Promlems
1) Find charge distribution on a grounded conducting sphere with radious R centered
at the origin due to a charge q at a position (r,θ,Φ) outside of the sphere. Plot the
charge distribution as a function of θ for different r/R values.
2) Find potential of a charge q which is near the charged (Q) conducting and
insulated sphere. Also find the force on the charge q, and charge distribution on
the sphere.
3) Find potential of a charge q near the conducting sphere at a fixed potential V. Also
find the charge distribution on the sphere and force on the charge q.
4) A conducting sphere is in a uniform electric field E. By using only method of
images, find potential outside of the sphere and charge distribution on the sphere.
5) Two large metal plates are held a distance h apart, one at a potential zero, the other
at a potential V0. A metal sphere of radius R (R<<h) is sliced in two, and one
hemisphere is placed on the grounded plate so that its potential is approximately
zero. If the region between the plates is filled with weakly conducting materials of
uniform conductivity σ, what current flows to the hemisphere?
6) A well collimated beam of protons moves through space in the form of a cylinder of
radius R. The velocity of the protons is v and the number per unit volume is ρ. Find
the forces on a proton at a radius r from the beam axis. Is the beam stable?
7) A small sphere of polarizability α and radius R1 is placed at a large distance from a
conducting sphere of radius R2, which is maintained at a potential V. Find an
approximate expression for the force on the dielectric sphere valid for R1<<r.
8) A current I passes on a flat, thin infinitely long ribbon of width w. Determine the
magnetic field at a point P that is in the plane of the ribbon at a distance x from one
edge, as shown in the diagram. Evaluate the case for w goes to ∞.
9) Two identical ideal parallel plate capacitors in air, each with capacitance C, are
connected to a battery of voltage V as shown schematically below. The plates are
separated by a distance d.
i) What is the charge on each plate in terms of C, V, d?
ii) What is the electric field (direction and magnitude) between the plates of each
capacitor?
Without changing anything else, fill gap of the capacitor on the left with a
dielectric κ=3.
iii) What is the charge on each plate in terms of C, V, d and κ?
iv) What is the electric field (direction and magnitude) between the plates of each
capacitor?
10)
An infinite, non-conducting sheet of negligible
thickness carrying a uniform surface charge density –σ and, next to it, an infinite
parallel slab of thickness h with uniform volume charge density +ρ.
Calculate the direction and magnitude of the electric field,
i) A distance L above the negatively charged sheet?
ii) Inside the slab at a distance d below the negatively charged sheet? (d<h)
iii) A distance H below the bottom of the slab?
11) A capacitor is formed by two rectangular conducting plates having edges L1 and L2.
The plates are not parallel. One pair of edges of length L1 is separated by a distance
d1 everywhere, and the other pair of edges of length L1 is separated by d2
everywhere, d2>d1.
i) Neglecting the edge effects when a voltage difference V is placed across the two
conductors, find potential everywhere between the plates.
ii) Determine the capacitance.
12) A and B are opposite ends of a diameter AOB of a very thin spherical shell of radius
R and thickness t. Current I enters and leaves by two small circular electrodes of
radius b whose centers are at A and B. If I is the total current and P is a point on
the shell such that the angle POA=θ, show that the magnitude of the current
density vector at P is proportional to
1
2πR t sin 𝜃
. Hence find the resistance of the
conductor.
13) The current I flowing along the edges of one face of a cube produces magnetic field
B0 in the center of the cube. Consider another cube where the current I flows along
a path shown in figure b below. What magnetic field will now exist at the center of
the cube?
a)
b)
14)A long hollow conducting cylinder with radius R is halved in two parts and
separated by a small gap. They are kept at different potentials V1 and V2. Find the
potential inside of the cylinder in terms of V1, V2, and R. Also calculate the surface
charge density on each half of the cylinder.
15) A sphere with radius a is placed into another sphere of radius b and they carry
charges ±Q respectively. A dielectric material of dielectric constant
filled between the spheres as shown in the figure below.
i) Find the electric field everywhere between the spheres.
ii) Find surface charge distribution on the inner sphere.
iii) Find bound charges on the dielectric material.
𝜀/𝜀0 is half