Problem: A non-conducting sphere of radius r has a spherical cavity of radius r/2. The solid part of the sphere has a uniform volume charge density ρ. Find the magnitude and direction of electric field at point (a) O and (b) A.

Solution:

Electric field at point O:
To find the electric field at point O, we need to consider the electric fields due to both the solid and the hollow parts of the sphere.

Electric field due to solid part:
Let's consider a small element of volume dv at a distance r' from the center of the sphere. The charge enclosed in this volume is given by dq = ρ dv. The electric field due to this small element of charge at point O is given by:

dE = k dq / r'^2

where k is the Coulomb's constant.

Integrating this expression over the entire volume of the solid part of the sphere, we get:

E_solid = k ρ (4π/3) r^3 / r^2

E_solid = (4/3) k ρ r

Electric field due to hollow part:
The electric field due to the hollow part of the sphere at point O is zero, as there is no charge present inside the cavity.

Net electric field at point O:
The net electric field at point O is given by the vector sum of the electric fields due to the solid and hollow parts of the sphere. As the electric field due to the hollow part is zero, the net electric field at point O is simply given by:

E_O = (4/3) k ρ r

The direction of the electric field is radially outward from the center of the sphere.

Electric field at point A:
To find the electric field at point A, we need to consider the electric field due to the solid part of the sphere only, as point A lies outside the sphere.

Electric field due to solid part:
The electric field due to the solid part of the sphere at point A is given by:

E_solid = kq / r^2

where q is the total charge enclosed by the sphere. The total charge enclosed by the sphere is given by:

q = ρ (4π/3) r^3

Substituting this value, we get:

E_solid = k ρ (4π/3) r^3 / r^2

E_solid = (4/3) k ρ r

Net electric field at point A:
The net electric field at point A is given by the electric field due to the solid part of the sphere only, as there is no charge present outside the sphere. Thus, the net electric field at point A is given by:

E_A = (4/3) k ρ r

The direction of the electric field is radially outward from the center of the sphere.