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Volume Charge Density
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Sketch the volume charge density: \begin{equation} \rho (x,y,z)=c\,\delta (x-3) \end{equation}

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Electric Field from a Rod
AIMS Maxwell AIMS 21 AIMS Maxwell AIMS 21 Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(xy\)-plane. The charge density \(\lambda\) is constant. Find the electric field at the point \((0,0,2L)\).

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Power Series Coefficients 3
AIMS Maxwell AIMS 21 Static Fields Winter 2021 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 \(z=2\).

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Power Series Coefficients 2
AIMS Maxwell AIMS 21 Static Fields Winter 2021 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 \(z=3\).

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Gradient Practice
AIMS Maxwell AIMS 21

Find the gradient of each of the following functions:

  1. \begin{equation} f(x,y,z)=e^{(x+y)}+x^2 y^3 \ln \frac{x}{z} \end{equation}
  2. \begin{equation} \sigma(\theta,\phi)=\cos\theta \sin^2\phi \end{equation}
  3. \begin{equation} \rho(s,\phi,z)=(s+3z)^2\cos\phi \end{equation}

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Gauss's Law for a Rod inside a Cube
AIMS Maxwell AIMS 21 Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(x,y\)-plane. The charge density \(\lambda_0\) is constant. Find the total flux of the electric field through a closed cubical surface with sides of length \(3L\) centered at the origin.

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Current in a Wire
AIMS Maxwell AIMS 21 The current density in a cylindrical wire of radius \(R\) is given by \(\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}\). Find the total current in the wire.

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Series Notation 1

Power Series Sequence (E&M)

AIMS Maxwell AIMS 21 Static Fields Winter 2021

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}

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Series Notation 2

Power Series Sequence (E&M)

AIMS Maxwell AIMS 21 Static Fields Winter 2021

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. \[\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots\]

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Volume Charge Density Practice
charge density delta function AIMS Maxwell AIMS 21 Static Fields Winter 2021

You have a charge distribution 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.

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Spherical Shell Step Functions
step function charge density AIMS Maxwell AIMS 21 Static Fields Winter 2021

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://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.

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Theta Parameters
AIMS Maxwell AIMS 21 Static Fields Winter 2021

The function \(\theta(x)\) (the Heaviside or unit step function) is a defined as: \begin{equation} \theta(x) =\begin{cases} 1 & \textrm{for}\; x>0 \\ 0 & \textrm{for}\; x<0 \end{cases} \end{equation} This function is discontinuous at \(x=0\) and is generally taken to have a value of \(\theta(0)=1/2\).

Make sketches of the following functions, by hand, on axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{align} y &= \theta (x)\\ y &= 2+\theta (x)\\ y &= \theta(2+x)\\ y &= 2\theta (x)\\ y &= \theta (2x) \end{align}

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Sphere in Cylindrical Coordinates
AIMS Maxwell AIMS 21 Find the surface area of a sphere using cylindrical coordinates.

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Total Current, Circular Cross Section

Integration Sequence

AIMS Maxwell AIMS 21

A current \(I\) flows down a cylindrical wire of radius \(R\).

  1. If it is uniformly distributed over the surface, give a formula for the surface current density \(\vec K\).
  2. If it is distributed in such a way that the volume current density, \(|\vec J|\), is inversely proportional to the distance from the axis, give a formula for \(\vec J\).

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Vector Sketch I
vector fields AIMS Maxwell AIMS 21 Sketch each of the vector fields below.
  1. \(\boldsymbol{\vec F} = y\,\boldsymbol{\hat x} - x\,\boldsymbol{\hat y}\)
  2. \(\boldsymbol{\vec G} = x\,\boldsymbol{\hat x} + y\,\boldsymbol{\hat y}\)
  3. \(\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)
  4. \(\boldsymbol{\vec K}=s\,\boldsymbol{\hat s}\)
  5. \(\boldsymbol{\vec L}=\frac1s\boldsymbol{\hat\phi}\)
  6. \(\boldsymbol{\vec M}=\sin\phi\,\boldsymbol{\hat s}\)
  7. \(\boldsymbol{\vec N}=\sin(2\pi s)\,\boldsymbol{\hat\phi}\)

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Mass Density
AIMS Maxwell AIMS 21 Consider a rod of length \(L\) lying on the \(z\)-axis. Find an algebraic expression for the mass density of the rod if the mass density at \(z=0\) is \(\lambda_0\) and at \(z=L\) is \(7\lambda_0\) and you know that the mass density increases
  • linearly;
  • like the square of the distance along the rod;
  • exponentially.

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Line Sources Using the Gradient
AIMS Maxwell AIMS 21 Static Fields Winter 2021
  1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}

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Divergence
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Shown above is a two-dimensional vector field.

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

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Flux through a Plane
AIMS Maxwell AIMS 21 Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).

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Total Current, Square Cross-Section

Integration Sequence

AIMS Maxwell AIMS 21 Static Fields Winter 2021
  1. Current \(I\) flows down a wire (length \(L\)) with square cross-section (side \(a\)). If it is uniformly distributed over the entire area, what is the magnitude of the volume current density \(\vec{J}\)?
  2. If the current is uniformly distributed over the outer surface only, what is the magnitude of the surface current density \(\vec{K}\)?