91Ó°ÊÓ

Find the interaction energy $\left({∈}_0∫E_1.E_{2}d\mathrm{τ}{∈}_0∫E_1-E_{2}d\mathrm{τ}inEq.2.47\right)$

for two point

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

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 $\mathit{σ}$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 aand 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 ${\mathbf{σ}}_{\mathbf{a}}\mathbf{,}{\mathbf{σ}}_{\mathbf{b}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{σ}}_{\mathbf{R}}$

(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?

Find the electric field a distance zabove 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.

(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${}^{\mathbf{∈}}$, 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.)

An inverted hemispherical bowl of radius Rcarries a uniform surface charge density .Find the potential difference between the "north pole" and the center.