D Laplace equations, such as $$\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}=0$$ We need solutions in the form of $$V(x,y)=X(x)Y(y)$$ This can be accomplished through some mathematical trickery to find our separated variables ... $$\frac{1}{X}\frac{d^{2}X}{d{x}^2}+\frac{1}{Y}\frac{d^{2}Y}{d{y}^2}=0$$ ... which is of the form $f(x)+g(y)=0$. Thus, both $f(x)$ and $g(y)$ must be constant (we can't hold one constant and change the other with this solution still holding).

So, $$\frac{1}{X}\frac{d^{2}X}{d{x}^2}=C_1\text{and}\frac{1}{Y}\frac{d^{2}Y}{d{y}^2}=C_2{C}_1+C_2=0$$ Converting each equation into an ODE, $$\frac{d^{2}X}{d{x}^2}=k^{2}X\frac{d^{2}Y}{d{y}^2}=-k^{2}Y$$ ... we converted a PDE into two ODEs, which are much easier to solve. Our solutions will be a constant coefficient set: $$X(x)=c_1{e}^{kx}+c_2{e}^{-kx},Y(y)=c_3\sin (ky)+c_4\cos (ky)$$ We can find our constants based on our boundary conditions now.

Multipole Expansion

Griffiths 3.4

Say you have some charge $Q=Q_{\text{net}}$ that you can see in space, far away. $Q$ is almost like an exoplanet in a sense: we know it exists, we know its total mass, and maybe we know stuff like how far away it is.

But, we don't know what the surface looks like; what it's composed of, be it rock or ice or water. Just its net mass.

Similarly, our charge distribution might be crazy complicated, and it's net charge $Q_{\text{net}}$ only describes a tiny part of the story, albeit an important one - the multipole expansion is used to describe this fuller story, and is defined in terms of voltage as

$$\begin{array}{rl}V(\vec{r})& =\frac{1}{4\pi {ϵ}_0}\left[\frac{1}{r}(\text{monopole moment})+\frac{1}{r^2}(\text{dipole moment})+\frac{1}{r^3}(\text{quadrupole moment})\dots \right]\end{array}$$ Thus, the monopole moment is $(1/r)$ stronger than the dipole moment, the dipole $1/r$ stronger than the quadrupole moment, and so on.

Mathematically, it is $$V(\vec{r})=\frac{1}{4\pi {ϵ}_0}\sum _{n=0}^{\infty }\frac{1}{r^{n+1}}\int (r^{\prime }{)}^n{P}_n(\cos \alpha )\rho ({\vec{r}}^{\prime })d{\tau }^{\prime }$$ where $\alpha$ is the angle between $\vec{r}$ and $\overrightarrow{r^{\prime }}$, $\vec{r}$ is the reference point (from origin) and ${\vec{r}}^{\prime }$ the charge (from origin).

Electric field of moments: the electric field is defined as $E=-\nabla V$.

To find the electric field i.e. caused by the dipole moment, find the voltage term for that dipole moment, then take the negative gradient of it.

Monopole and dipole terms

At large $\vec{r}$, a charge distribution just looks like a net charge (like an exoplanet). Thus, the monopole moment $Q$ is just the net charge: $$\begin{array}{r}Q\equiv \int \rho d\tau V_{\text{mon}}(\vec{r})=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}\end{array}$$ The dipole moment $\vec{p}$ describes the individual distribution of charges: $$\vec{p}=\sum _{i=1}^n{q}_i{\overrightarrow{r_i}}^{\prime }\equiv \int {\vec{r}}^{\prime }\rho (\overrightarrow{r^{\prime }})d{\tau }^{\prime }{V}_{\text{dip}}(\vec{r})=\frac{1}{4\pi {ϵ}_0}\frac{\vec{p}}{r^2}$$

${\vec{r}}^{\prime }$ points from the origin to some charge distribution. $\vec{r}$ is from the origin to some reference point.

Origin of coordinates

Changing the origin will never change the monopole moment $Q$, but will change the dipole moment $\vec{p}$ as long as the total charge $Q\ne 0$.

For example: if our system has $+q$ and $-q$ as its point distribution, $Q=+q+(-q)=0$ and the dipole moment is origin-independent.

