Explanation:

When a linear isotropic dielectric is placed in an electric field of strength E, the polarization P is defined as the electric dipole moment per unit volume of the dielectric material.

Definition:

P = (ΔD) / V

Where,
ΔD = Change in electric displacement
V = Volume of the dielectric material

Electric displacement:

The electric displacement D is defined as the electric flux per unit area produced by free charges in a dielectric material.

D = ε0E + P

Where,
ε0 = Permittivity of free space
E = Electric field strength
P = Polarization

Substituting the value of D in the equation for polarization:

P = (ΔD) / V
= (Δ(ε0E + P)) / V
= ε0(ΔE) / V + (ΔP) / V

Applying the continuity equation:

The continuity equation states that the net charge density in a dielectric material is conserved. Therefore, any change in polarization must be accompanied by a corresponding change in free charge density.

Δρf = -∇.J

Where,
Δρf = Change in free charge density
J = Free current density

Substituting the value of Δρf in the expression for polarization:

(ΔP) / V = -∇.J / ε0

Conclusion:

Therefore, the polarization P of a linear isotropic dielectric placed in an electric field of strength E is given by the equation P = ε0(ΔE) / V - ∇.J / ε0, where ΔE is the change in electric field strength and J is the free current density.