A spherical cavity is excised from the inside of the sphere. The cavity has radi

Question

A spherical cavity is excised from the inside of the sphere. The cavity has radius a4and is centered at position h? , where |h? |<34a, so that the entire cavity is contained within the larger sphere. Find the electric field inside the cavity. Express your answer as a vector in terms of any or all of ? (Greek letter rho), ?0, r? , and h? . A spherical cavity is excised from the inside of the sphere. The cavity has radius a4and is centered at position h? , where |h? |<34a, so that the entire cavity is contained within the larger sphere. Find the electric field inside the cavity. Express your answer as a vector in terms of any or all of ? (Greek letter rho), ?0, r? , and h? .

Explanation / Answer

Throughout this E, r, h, and d will denote vectors.

First find the electric field a a sphere without the cavity:
q = ?V = ?(4/3*?r^3)
E = kq/r^2 = k?(4/3?r^3)/r^2 = 4k??r/3 = ?r/(3?_0)

A cavity has no charge, which is the same as having two equal and opposite charges. Find the electric field of such an opposite sphere:
E_opp = -?d/(3?_0)
This is the same as above, just using d to denote the distance from the center of the opposite sphere. Because the sphere is centered at h in the sphere, d = r - h:
E_opp = -?(r - h)/(3?_0)

E_cav = E + E_opp = ?r/(3?_0) - ?(r - h)/(3?_0) = ?h/(3?_0)

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