Exercise: Estimate the capacitance of the two "potato-shaped" objects
in Figure 27.2. Assume they are far apart.
Following the standard plan, we put a charge +Q on one conductor and a charge -Q on the other. Now we must find the electric field that is produced.
For our estimate, we can model each potato as a sphere. (They are clearly
more like spheres than like infinite cylinders or like sheets.) Being far
apart, we may assume that the conductors do not have much influence on
each other. Thus the electric field produced by each sphere looks like
a point-charge field: E=kQ/r2. Now we need to superpose
the fields produced by the two spheres.
First set up a coordinate system with origin at the center of one sphere and x-axis along the line joining their centers. Then at any point with coordinate x, the sphere on the left, with positive charge, produces electric field directed to the right. The sphere on the right, with negative charge, also produces field pointing right. The point is a distance x from the center of the left sphere, but a distance d-x from the right sphere. Then we have:
Next we find the potential difference between the two spheres. We integrate along a straight line path from the surface of the left sphere, at x = a to the surface of the right sphere at x=d-a. The result is:
Thus the potential difference is:
and thus the capacitance is:
When d >> a, the relevant dimension in our estimation rule is
the radius of each sphere.
How would the result change if the two spheres had different radii?