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(Solved): A single isolated conductor can store charge. We can define its capacitance as the ratio of the cha ...
A single isolated conductor can store charge. We can define its capacitance as the ratio of the charge to the potential at the surface. (Alternatively, one can think of a second conductor at infinity storing a negative charge of equal magnitude. This gives the same result, since at infinity and the potential difference is therefore just the potential on the first conductor's surface.) (a) Find an expression for the capacitance of an isolated spherical conductor of radius R. (HINT: The field outside the sphere is the same as though all of the charge were concentrated at the center. Gauss's law can be used to prove this result. Use the following as necessary: for the electrostatic constant, and R.) (b) Find the capacitance of a copper sphere of radius . (c) Find the capacitance of an isolated spherical conductor the size of the earth. (Although the earth stores charge, it is not a single isolated conductor. Negative charge is stored on the ground and a roughly equal quantity of positive charge is spread through the lower atmosphere.)
Expert Answer
Solution:(a) To find the capacitance of an isolated spherical conductor of radius R, we can use the formula for capacitance:C = Q / VWhere C is the capacitance, Q is the charge stored on the conductor, and V is the potential at the surface of the conductor.To find the charge Q, we can use Gauss's law. According to Gauss's law, the electric field outside a conducting sphere is the same as if all of the charge were concentrated at the center of the sphere.Using this result, we can say that the electric field at a distance r (where r > R) from the center of the sphere is given by: