## 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. \begin{equation} \sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}} \end{equation}

• 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.

##### Visualising the Gradient
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: \begin{equation} \rho (x,y,z)=c\,\delta (x-3) \end{equation}