Question

Give the definition of electric potential energy. Find out the expression for electric potential energy of a dipole in uniform electric field. In which condition get the position of stable or unstable equilibrium

Answer 1

Electric Potential Energy of an Electric Dipole in an External Electric Field:

Electric potential energy of an electric dipole in an electric field is equal to work done in bringing an electric dipole from infinity to that field.

In figure, an electric dipole is brought from infinity to a uniform electric field \(\vec E\) so that the dipole moment \(\vec E\) is always along electric field \(\vec E\). The force on charge +q due to electric field \(\vec E\), \(\vec F\) = q \(\vec E\) along the field and force on charge -q, \(\vec F\) = -q \(\vec E\) is opposite to the field. Thus, an external work is done for bringing charge q of dipole in electric field where as the electric field itself Work on charge -q. On coming from infinity to electric field, -q charge has to move 2a more distance that by q. Thus, the work done by the charge -q is more and negative. Thus, work done by the electric field,

W = Force on charge (-q) × Distance covered
W = -qE × 2a = -2 qaE
W = -pE ∵ p = 2qa
Thus, potential energy of electric dipole placed parallel to electric field .
U₁ = -pE ……………. (1)
Now, work done in rotating by angle θ from the parallel position of electric field .
U₂ = pE (1 - cosθ) ………….. (2)
Thus, potential energy of electric dipole at angle θ in electric field,
U = U₁ + U₂
U = -pE + pE (1 - cosθ)
U = -pE cosθ
In vector form, U = \(-\vec{p} \cdot \overrightarrow{{E}}\) …………. (3)
Equation (2) is the expression for potential energy of electric dipole.

Special Cases
(a) If the electric dipole is at 0° angle from the electric field, then the potential energy
U = -pE cosθ
U = -pE cos0°
i.e., U = -pE …………. (4)
(It is a condition of stable equilibrium)

(b) If electric dipole moment is perpendicular to the electric field, then the potential energy
U = -pEcos 90°
U = 0 ………….. (5)

(c) If the electric dipole moment is at 180° angle from the electric field, then the potential energy,
U = -pE cos 180°
U = pE …………….. (6)
(It is a condition of unstable equilibrium)

Case (a) is called the state of stable equilibrium because in this case, energy is minimum and case (c) is called the unstable equilibrium because in this case, the energy is maximum.