A spherically symmetric charge distribution is characterised by a charge density having the following variations: (r)= _o (1- r R ) for r < R ( r )=0 for r R Where r is the distance from the centre of the charge distribution _ o is a constant. The electric field at an internal point (r < R) is:

Step-by-step solution

Let us consider a spherical shell of radius x and thickness dx . Charge on this shell d q= .4 x^2 d x= ₀ (1- x R ) 4 x^2 d x Total charge in the spherical region from centre to r(r < R) is aligned &q= d q=4 ₀ ₀^r (1- x R ) x^2 d x &=4 ₀ [ x^3 3 - x^4 4 R ]₀^r &=4 ₀ [ r^3 3 - r^4 4 R ] &=4 ₀ r^3 [ 1 3 - r 4 R ] aligned Electric field at r , E = 1 4 ₀ q r ^2 aligned &= 1 4 ₀ 4 ₀ r^3 r^2 [ 1 3 - r 4 R ] &= ₀ ₀ [ r 3 - r^2 4 R ] aligned