Power Series Coefficients A
- assignment Power Series Coefficients B
assignment Homework
Power Series Coefficients B
Static Fields 2023 (6 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 series expansion for \(f(z)=\cos(kz)\) to second order around \(z=2\). - assignment Volume Charge Density
assignment Homework
Volume Charge Density
Static Fields 2023 (6 years)Sketch the volume charge density: \begin{equation} \rho (x,y,z)=c\,\delta (x-3) \end{equation}
- assignment Series Notation 2
assignment Homework
Series Notation 2
Static Fields 2023 (6 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.)
\[1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots\]
- \[\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots\]
- assignment Spherical Shell Step Functions
assignment Homework
Spherical Shell Step Functions
step function charge density Static Fields 2023 (6 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 2023 (6 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
Static Fields 2023 (6 years)Write out the first four nonzero terms in the series:
\[\sum\limits_{n=0}^\infty \frac{1}{n!}\]
- \[\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\]
- \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 2023 (6 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 \(x=+d\).
- Sketch the charge distribution.
- 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 2023 (6 years) Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields. - assignment Cross Triangle
assignment Homework
Cross Triangle
Static Fields 2023 (6 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 \(\{(1,0,0),(0,1,0),(0,0,1)\}\).
- assignment Total Current, Square Cross-Section
assignment Homework
Total Current, Square Cross-Section
Static Fields 2023 (6 years)- Current \(I\) flows down a wire with square cross-section. The length of the square side is \(L\). If the current is uniformly distributed over the entire area, find the current density .
- If the current is uniformly distributed over the outer surface only, find the current density .
- Static Fields 2023 (6 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 series expansion for \(f(z)=e^{-kz}\) to second order around \(z=3\).