Name: Electric Energy Density
Uploaded: 2019-07-19
Duration: 4 min 11 s
Description: There is an energy associated with the electrostatic field. The expression of this electric field energy density is derived using the energy stored in a para...

Question

Derive an expression for the potential energy of an electric dipole placed in a uniform electric field. Hence discuss the conditions of its stable and unstable equilibrium.

Accepted Answer

Consider an electric dipole placed in a uniform electric field $\vec{E}$ with its dipole moment $\vec{p}$ making an angle $θ$ with the field. Two equal and opposite forces $+ q \vec{E}$ and $- q \vec{E}$ act on its two ends. The two forces form a couple. The torque exerted by the couple will be

$τ = q E \times 2 a \sin θ = p E \sin θ$

where $q \times 2 a = p$ , is the dipole moment.

If the dipole is rotated through a small angle $d θ$ against the torque acting on it, then small work done is

$d W = τ d θ = p E \sin θ d θ$

The total work done in rotating the dipole from its orientation making an angle $θ_{1}$ , with the direction of the field to $θ_{2}$ will be

$W = \int d W = \int_{θ_{1}}^{θ_{2}} p E \sin θ d θ$

$W = p E {[- \cos θ]}_{θ_{1}}^{θ_{2}} = p E (\cos θ_{1} - \cos θ_{2})$

This work done is stored as the potential energy $u$ of the dipole.

$∴ u = p E (\cos θ_{1} - \cos θ_{2})$

(i) When the dipole moment $\vec{p}$ parallel to the electric field $\vec{E} (θ = 0^{o})$ ), the dipole is in stable equilibrium.

(ii) When the dipole moment $\vec{p}$ is antiparallel to the electric field $\vec{E} (θ = 180^{o})$ ), the dipole is in unstable equilibrium.

Derive an expression for the potential energy of an electric dipole placed in a uniform electric field. Hence discuss the conditions of its stable and unstable equilibrium.

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