Question

An insulating sphere of radius a, centered at the origin, has a uniform volume charge density p. Find the electric field E(F) inside the

sphere (for r < a) in terms of the position vector T. spherical cavity is excised from the inside of the sphere. The cavity has radius a/4 and is centered at position h = hi, where |<< a, so that the entire cavity is contained within the larger sphere. Find the electric field inside the cavity. Express your answers in terms of any or all of p, r,h.

Answer 1

To find the electric field within a charged sphere and a cavity within it, Gauss's law is applied to the symmetrical charge distributions. The electric field inside the sphere and cavity can be determined by the superposition of the fields.

Step-by-step explanation:

The student is asking about the electric field inside a uniformly charged insulating sphere with a spherical cavity removed from within. To find the electric field inside the cavity, the effects of the removed charge and the remaining charge distribution must be considered. Due to the spherical symmetry of the problem and by using Gauss's law, one can deduce the field inside a uniform sphere of charge as well as inside the cavity.

Finding the Electric Field inside the Sphere

For a sphere with uniform volume charge density ρ and radius a, the electric field E inside the sphere (for r < a) can be found using Gauss's law:

E(r) = (1/4πε_{0}) * (ρ/a^3) * r

Finding the Electric Field inside the Cavity

For the cavity of radius a/4 and centered at h = hi, one must superpose the effects from the region with the charge density ρ and the cavity where the charge density is effectively '-ρ'. The total electric field E inside the cavity is:

E_{cavity} = E_{sphere} - E_{excised sphere}

The sum of these fields, considering their appropriate directions, yields the electric field inside the cavity.