Derive an expression for the force acting on electric dipole placed in a non uniform electric field.
Answers
If a dipole placed in non-uniform electric field, it will experience net force of attraction toward electric field of large magnitude that is, F(net)=p(E2-E1).
Torque is non zero when dipole is placed in uniform as well as non uniform electric field but in non-uniform electric field, dipole will experience net force of attraction whereas in uniform magnetic field, it doesn't.
This is best understood by approximating the dipole as a pair of finite charges ±q separated by a finite distance d. In a uniform electric field, the electrostatic forces on each of the charges will cancel out exactly, but in a non-uniform one the forces on the two will be slightly different, leading to a slight imbalance and therefore a non-zero net force. As you take the distance to zero, the difference in electric field goes to zero, but the charge also grows to exactly cancel it out.
To be more quantitative, suppose the negative charge is at r and the positive charge at r+dn. The total force is then
F=q[E(r+dn)−E(r)].
To get the correct form for the limit, change from the charge q to the electric dipole p=qd, to get
F=pE(r+dn)−E(r)d.
The true force on a point dipole is the limit of this as d→0,
F=plimd→0E(r+dn)−E(r)d,
and this is exactly the directional derivative along n, typically denoted n⋅∇, so
F=pn⋅∇E=p⋅∇E.