Potential energy of a dipole in an external field

Potential energy of a dipole

Consider a dipole with charges q₁ = +q and q₂ = -q placed in a uniform electric field as shown in the figure above. The charges are separated by a distance d and the magnitude of an electric field is E. The force experienced by the charges is given as –qE and +qE, as can be seen in the figure.

As we know that, when a dipole is placed in a uniform electric field, both the charges as a whole do not experience any force, but it experiences a torque equal to τ which can be given as,

This torque rotates the dipole unless it is placed parallel or anti-parallel to the field. If we apply an external and opposite torque, it neutralizes the effect of this torque given by τ_ext and it rotates the dipole from the angle Ɵ₀ to an angle Ɵ₁ at an infinitesimal angular speed without any angular acceleration.

The amount of work done by the external torque can be given by

As we know that the work done in bringing a system of charges from infinity to the given configuration is defined as the potential energy of the system, hence the potential energy U(Ɵ)  can be associated with the inclination Ɵ of the dipole using the above relation.

From the above equation, we can see that the potential energy of dipole placed in an external field is zero when the angle Ɵ is equal to 90° or when the dipole makes an angle of 90°.

Considering the initial angle to be the angle at which the potential energy is zero, the potential energy of the system can be given as,

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