JEE Main · 2012 · 29 Apr 2012 · Offline · Electrostatics

In a uniformly charged sphere of total charge

Q

and radius

R

, the electric field

E

is plotted as a function of distance from the centre. The graph which would correspond to the above will be

Answer: (C)

Solution

For a sphere carrying charge spread uniformly through its volume, Gauss's law gives two separate behaviours. Inside, at radius

r < R

, only the enclosed charge grows, as

r^{3}

, so

E = \frac{1}{4 π ε _{0}} \frac{Q r}{R ^{3}}

and the field rises in direct proportion to

r

. Outside, at

r > R

, the whole charge acts as though at the centre, so

E = \frac{1}{4 π ε _{0}} \frac{Q}{r ^{2}}

and the field decays as

1/ r^{2}

. The correct plot therefore climbs as a straight line from the centre to the surface, reaches its greatest value exactly at

r = R

, and then falls away as an inverse-square curve beyond it.

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option