Question

The nuclear charge (Ze) is non- uniformly distributed within a nucleus of radius R. The charge density ρ(r) (Charge per unit volume) is dependent only on the radial distance ‘r’ from the center of the nucleus as shown in figure. The electric field is only along the radial direction.

(1) The electric field at r = R is:

(a) Independent of a 

(b) Directly proportional to a 

(c) Directly proportional to a² 

(d) Inversely proportional to a

(2) For a = 0, then value of d(maximum value of as shown in figure) is :

(a) \(\frac{3Ze}{4\pi R^3}\)

(b) \(\frac{3Ze}{\pi R^3}\)

(c) \(\frac{4Ze}{3\pi R^3}\)

(d) \(\frac{Ze}{3\pi R^3}\) 

(3) The electric field within the nucleus is generally observed to be linearly dependent on ‘r’. This implies: 

(a) a = 0 

(b) a = \(\frac{R}{2}\)

(c) a = R 

(d) a = \(\frac{2R}{3}\)

Answer 1

(1) Option : (A)

According to gauss’s law :

Also from graph, 

for a<R,

ρ is independent of a.

∴ E at r = R, will be independent of ‘a’ as the distribution of charge with in the surface is immaterial.

(2) Option : (B) 

For a = 0, 

means for r = 0, 

ρ is maximum and minimum (zero) at r = R 

Charge with in a spherical shell of radius ‘r’ and thickness dr,

(3) Option : (c)

As given, 

E∝ r

∴ a = R, from graph.