Download 1. Find the potential a distance s from an infinitely long straight wire

1. Find the potential a distance s from an infinitely long straight wire that carries a uniform line
charge λ. Compute the gradient of your potential, and check that it yields the correct field.
2. Consider an infinite chain of point charges, ±q (with alternating signs), strung out along the x axis,
each a distance a from its nearest neighbors. Find the work per particle required to assemble this
system.
3. Two positive point charges, 𝑞𝐴 and 𝑞𝐵 (mass 𝑚𝐴 and 𝑚𝐵 ) are at rest, held together by a massless
string of length a. Now the string is cut, and the particles fly off in opposite directions. How fast is
each one going, when they are far apart?
4. A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner
radius a, out radius b, as in Fig. 2.1). The shell carries no net charge.
(a) Find the surface charge density 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 the infinity). How do your answers to (a) and (b) change?
5. An inverted hemispherical bowl of radius R carries a uniform surface charge density σ. Find the
potential difference between the “north pole” and the center.
6. The electric potential of some configuration is given by expression
𝑒 −𝜆𝑟
𝑟
Where A and λ are constants. Find the electric field E(r), the charge density ρ(r), and the total
charge Q.
𝑉(𝒓) = A
7. A point charge q is at the center of an uncharged spherical conducting shell, of inner radius a and
outer radius b. Question: How much work would it take to move the charge out to infinity (through
a tiny hole drilled in the shell)?
8. Four equal positive charges Q are fixed in space and form a square. Another positive charge q of
mass m is trapped in the middle of the square by repulsive forces (no gravity). The distance from
charge q to other charges is L. The position of this charge is then perturbed along the horizontal
axis causing it to fluctuate around the equilibrium position. Assuming small perturbations, show
that the oscillations are harmonic and find their period.
Q
q