Solveeit Logo
Login

Question

Question: What work must be done to rotate an electric dipole through an angle \(\theta \) with the electric f...

What work must be done to rotate an electric dipole through an angle θ\theta with the electric field, if an electric dipole of moment pp is placed in a uniform electric field EE with pp parallel to EE?

Explanation

Solution

A dipole is a pair of two electric charges of equal magnitude and opposite signs. To solve the problem, apply the formula for external work required to rotate the dipole in the given external electric field EE i.e., dW=τdθdW = \tau d\theta .
τ\tau is the external torque acting on the dipole and it is given by, τ⃗=p⃗×E⃗\vec \tau = \vec p \times \vec E
pp is the dipole moment.

Complete step by step answer:
The given dipole is placed in a uniform electric field E⃗\vec E.
The torque on a dipole is given by τ→=p→×E→\overrightarrow \tau = \overrightarrow p \times \overrightarrow E .
τ=pEsin⁡θ\tau = pE\sin \theta
Where, θ\theta is the angle between the dipole moment pp and the electric field EE.
The work to be done to rotate the dipole by an angle dθd\theta is dW=τdθdW = \tau d\theta
The total work done to rotate the dipole from θ1{\theta _1} to θ2{\theta _2}is W=∫θ1θ2dWW = \int_{{\theta _1}}^{{\theta _2}} {dW}
W=∫θ1θ2τdθW = \int_{{\theta _1}}^{{\theta _2}} {\tau d\theta }
Substitute the magnitude of torque τ\tau
⇒W=∫θ1θ2pEsin⁡θdθ\Rightarrow W = \int_{{\theta _1}}^{{\theta _2}} {pE\sin \theta d\theta }
Integrate the above equation
⇒W=pE[−cos⁡θ]θ1θ2\Rightarrow W = pE\left[ { - \cos \theta } \right]_{_{{\theta _1}}}^{{\theta _2}}
⇒W=pE(cos⁡θ1−cos⁡θ2)\Rightarrow W = pE\left( {\cos {\theta _1} - \cos {\theta _2}} \right)
It is given that initially the dipole is placed parallel to the external electric field EE.
θ1=00{\theta _1} = {0^0} and later rotated with θ2=θ{\theta _2} = \theta
Therefore, Total work done W=pE(cos⁡00−cos⁡θ)W = pE\left( {\cos {0^0} - \cos \theta } \right)
Or W=pE(1−cos⁡θ)W = pE\left( {1 - \cos \theta } \right)
Hence, the correct option is (A) pE(1−cos⁡θ)pE\left( {1 - \cos \theta } \right).

Note: The total external force acting on a dipole in an external electric field is always zero. The external torque acting on a dipole is zero when the dipole moment is either parallel or antiparallel with the external electric field.
The total work done by a system is equal to the potential energy stored in the system.