A solid sphere of radius R has a non-uniform volume charge density given by = ₀ ( r R ) , where r is the distance from the center and ₀ is a positive constant. The magnitude of the potential difference between the center of the sphere and its surface is ₀ R^2 N ₀ . The value of N is ______.

Step-by-step solution

Consider a spherical Gaussian surface of radius r ( r The charge enclosed is: q_ encl = ₀^r (r') 4 r'^2 dr' q_ encl = ₀^r ₀ ( r' R ) 4 r'^2 dr' = 4 ₀ R ₀^r r'^3 dr' = ₀ r^4 R Applying Gauss's law: E(r) 4 r^2 = q_ encl ₀ = ₀ r^4 R ₀ E(r) = ₀ r^2 4 R ₀ The magnitude of the potential difference between the center and the surface is: V = ₀^R E(r) dr = ₀^R ₀ r^2 4 R ₀ dr V = ₀ 4 R ₀ [ r^3 3 ]₀^R = ₀ R^2 12 ₀ Comparing with the given expression, N = 12 . Answer: 12