Volume Charge Density

• assignment Power Series Coefficients 2

assignment Homework

Power Series Coefficients 2
Static Fields 2022 (5 years) Use the formula for a Taylor series: $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n$ to find the first three non-zero terms of a series expansion for $f(z)=e^{-kz}$ around $z=3$.
• assignment Series Notation 2

assignment Homework

Series Notation 2

Power Series Sequence (E&M)

Static Fields 2022 (5 years)

Write (a good guess for) the following series using sigma $\left(\sum\right)$ notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

1. $1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots$

2. $\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots$

• assignment Power Series Coefficients 3

assignment Homework

Power Series Coefficients 3
Static Fields 2022 (5 years) Use the formula for a Taylor series: $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n$ to find the first three non-zero terms of a series expansion for $f(z)=\cos(kz)$ around $z=2$.
• assignment Spherical Shell Step Functions

assignment Homework

Spherical Shell Step Functions
step function charge density Static Fields 2022 (5 years)

One way to write volume charge densities without using piecewise functions is to use step $(\Theta)$ or $\delta$ functions. If you need to review this, see the following link in the math-physics book: https://paradigms.oregonstate.eduhttps://books.physics.oregonstate.edu/GMM/step.html

Consider a spherical shell with charge density $\rho (\vec{r})=\alpha3e^{(k r)^3}$ between the inner radius $a$ and the outer radius $b$. The charge density is zero everywhere else. Use step functions to write this charge density as a single function valid everywhere in space.

• assignment Divergence

assignment Homework

Divergence
Static Fields 2022 (5 years)

Shown above is a two-dimensional vector field.

Determine whether the divergence at point A and at point C is positive, negative, or zero.

• assignment Series Notation 1

assignment Homework

Series Notation 1

Power Series Sequence (E&M)

Static Fields 2022 (5 years)

Write out the first four nonzero terms in the series:

1. $\sum\limits_{n=0}^\infty \frac{1}{n!}$

2. $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}$
3. $$\sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}}$$

• assignment Volume Charge Density, Version 2

assignment Homework

Volume Charge Density, Version 2
charge density delta function Static Fields 2022 (5 years)

You have a charge distribution on the $x$-axis composed of two point charges: one with charge $+3q$ located at $x=-d$ and the other with charge $-q$ located at $x=+d$.

1. Sketch the charge distribution.
2. Write an expression for the volume charge density $\rho (\vec{r})$ everywhere in space.

• computer Visualising the Gradient

computer Mathematica Activity

30 min.

Static Fields 2022 (5 years)

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
• assignment Memorize Power Series

assignment Homework

Memorize Power Series

Power Series Sequence (E&M)

Static Fields 2022 (2 years)

Look up and memorize the power series to fourth order for $e^z$, $\sin z$, $\cos z$, $(1+z)^p$ and $\ln(1+z)$. For what values of $z$ do these series converge?

• assignment Cross Triangle

assignment Homework

Cross Triangle
Static Fields 2022 (5 years)

Use the cross product to find the components of the unit vector $\mathbf{\boldsymbol{\hat n}}$ perpendicular to the plane shown in the figure below, i.e.  the plane joining the points $\{(1,0,0),(0,1,0),(0,0,1)\}$.

• Static Fields 2022 (5 years)

Sketch the volume charge density: $$\rho (x,y,z)=c\,\delta (x-3)$$