Question

If a linear isotropic dielectric is placed in an electric field of strength E. Then the polarisation p is

Answer 1

you should understand linear isotropic dielectric is that in which polarisation of dielectric is parallel to the applied electric field or you can say say polarisation is directly dpeends upon electric field.

We know,
Volume Charge density = volume charge density in case of free + volume charge density in case of bound
e.g.,$\bold{\rho=\rho_f+\rho_b}$
Volume charge density also rewritten as function of polarisation
e.g.,$\bold{\rho=\rho_f} - \bold{\nabla P}$

∵ electric field is the ratio of volume charge density and permittivity of medium.
so, $\bold{\nabla.E=\frac{\rho}{\epsilon_0}}$
​​
$\bold{\nabla.E=\frac{\rho_f-\nabla.P}{\epsilon_0}}$
Hence, $\bold{P_f=\epsilon_0E+P}$
Hence, polarisation in case of free charge, $\bold{P_f=\epsilon_0E+P}$

Answer 2

directly proportional to E