A sphere of radius R carries a nonuniform but spherically symmetric volume charge density that results in an electric field in the sphere given by E=E_0(r/R)^2 * r_hat, where E_0 is a constant. Find the potential difference from the sphere's surface to its center.
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v = ∫E·dl
V = ∫((E_0)(r/R)^2)dr since dl = dr and since dr ·r hat = rdr
V = (E_0)(r^3/(3R^3) from r= 0 to R
at r =0, V =0 at r = R, V = (E_0)/3
So V = (E_0)/3 from the surface to the center
V = ∫((E_0)(r/R)^2)dr since dl = dr and since dr ·r hat = rdr
V = (E_0)(r^3/(3R^3) from r= 0 to R
at r =0, V =0 at r = R, V = (E_0)/3
So V = (E_0)/3 from the surface to the center