assignment Homework
The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}
This problem explores the consequences of the divergence theorem for this shell.
group Small Group Activity
30 min.
charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.keyboard Computational Activity
120 min.
assignment Homework
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 Homework
For each case below, find the total charge.
assignment Homework
For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.
group Small Group Activity
120 min.
group Small Group Activity
30 min.
assignment Homework
Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).
assignment Homework
Shown above is a two-dimensional vector field.
Determine whether the divergence at point A and at point C is positive, negative, or zero.