1. << The Path | Gradient Sequence | Gradient Point Charge >>
assignment Homework
Calculate the divergence of each of the following vector fields. You may look up the formulas for divergence in curvilinear coordinates.
assignment Homework
Calculate the curl of each of the following vector fields. You may look up the formulas for curl in curvilinear coordinates.
assignment Homework
Determine the total mass of each of the slabs below.
assignment Homework
Consider the fields at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\).
assignment Homework
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 \(x=+d\).
assignment Homework
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}
assignment Homework
A current \(I\) flows down a cylindrical wire of radius \(R\).
assignment Homework
assignment Homework
Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).
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.
Find the gradient of each of the following functions: