Statement
(a) Find the capacitance per unit length between a cylindrical conductor of radius a and a ground plane parallel to the conductor axis, and a distance h from it. The conductor carries a charge distribution r. (b) Double the conductor diameter to see the effect on the capacitance.
System Parameters
Permittivity of free space:
Solution
A useful technique in problems of this kind is the method of images. Take the mirror image of the conductor in the ground plane, and let this image conductor carry the negative of the charge distribution on the actual conductor. Now suppose the ground plane is removed. The electric field of the conductors and it image still obeys the correct boundary condition at the location of the original conductor, and, by symmetry, has an equipotential surface where the ground plane was. The field found in this way is therefore the field in the region between the actual conductor and the ground plane. See the figure below for clarification.
(a)
P.21 |
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P.22 Method of Images |
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Approximating the actual and image charge distributions by line charges at their centers,
Potential at radius a due to the actual conductor:
(See Problem 5.3 for the potential from a line charge.)
To find the potential due to the image conductor, find the distance from the imaginary image line charge to any point on the circumference of the real conductor.
Potential at point P due to the image conductor:
Similarly, the potential of the image conductor is -Va. Thus, the potential difference between the two conductors is 2Va, and the potential difference at the ground plane is 2Va/2 = Va.
The capacitance per unit length is then given by C = Q/(LV):
Observe that the capacitance per unit length for the whole source-image system (and more generally, for any pair of parallel cylindrical conductors with center-to-center separation 2h) is one-half the value found above, since it contains the same charge, but twice the voltage drop.
(b)
If the radius is doubled in the original problem, the resulting capacitance is
Recall that
for comparison
