JEE Advanced · 2008 · 13 Apr 2008 · Paper 2 · Electrostatics

The nuclear charge

(Z e)

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 centre of the nucleus as shown in figure. The electric field is only along the radial direction.

For

a = 0

, the value of

d

(maximum value of

ρ

as shown in the figure) is

Answer: (B)

Solution

When

a = 0

, the charge density varies linearly from

d

at the center to 0 at

r = R

ρ (r) = \frac{d}{R} (R - r)

.

The total charge must equal

Z e

Z e = \int_{0}^{R} ρ (r) \cdot 4 π r^{2} d r = \int_{0}^{R} \frac{d}{R} (R - r) \cdot 4 π r^{2} d r

Z e = \frac{4 π d}{R} \int_{0}^{R} (R r^{2} - r^{3}) d r = \frac{4 π d}{R} [\frac{R r ^{3}}{3} - \frac{r ^{4}}{4}]_{0}^{R}

Z e = \frac{4 π d}{R} (\frac{R ^{4}}{3} - \frac{R ^{4}}{4}) = \frac{4 π d}{R} \cdot \frac{R ^{4}}{12} = \frac{π d R ^{3}}{3}

d = \frac{3 Z e}{π R ^{3}}

JEE Advanced · 2008 · 13 Apr 2008 · Paper 2 · Single correct option