Question

An electric field of 1000V/m is applied to an electric dipole at angle of ${45^0}$. The value of electric dipole moment is ${10^{ - 29}}C.m.$. What is the potential energy of the electric dipole moment?
A) -9$ \times {10^{ - 20}}J$
B) -7$ \times {10^{ - 27}}J$
C) -10$ \times {10^{ - 29}}J$
D) -20$ \times {10^{ - 18}}J$

Answer 1

Hint- When a dipole is placed in an electric field both of the forces will experience torque. They do not experience any force.

Formula used: To solve this type of question we use the following formula.
\[U = - \vec p.\vec E = - pE\cos \theta \]; Here p is the dipole moment; E is the electric field and $\theta $ is the angle between dipole moment and electric field.

Complete step by step answer:
Let us write the information given in the question.
$E = 1000V/m,\theta = {45^ \circ },\left| {\vec p} \right| = {10^{ - 29}}C.m.$
Now, let us use the formula \[U = - \vec p.\vec E = - pE\cos \theta \]to calculate the potential energy of the electric dipole moment.
\[U = - {10^{ - 29}} \times 1000 \times \cos 45\]
Now, let us simplify the above expression.
\[U = - {10^{ - 26}} \times \dfrac{1}{{\sqrt 2 }}J = - 0.707 \times {10^{ - 26}}J = 7.07 \times {10^{ - 27}}J\]
Hence, option (B) \[ - 7.07 \times {10^{ - 27}}J\] 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.
                  $W = pE(1 - \cos \theta )$.
*The negative sign represents that work must be done against the electric field to move the charges.