Chapter 4 - Electric Fields in Matter

Reference "Introduction to Electrodynamics" by David Griffiths.

This chapter looks at electric fields within matter that aren't necessarily conductors - i.e.most matter in the universe.

Polarization

Griffith's 4.1

If we place a neutral atom in an electric field, the net force on it is zero (as it is uncharged) - however, the positively-charged nucleus is pushed along into the field, while the negatively charged electron cloud is pulled down into the field.

The neutral atom is now polarized, with some tiny dipole moment $\vec{p}$: $$\vec{p}=\alpha \vec{E}$$ $\alpha$ is the atomic polarizability and varies from atom to atom.

If the electric field is strong enough, the $e^{-}$ cloud will be ripped away from the nucleus and the atom will become ionized. Coulomb's law determines both.

Molecules become more complicated, since they're a few individual atoms coupled together - CO${}_2$ for instance has different polarizabilities parallel or orthogonal to its axis ($\vec{p}={\alpha }_{\perp }{\vec{E}}_{\perp }+{\alpha }_{\parallel }{\vec{E}}_{\parallel }$), so for molecules, a polarization tensor might be more useful. $$\vec{p}=[p_x,p_y,p_z]=\left(\begin{array}{ccc}{\alpha }_{xx}{E}_x& 0& 0\\ 0& {\alpha }_{yy}{E}_y& 0\\ 0& 0& {\alpha }_{zz}{E}_z\end{array}\right)$$

Force on dipoles

In the presence of an electric field, a dipole will rotate to point in the direction of the $E$-field with a torque given by $$\vec{N}=\vec{p}\times \vec{E}$$

At any other point on the dipole than the origin, $\vec{N}=(\vec{p}\times \vec{E})+(\vec{r}\times \vec{F})$.

The net force on a dipole in a uniform field is zero as seen in the first image. However, in a nonuniform electric field, the force on a dipole is $$\vec{F}=(\vec{p}\cdot \vec{\nabla })\vec{E}$$ The potential energy of a dipole in an electric field is $$U=-\vec{p}\cdot \vec{E}$$

The Field of a Polarized Object

Griffiths 4.2

In the presence of an external electric field, the individual atoms of a given material will become polarized, leading to a lot of little baby dipoles, all pointing in the direction of the applied field.

Since all of its constituent atoms are polarized, the material itself is said to be polarized with some polarization density $\vec{P}$. $$\vec{P}=\frac{\text{dipole moment}}{\text{unit volume}}$$

Potential

Since the potential for an individual dipole is $$V(\vec{r})=\frac{1}{4\pi {ϵ}_0}\frac{\vec{p}}{{\mathcal{R}}^2}\hat{\mathcal{R}}=\frac{1}{4\pi {ϵ}_0}\frac{\vec{p}}{|\vec{r}-{\vec{r}}^{\prime }{|}^3}(\vec{r}-\overrightarrow{r^{\prime }})$$ The electric potential for the entire object with polarization $\vec{P}$ is $$V(\vec{r})=\frac{1}{4\pi {ϵ}_0}\int \frac{\vec{P}({\vec{r}}^{\prime })\cdot \hat{\mathcal{R}}}{{\mathcal{R}}^2}d{\tau }^{\prime }$$ With surface and volume bound charge densities of $${\sigma }_b=\vec{P}\cdot \hat{n}{\rho }_b=-\vec{\nabla }\cdot \vec{P}$$ We can rewrite the potential of our whole polarized object in terms of these densities: $$V(\vec{r})=\frac{1}{4\pi {ϵ}_0}\left[{\oint }_S\frac{{\sigma }_b}{\mathcal{R}}d{a}^{\prime }+{\int }_V\frac{{\rho }_b}{\mathcal{R}}d{\tau }^{\prime }\right]$$

The net potential is just the sum of the potentials produced by the bound surface and volume charges.

$\vec{P}\cdot \hat{n}=P\cos \theta$, where $\theta$ is the angle between $\vec{P}$ and $\vec{n}$. $r\cos \theta =z$ in spherical.

