Electrostatics

(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 Q at the center?

(b) Suppose one of 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 Q at the center?

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

Find the electric field (magnitude and direction) a distance z above the midpoint between equal and opposite charges ( $\mathbf{+}\mathit{q}$ ), a distanced apart (same as Example 2.1, except that the charge at $\mathit{x}\mathbf{=}\mathbf{+}\frac{\mathbf{d}}{\mathbf{2}}\mathbf{is}\mathbf{-}\mathit{q}\mathbf{)}$.

Find the electric field a distance z above one end of a straight line segment of length L (Fig. 2.7) that carries a uniform line charge A. Check that your formula is consistent with what you would expect for the case z » L.

Find the electric field a distance z above the center of a square loop (side a) carrying uniform line charge A (Fig. 2.8). [Hint: Use the result of Ex. 2.2.]

Find the electric field a distance z above the center of a circular loop of radius $\mathit{r}$ (Fig. 2.9) that carries a uniform line charge $\mathit{\lambda }$

Find the electric field a distance $\mathit{z}$ above the center of a flat circular disk of radius $\mathit{R}$ (Fig. 2.1 0) that carries a uniform surface charge $\mathit{a}$. What does your formula give in the limit $\mathit{R}\mathbf{\to }\mathbf{\infty }$? Also check the case $\mathit{z}\mathbf{\gg }\mathit{R}$.

Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11) that carries a uniform charge density $\mathit{\sigma }$. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write $\mathit{r}$ in terms of R and $\mathit{\theta }$. Be sure to take the positive square root: $\sqrt{{\mathbf{R}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{R}\mathbf{z}}\mathbf{=}\left(R-z\right)$ if $\mathit{R}\mathbf{>}\mathit{z}$, but it's $\left(z-R\right)\mathbf{ }$if $\mathit{R}\mathbf{<}\mathit{z}$.]

Use your result in Prob. 2.7 to find the field inside and outside a solidsphere of radius $\mathit{R}$that carries a uniform volume charge density $\mathit{p}$. Express your answers in terms of the total charge of the sphere, $\mathit{q}$. Draw a graph of lEIas a function of the distance from the center.

Suppose the electric field in some region is found to be${}^{\mathbf{E}\mathbf{=}\mathbf{ }\mathbf{K}{\mathbf{r}}^{\mathbf{3}}}\hat{\mathbf{r}}$

in spherical coordinates (k is some constant).

(a) Find the charge density ${}^{\mathbf{P}}$ 

(b) Find the total charge contained in a sphere of radius ${}^{\mathit{R}}$centered at the origin.(Do it two different ways.)

A charge q sits at the back comer of a cube, as shown in Fig. 2.17.What is the flux of E through the shaded side?

Use Gauss's law to find the electric field inside and outside a spherical shell of radius R that carries a uniform surface charge density ${}^{\sigma }$Compare your answer to Prob. 2.7.

Find the electric field a distance${}^s$ from an infinitely long straight wire that carries a uniform line charge${}^{\lambda }$) ., Compare Eq. 2.9

Find the electric field inside a sphere that carries a charge density proportional to the distance from the origin,${}^{\mathbf{P}\mathbf{=}\mathbf{K}\mathbf{r}}$for some constant k. [Hint: This charge density is not uniform, and you must integrate to get the enclosed charge.]

A thick spherical shell carries charge density

$\mathit{p}\mathbf{=}\frac{\mathbf{k}}{{\mathbf{r}}^{\mathbf{2}}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathit{a}\mathbf{<}\mathit{r}\mathbf{<}\mathit{b}\mathbf{)}$

(Fig. 2.25). Find the electric field in the three regions: (i) r < a, (ii) a < r < b, (iii) r> b. Plot lEI as a function of r, for the case b = 2a.

A long coaxial cable (Fig. 2.26) carries a uniform volume charge density p on the inner cylinder (radius a ), and a uniform surface charge density on the outer cylindrical shell (radius b ). This surface charge is negative and is of just the right magnitude that the cable as a whole is electrically neutral. Find the electric field in each of the three regions: (i) inside the inner cylinder$(s<a)$,  (ii) between the cylinders $(a < s < b)$ (iii) outside the cable $(s>b)$Plot lEI as a function of  s . 

An infinite plane slab, of thickness 2d, carries a uniform volume charge density p  (Fig. 2.27). Find the electric field, as a function of y, where y = 0  at the center. Plot E versus y, calling E positive when it points in the +y direction and negative when it points in the -y direction.

Calculate the divergence of the following vector functions:

Two spheres, each of radius R and carrying uniform volume charge densities +p and -p , respectively, are placed so that they partially overlap (Fig. 2.28). Call the vector from the positive center to the negative center d. Show that the field in the region of overlap is constant, and find its value. [Hint: Use the answer to Prob. 2.12.]

