Static Fields: Fall-2025
HW 04: Due W2 D5

1. Gradient Practice

   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}

2. Linear Quadrupole (w/ series)

   Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

   1. Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole. The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r}'\) is given by: \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r}'\vert} \]

   2. Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.

   3. A series of charges arranged in this way is called a linear quadrupole. Why?