Polarisation & Boundary Condition: Assignment | Electricity & Magnetism - Physics PDF Download

Q.1. Assume that z = 0 plane is the interface between two linear and homogenous dielectrics (see figure). The relative permittivities are ε_r = 5 for z > 0 and ε_r = 4 for z<0. The electric field in the region  Assume that there are no free charges on the interface.

(a) Find the electric field  in the region z < 0.
(b) Find the electric field in the region z < 0.

(a)
and

(b)

Q.2. A unit cube made of a dielectric material has a polarization units. The edges of the cube are parallel to the Cartesian axes. Find
(a) the volume bound charge.
(b) the bound charge on both the surfaces parallel to the y - z plane.
(c) the bound charge on both the surfaces parallel to the x - z plane.

(a)
(b)
(c)

Q.3. The half space region x>0 and x<0 are filled with dielectric media of dielectric constants ε₁ and ε₂ respectively. There is a uniform electric field in each part. In the right half, the electric field makes an angle θ₁ to the interface and the corresponding angle to the left half θ₂. Show that ε₁ tan θ₁ = ε₂ tan θ₂ .

Q.4. A spherical conductor of radius a is placed in a uniform electric field  The potential at a point P(r ,q ) for r>a, is given by

where k is some constant.
Where r is the distance of P from the centre O of the sphere and θ is the angle OP makes with the z -axis. Find the charge density on the sphere at θ = 60º.

Q.5. A conducting spherical shell of radius R₁ carries a total charge Q. A spherical layer of a linear, homogeneous and isotropic dielectric of dielectric constant K and outer radius R₂ (>R₁) covers the shell as shown in figure.
(a) Find the electric field and the polarization vector  inside the dielectric. From this , calculate the surface bound charge density, σ_b, on the outer surface of the dielectric layer and the volume bound charge density ρ_b, inside the dielectric.
(b) Calculate the electrostatic energy of the system

 
(a) 


The surface bound charge density at r


(b)

Q.6. A polarized dielectric cube of side ℓ is kept on the x -y plane as shown. If the polarization in the cube is  where k is a positive constant, and then finds all the bound surface charge densities and volume charge density.

Volume charge density
Bound surface charge density on the surface OADGO (y = 0 plane)

Bound surface charge density on the surface EBCFE (y = l plane)

Bound surface charge density on the surface OGFEO (x = 0 plane)

Bound surface charge density on the surface ADCBA (x = l plane);

Bound surface charge density on the surface OABEO (z = 0 plane)

Bound surface charge density on the surface GFCDG ( z = l plane)

Q.7. A ray of light inside Region 1 in the xy -plane is incident at the semicircular boundary that carries no free charges. The electric field at the point  in plane polar coordinates is  where  are the unit vectors. The emerging ray in Region 2 has the electric field  parallel to x -axis. If ε₁ and ε₂ are the dielectric constants of Region-1 and Region-2 respectively then find ε₂/ε₁.

 
Thus  makes an angle

Q.8. A spherical shell of inner and outer radii R₁ and R₂, respectively, is made of a dielectric material with frozen polarization  where a is a constant and r is the distance from its centre. Find the electric field in the region R₁ < r< R₂.

  and

For, R₁ < r< R₂ ;

⇒ Q_enc = 4παR₁ + 4πα(r - R₁) = 4παr

Q.9. A conducting sphere of radius R_A has a charge Q. It is surrounded by a dielectric spherical shell of inner radius R_A and outer radius R_B (as shown in the figure below) having electrical permittivity ε(r ) = ε₀r.
(a) Find the surface bound charge density at r =R_B.
(b) Find the electrostatic energy of the system.
 

 
or

(a)


The surface bound charge density at r

(b)

Q.10. In a parallel plate capacitor the distance between the plates is 2d. Two dielectric slabs of thickness d each and dielectric constants K₁ and K₂ respectively, are inserted between the plates. A potential of V is applied across the capacitor as shown in the figure. Show that the value of the net bound surface charge density at the interface of the two dielectrics is