Question

An electric dipole when placed in a uniform electric field E will have a minimum potential energy if the dipole moment makes the following angle with E.
(A) $\pi $
(B) $\dfrac{\pi }{2}$
(C) Zero
(D) $\dfrac{{3\pi }}{2}$

Answer 1

Hint: The dipole moment arises due to the separation of the charges. It occurs due to the change in the electronegativity between the two chemically bound atoms. It involves the concept of the measure of the separation of the positive and negative charges.
Formula used:
To solve this type of problem we have the following formula.
\[U = - \vec p.\vec E = - pE\cos \theta \]; Here p is the dipole moment; the units of the dipole moments are: In Debye system it is measured in D and in S.I units’ system the units are Coulomb per meter.
E is the electric field and $\theta $ is the angle between the dipole moment and electric field.

Complete step by step answer:
We have to find the angle for which potential energy will be minimum.
We have the formula for potential energy of a dipole in an electric field as follows.
\[U = - \vec p.\vec E = - pE\cos \theta \]
Now, the minimum or maximum value depends on the minimum possible values of $\cos \theta $.
We also know the range of cosine is -1 to +1. Now, due to the already negative sign involved in the formula, we will take that angle for which $\cos \theta $gives +1.
So, the angle for which $\cos \theta $gives +1 is zero.
Hence, we can write.
\[{U_{minimum}} = - pE\cos 0 = - pE\]
$ \Rightarrow \theta = {0^ \circ }$

Hence, option (C) $ \Rightarrow \theta = {0^ \circ }$is the correct option.

Additional information:
When two charges, one positive and other negative, are kept at some distance. This setup is called electric dipole. It measures the polarity of the system.
When we place a dipole in a uniform magnetic field in equilibrium position. Now when we start rotating it from the equilibrium position with some angle$\theta $, work needs to be done against the electric field. This work done is saved in the form of potential energy of dipole.
Now, when we take the initial angle $\theta = \dfrac{\pi }{2}$ , i.e., when potential energy is zero, to some other angle $\theta $, we have the formula \[U = - \vec p.\vec E = - pE\cos \theta \].


Note:
The total work done in rotating the dipole by an angle $\theta $ is given by the following.
$W = pE(1 - \cos \theta )$.