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## Homework for Symmetries

- (GradientPtCharge)
Consider the fields at a point $\rr$ due to a point charge located at $\rr'$.

Write down an expression for the electrostatic potential $V(\rr)$ at a point $\rr$ due to a point charge located at $\rr'$. (There is nothing to calculate here.)

Write down an expression for the electric field $\EE(\rr)$ at a point $\rr$ due to a point charge located at $\rr'$. (There is nothing to calculate here.)

Working in rectangular coordinates, compute the gradient of $V$.

Write several sentences comparing your answers to the last two questions.

- (LineSources)
Consider the fields around both finite and infinite uniformly charged, straight wires.

Find the electric field around an infinite, uniformly charged, straight wire, starting from the expression for the electrostatic potential that we found in class:

$$V(\Vec r)={2\lambda\over 4\pi\epsilon_0}\, \ln{ r_0\over r}$$

Compare your result to the solution found from Coulomb's law. Which method is easier?

Find the electric field around a finite, uniformly charged, straight wire, at a point a distance $r$ straight out from the midpoint, starting from the expression for the electrostatic potential that we found in class:

$$V(\Vec r)={\lambda\over 4\pi\epsilon_0} \left[\ln{\left(L + \sqrt{L^2+r^2}\right)}- \ln{\left(-L + \sqrt{L^2+r^2}\right)}\right]$$

Compare your result to the solution found from Coulomb's law. Which method is easier?

Find the electric field around an infinite, uniformly charged, straight wire, starting from Coulomb's Law.

Find the electric field around a finite, uniformly charged, straight wire, at a point a distance $s$ straight out from the midpoint, starting from Coulomb's Law.

- (ConductorsGem235ModB)
Find $E_r$, the radial component of the electric field and plot it as a function of $r$. %Are the discontinuities in the electric field what you expect from our unit on boundary conditions? Explain.