Consider the electric field E = γrr/εo.
(a) What are the units of γ?
(b) Verify that its curl is zero.
(c) Find its divergence.
(d) Find the integral ∫ E⋅dl from the origin O to the point A at x=y=z=R, along the following paths:
(i) Straight line from O to A
(ii) From O to (R,0,0) to (R,R,0) to A
(e) Find the volume integral of the divergence of E inside a sphere of radius R.
(f) Find the flux of E through the surface of the sphere in part (e).
A conducting sphere with radius R has within it a spherical cavity with radius a. The conducting sphere itself has no net charge on it but the volume in the cavity is filled with a material with uniform charge density ρ. (a) Find the surface charge on all surfaces associated with the conductor. (b) Find the electric field at all points in space. (c) Find the electric potential at all points in space. | 
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A capacitor is constructed using three thin coaxial conducting cylindrical shells of length L as shown in the figure. The shells have radii a < b < c. The outermost shell is connected with a thin conducting wire W to the innermost shell. A charge Q is placed on the middle conductor, and -Q on the outer conductor. (Some of the charge on the outer conductor may go to the inner conductor through the wire, until the potential of those two conductors become equal. You can neglect the presence of the wire after that point.) (a) Find the charge distributions and charges on all surfaces. (b) Find the potential difference between conductors and determine the capacitance. (c) Find the electric field at all points in space. (d) Integrate the energy density associated with the electric field to verify that U = ½CV2. | 
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Find the capacitance of two conducting spheres, one with a radius R and second with radius r, separated a distance L apart. Assume that r ≪ R and r ≪ L so that the potential of the smaller sphere may be assumed not to be influenced by the larger sphere. | 
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A channel with conducting walls is constructed as shown in the figure. The channel has a
square cross section of width and heigth a, and extends to infinity in the +z direction.
At z=0, the channel is capped with a square shaped material which maintains a potential
v(x,y) = α x(x-a)y(y-a)
where α is a constant.
(a) Find the potential at all points inside the channel.
(b) Obtain computer plots of the potential in the channel. You can display the potential
in two forms:
o You can plot V versus z for x=a/2 and y=a/2
and V versus x for z=a and y=a/2
o Or (extra credit) plot V(x,y,a/2) and V(x,y,a) as contour or "heat map" graphs.
If you do not know how to construct such plots, search the internet for more
information: contour plots heat map plots
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A circular line charge with total magnitude Q and radius R has been placed
on the z-axis as shown. The center of the cirle is on the z-axis, at z=a
and the plane of the circle is parallel to the x-y plane.
(a) Find the potential at points on the z-axis.
(b) Expand this potential in powers of z (or 1/z where appropriate), to
6th order.
(c) Does this problem have azimuthal symmetry? If so, find the series expansion
for V(r,θ) at a general point in space, to the same order as in part (b).
Extra-credit part
(d) Using the result of part (c) plot V(r,θo) as a function of r.
Use cos(θo) = 1.1 a/(a2+R2)1/2 .
This choice of θo corresponds to a path which passes very close to
the charge so that you should get high potentials for r ~ (a2+R2)1/2.
Does this path go through inside or outside the circular charge?
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Two dipoles of magnitudes p1 and p2 are placed such that the first dipole is at the origin and is pointing in the +z direction, while the second dipole is on the x-axis, a distance R from the origin and pointing in the +x direction. Find all forces and torques in the system and verify that they all add up to zero.