Static Fields: Fall-2025
HW 06 Practice: Due W3 D5
- Electric Field from a Rod 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 \((0,0,2L)\).
- Electric Field Due to a Ring - Limiting Cases
In class, we considered the electric field due to a charged ring with total charge \(Q\) and radius \(R\).
Find the first two non-zero terms of a series expansion for the electric field at each of these special locations:
\begin{align} \vec{E}(\vec{r})=\frac{kQ}{2\pi} \int \frac{\left(\left(s \cos\phi -R \cos\phi'\right) \hat x + \left(s \sin\phi -R \sin\phi'\right)\hat y + z \hat z\right)}{\left(s^2+R^2-2sR\cos(\phi-\phi')+z^2\right)^\frac{3}{2}}d\phi' \end{align}
Near the center of the ring, on the axis perpendicular to the plane of the ring;
Far from the ring, in the plane of the ring;
- Cross Triangle
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 \(\{(1,0,0),(0,1,0),(0,0,1)\}\).
- Directional Derivative
You are on a hike. The altitude nearby is described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.
- Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
- In which direction in space does the water flow?
- At the spot you're standing, what is the slope of the ground in the direction of the cottage?
- Does your result to part (c) make sense from the graph?