To calculate the electric field caused by polarization, use $$\vec{E}=-\vec{\nabla }V$$

Physical interpretation of bound charge

Say we have a long string of dipoles, as we might in a polarized dielectric in some field $\vec{E}$. The head of one dipole will cancel the tail of its neighbor, except at the ends, so that:

No one electron makes the whole trip - but the magnitude of the $\Delta -$ in some field will be matched by $\Delta +$, such that it's almost conductor-like - in a uniform field, we see only accumulations of bound charge on the surface, such that $${\sigma }_b=\vec{P}\cdot \hat{n}\equiv P\cos \theta \text{(for some oblique surface)}$$ If, however, the polarization is non-uniform, then we'll see accumulations of bound charge within the object, as well as on the surface, $${\rho }_b=-\vec{\nabla }\cdot \vec{P}$$

The Electric Displacement

Griffiths 4.3

Most materials aren't perfect conductors or perfect insulators. Conductors have free flowing electrons $({\rho }_f)$, while insulators have exclusively bound charges $({\rho }_b)$ - a normal dielectric will have both free and bound charges, such that $$\rho ={\rho }_b+{\rho }_f$$ Let the displacement field take into account the fields of the polarization field and electric field, such that $$\vec{D}={ϵ}_0{\vec{E}}_{\text{net}}+\vec{P}$$ Gauss's law for $\vec{D}$ is $$\vec{\nabla }\cdot \vec{D}=\rho -{\rho }_b={\rho }_f\oint \vec{D}\cdot d\vec{a}=Q_{f_{\text{enc}}}$$

We only have control over free charge - bound charge is a consequence of the properties of the material itself. Free charges themselves generate a field.

Though Gauss's law has a parallel for $\vec{D}$, Coulomb's law does not.

The curl of $\vec{D}$ is $$\vec{\nabla }\times \vec{D}=\vec{\nabla }\times \vec{P}$$

Boundary conditions

For a given dielectric, the boundary conditions are: $$D_{\text{above}}^{\perp }-D_{\text{below}}^{\perp }={\sigma }_f{\vec{D}}_{\text{above}}^{\parallel }-{\vec{D}}_{\text{below}}^{\parallel }={\vec{P}}_{\text{above}}^{\parallel }-{\vec{P}}_{\text{beow}}^{\parallel }$$

Note that since the curl of $\vec{E}$ is zero $(\vec{\nabla }\times \vec{E}=0$), $${\vec{E}}_{\text{above}}^{\parallel }-{\vec{E}}_{\text{below}}^{\parallel }=0$$

Linear dielectrics

Griffiths 4.4

For linear substances, the polarization is linearly proportional to the total electric field: $$\vec{P}={ϵ}_0{\chi }_e{\vec{E}}_{\text{net}}$$

${\chi }_e$ is the electric susceptibility of the medium (dimensionless), with $$ϵ={ϵ}_0(1+{\chi }_e)$$ being the permittivity of the material (with ${\chi }_e=0$ in vacuum, ${ϵ}_0$ the permittivity of free space). A unit-less version is the dielectric constant, ${ϵ}_r=ϵ/{ϵ}_0$.

The displacement field in linear media is thus $$\begin{array}{rl}\vec{D}& ={ϵ}_0\vec{E}+\vec{P}\\ & ={ϵ}_0\vec{E}+{ϵ}_0{\chi }_e\vec{E}\\ & ={ϵ}_0(1+{\chi }_e)\vec{E}\\ & =ϵ{\vec{E}}_{\text{net}}\end{array}$$

Notes on examples

Example 4.5: conducting sphere of radius $a$ and charge $Q$ surrounded by linear dielectric of radius $b$ and permittivity $ϵ$, find potential at the center.

• The conducting sphere of charge $Q$ is our enclosed free charge, so $$\vec{D}=\frac{Q}{4\pi r^2}\hat{r}r>a$$
• Inside the sphere, $\vec{E}=\vec{P}=\vec{D}=\vec{0}$
• Integral to find potential is (with reference to infinity): $$V=-{\int }_{\infty }^0\vec{E}\cdot d\vec{l}$$

Example 4.6: parallel plate capacitor filled with dielectric material of dielectric constant ${ϵ}_r$, what effect does this have on capacitance?

• Dielectric will reduce $\vec{E}$, and hence $V$, by factor of $1/{ϵ}_r$
• Since $C=Q/V$, capacitance increased by $1/{ϵ}_r$: $C={ϵ}_r{C}_{\text{vac}}$