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Example Questions
Example Question #91 : Electrostatics
Tom is given an uncharged, solid sphere of radius . He then decides to charge all the material within a distance from the center uniformly. He gives that part of the sphere a total charge of . He then charges the rest of the material uniformly, giving that part of the sphere a total charge of . Besides the very center point of the sphere, at what distance from the center will there be no electric field?
We need the charge enclosed by a gaussian surface (in this case, we'll use a sphere because there's spherical symmetry) to be zero. When this happens, there cannot be any electric field as
All the charge is uniformly distributed, so the charge density of the inner sphere is and on the outer sphere is . We only need to use the outer sphere's surface density since the zero electric field point cannot enclose zero charge in the inner sphere since it's only enclosing negative charge.
The distance where there is no electric field will be when the gaussian surface encloses from the outer sphere in order to cancel the from the inner sphere.
Because everything is uniformly distributed:
. is charge, is charge density, and is volume. We need this to equal
This is the volume of just the outer surface needed. To find the distance, we just use the formula for the volume of a thick spherical shell and algebra.
Let be the distance where there will be zero electric field.
This gives