Calculate$\mathbf{\nabla }\mathbf{\times }\mathit{E}$directly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.63 if you get stuck.

Use Gauss's law to find the electric field inside a uniformly charged solid sphere (charge density p) Compare your answer to Prob. 2.8.

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 $\mathbf{\nabla }\mathit{V}$. [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 distance s from an infinitely long straight wire

that carries a uniform line charge $\lambda$. Compute the gradient of your potential, and

check that it yields the correct field.

For the charge configuration of Prob. 2.15, find the potential at the center, using infinity as your reference point.

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.

Using Eqs. 2.27 and 2.30, find the potential at a distance z above the

center of the charge distributions in Fig. 2.34. In each case, compute $\mathit{E}\mathbf{ }\mathbf{⃗}\mathbf{=}\mathbf{-}\mathbf{\nabla }\mathbf{ }\mathbf{⃗}\mathit{V}$ , and compare your answers with Ex. 2.1, Ex. 2.2, and Prob. 2.6, respectively. Suppose that we changed the right-hand charge in Fig. 2.34a to -q; what then is the potential at P? What field does that suggest? Compare your answer to Pro b. 2.2, and explain carefully any discrepancy.

A conical surface (an empty ice-cream cone) carries a uniform surface charge . The height of the cone is  as is the radius of the top. Find the potential difference between points (the vertex) and (the center of the top).

Find the potential on the axis of a uniformly charged solid cylinder,

a distance  z from the center. The length of the cylinder is L, its radius is R, and

the charge density is p. Use your result to calculate the electric field at this point.

(Assume that $\mathit{z}\mathbf{>}\mathit{L}\mathbf{/}\mathbf{2}$.)

Use Eq. 2.29 to calculate the potential inside a uniformly charged

solid sphere of radius R and total charge q . Compare your answer to Pro b. 2.21.

Check that Eq. 2.29 satisfies Poisson's equation, by applying the Laplacian and using Eq. 1.102.

(a) Check that the results of Exs. 2.5 and 2.6, and Prob. 2.11, are consistent with Eq. 2.33.

(b) Use Gauss's law to find the field inside and outside a long hollow cylindrical

tube, which carries a uniform surface charge $\mathit{\sigma }$. Check that your result is consistent with Eq. 2.33.

(c) Check that the result of Ex. 2.8 is consistent with boundary conditions 2.34 and 2.36.

Three charges are situated at the comers of a square , as shown in Fig. 2.41. 

1. How much work does it take to bring in another charge, +q , from far away and place it in the fourth comer?
2.  How much work does it take to assemble the whole configuration of four charges?

Two positive point charges, and ${\mathit{q}}_{\mathbf{B}}$(masses ${\mathit{m}}_{\mathbf{A}}$and ${\mathit{m}}_{\mathbf{B}}$)are at rest, held together by a massless string of length . Now the string is cut, and the particles fly off in opposite directions. How fast is each one going, when they are far apart?${\mathit{q}}_{\mathbf{A}}$

Consider an infinite chain of point charges, $\mathbf{\pm }\mathit{q}$(with alternating signs), strung out along the  axis, each a distance  from its nearest neighbors. Find the work per particle required to assemble this system. [Partial Answer: $\mathbf{-}\mathit{\alpha }{\mathit{q}}^{\mathbf{2}}\mathbf{/}\mathbf{\left(}\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}\mathbf{a}\mathbf{\right)}$for some dimensionless number $\mathbf{\text{α}}$your problem is to determine it. It is known as the Madelung constant. Calculating the Madelung constant for 2- and 3-dimensional arrays is much more subtle and difficult.]

Find the energy stored in a uniformly charged solid sphere of radius R and charge q. Do it three different ways:

(a)Use Eq. 2.43. You found the potential in Prob. 2.21.

(b)Use Eq. 2.45. Don't forget to integrate over all space.

(c)Use Eq. 2.44. Take a spherical volume of radius  a . What happens as $a\to \infty$?

Here is a fourth way of computing the energy of a uniformly charged

solid sphere: Assemble it like a snowball, layer by layer, each time bringing in an infinitesimal charge $\mathit{d}\mathit{q}$from far away and smearing it uniformly over the surface, thereby increasing the radius. How much work$\mathit{d}\mathit{W}$does it take to build up the radius by an amount$\mathit{d}\mathit{r}$? Integrate this to find the work necessary to create the entire sphere of radius R and total charge q .

Consider two concentric spherical shells, of radii a and b. Suppose the inner one carries a charge q , and the outer one a charge -q (both of them uniformly distributed over the surface). Calculate the energy of this configuration, (a) using Eq. 2.45, and (b) using Eq. 2.47 and the results of Ex. 2.9.

Find the interaction energy $\left({\in }_0\int E_1.E_2 d\tau {\in }_0\int E_1-E_{2}d\tau in Eq.2.47\right)$

for two point

charges $q_1$and $q_2$a distance a apart.

A metal sphere of radius R , carrying charge q , is surrounded by a

thick concentric metal shell (inner radius a , outer radius b , as in Fig. 2.48). The

shell carries no net charge.

(a) Find the surface charge density  $\mathit{\sigma }$at R , at a , and at b .

(b) Find the potential at the center, using infinity as the reference point.

(c) Now the outer surface is touched to a grounding wire, which drains off charge

and lowers its potential to zero (same as at infinity). How do your answers to (a) and (b) change?

Two spherical cavities, of radii a and b, are hollowed out from the

interior of a (neutral) conducting sphere of radius (Fig. 2.49). At the center of

each cavity a point charge is placed-call these charges ${\mathit{q}}_{\mathbf{a}}$ and ${\mathit{q}}_{\mathbf{b}}$.

(a) Find the surface charge densities ${\mathbf{\sigma }}_{\mathbf{a}}\mathbf{,}{\mathbf{\sigma }}_{\mathbf{b}}\mathbf{ }\mathbf{and}\mathbf{ }{\mathbf{\sigma }}_{\mathbf{R}}$

(b) What is the field outside the conductor?

(c) What is the field within each cavity?

(d) What is the force on ${\mathit{q}}_{\mathbf{a}}$ and ${\mathit{q}}_{\mathbf{b}}$? 

(e) Which of these answers would change if a third charge,${\mathit{q}}_{\mathbf{c}}$ , were brought near

the conductor?

 (a) A point charge${}^{\mathbf{q}}$ is inside a cavity in an uncharged conductor (Fig. 2.45). Is the force on${}^{\mathbf{q}}$ necessarily zero?

(b) Is the force between a point charge and a nearby uncharged conductor always

attractive?

Two large metal plates (each of area $\mathbf{A}$) are held a small distance $\mathbf{d}$

a part. Suppose we put a charge${}^{\mathbf{Q}}$on each plate; what is the electrostatic pressure on the plates?

A metal sphere of radius${}^{\mathbf{R}}$carries a total charge${}^Q$.What is the force

of repulsion between the "northern" hemisphere and the "southern" hemisphere?

Find the capacitance per unit length of two coaxial metal cylindrical tubes, of radi${}^{\mathbf{a}}$and${}^{\mathbf{b}}$.

 Suppose the plates of a parallel-plate capacitor move closer together by an infinitesimal distance${}^{\mathbf{\in }}$, as a result of their mutual attraction.

(a) Use Eq. 2.52 to express the work done by electrostatic forces, in terms of the field ${}^{\mathbf{E}}$, and the area of the plates, ${}^{\mathbf{A}}$.

(b) Use Eq. 2.46 to express the energy lost by the field in this process.

(This problem is supposed to be easy, but it contains the embryo of an alternative derivation of Eq. 2.52, using conservation of energy.)

Two infinitely long wires running parallel to the x axis carry uniform

charge densities $\mathbf{+}\mathit{\lambda }$ and $\mathbf{-}\mathit{\lambda }$.

(a) Find the potential at any point $\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}$ using the origin as your reference.

(b) Show that the equipotential surfaces are circular cylinders, and locate the axis

and radius of the cylinder corresponding to a given potential ${\mathit{V}}_{\mathbf{0}}$.

Find the potential on the rim of a uniformly charged disk (radius R,

charge density u).

The electric potential of some configuration is given by the expression

$\mathit{V}\left(r\right)\mathbf{=}\mathit{A}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{\lambda }\mathbf{r}}}{\mathbf{r}}$

Where $\mathit{A}$ and $\mathit{\lambda }$ are constants. Find the electric field $\mathit{E}\mathbf{\left(}\mathbf{r}\mathbf{\right)}$, the charge density $\mathit{\rho }\mathbf{\left(}\mathbf{r}\mathbf{\right)}$, and the total charge $\mathit{Q}$.

Find the electric field at a height $Z$ above the center of a square sheet (side a) carrying a uniform surface charge $\sigma$. Check your result for the limiting

cases $a\to \infty$ and $z>>a$.

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

by the expression

$E\left(r\right)=\frac{k}{r}[3\hat{r}+2sin\theta cos\theta sin\varphi \hat{\theta }+sin\theta cos\varphi \hat{\varphi }]$

for some constant , what is the charge density?

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