(a) Deduce the expression for the torque acting on a dipole of dipole moment p in the presence of a uniform electric field E⃗.

(b) Consider two hollow concentric spheres, S₁ and S₂, enclosing charges 2Q and 4Q respectively as shown in the figure.

(i) Find out the ratio of the electric flux through them.

(ii) How will the electric flux through the sphere S₁ change if a medium of dielectric constant 'εr' is introduced in the space inside S₁ in place of air ?

 Deduce the necessary expression.  

a) Dipole in a uniform electric field:

Consider an electric dipole consisting of charges −q and +q and of length 2a placed in a uniform electric field making an angle θ with the electric field.

Forces acting on the two charges of the dipole, are +qE and -qE.

That is, the net force on the dipole is equal and opposite.

So, Force = 0

Two forces are equivalent to torque having magnitude given by,

 

Therefore, torque acting on the dipole is given by, 

b) i)  Charge enclosed by sphere S₁ = 2Q

Charge enclosed by sphere S₂ = 2Q + 4Q = 6Q

Now, using Gauss law, electric flux enclosed by sphere S₁ and S₂ is given by, 

 

ii) If a medium of dielectric constant 'εr' is introduced in the space inside S₁ in place of air, electric flux becomes



 
That is, electric flux decreases.