91Ó°ÊÓ

(a) Twelve equal charges, q,are situated at the comers of a regular 12-sided polygon (for instance, one on each numeral of a clock face). What is the net force on a test charge Qat the center?

(b) Suppose oneof the 12 q'sis removed (the one at "6 o'clock"). What is the force on Q?Explain your reasoning carefully.

(c) Now 13 equal charges, q,are placed at the comers of a regular 13-sided polygon. What is the force on a test charge Qat the center?

(d) If one of the 13 q'sis removed, what is the force on Q?Explain your reasoning.

One of these is an impossible electrostatic field. Which one?

(a) $\mathit{E}\mathbf{=}\mathit{k}\left[xy\hat{x}+2yz\hat{y}+3xz\hat{z}\right]$

(b) $\mathit{E}\mathbf{=}\mathit{k}\mathbf{[}{\mathbf{y}}^{\mathbf{2}}\hat{\mathbf{x}}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{yz}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\mathbf{)}\hat{\mathbf{y}}\mathbf{+}\mathbf{2}\mathbf{yz}\hat{\mathbf{z}}\mathbf{]}$.

Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your reference point. Check your answer by computing $\mathbf{∇}\mathit{V}$. [Hint: You must select a specific path to integrate along. It doesn't matter what path you choose, since the answer is path-independent, but you simply cannot integrate unless you have a definite path in mind.]

Find the potential inside and outside a uniformly charged solid sphere whose radius is and whose total charge is .Use infinity as your reference point. Compute the gradient of in each region, and check that it yields the correct field. Sketch$\mathit{V}\mathbf{\left(}\mathit{r}\mathbf{\right)}$.

Find the potential a distancesfrom an infinitely long straight wire

that carries a uniform line charge$\mathrm{λ}$. Compute the gradient of your potential, and

check that it yields the correct field.

Question: Find the electric field at a height z above the center of a square sheet (side a) carrying a uniform surface charge $\mathit{σ}$. Check your result for the limiting

cases $\mathit{a}\mathbf{→}\mathbf{∞}$and $\mathit{z}\mathbf{>}\mathbf{>}\mathit{a}$.

Question: If the electric field in some region is given (in spherical coordinates)

by the expression

$E\left(r\right)=\frac{k}{r}[3\hat{r}+2sin\mathrm{θ}cos\mathrm{θ}sin\mathrm{ϕ}\widehat{\mathrm{θ}}+sin\mathrm{θ}cos\mathrm{ϕ}\widehat{\mathrm{ϕ}}]$

for some constant , what is the charge density?

Find the net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere. Express your answer in terms of the radius R and the total charge Q.

An inverted hemispherical bowl of radius R carries a uniform surface charge density $\mathit{σ}$. Find the potential difference between the "north pole" and the center.

A sphere of radius R carries a charge density $\mathit{ÒÏ}\left(r\right)\mathbf{=}\mathit{k}\mathit{r}$(where $\mathit{k}$ is a constant). Find the energy of the configuration. Check your answer by calculating it in at least two different ways.

For the configuration of Prob. 2.16, find the potential difference between a point on the axis and a point on the outer cylinder. Note that it is not necessary to commit yourself to a particular reference point, if you use Eq. 2.22.