Question

A point dipole with dipole moment, $\vec{p} = p_0 \hat{k}$ , is kept at the origin. An external electric field given by, $\vec{E} = E_0 ( 2\hat{i} - 3\hat{j} + 4\hat{k})$ , is applied on it. Which one of the following statements is true ?

• A. The force on the dipole is zero while torque rotates the dipole on the xy-plane
• B. The force on the dipole moves it along the direction of electric field
• C. The interaction energy between the dipole and electric field is zero
• D. The potential due to the dipole alone on the xy-plane with z = 0 depends on the value of p0
• E. The application of the electric field orients the dipole along the $ - \hat{k}$ direction

Answer 1

Answer: (A)

The given electric field is uniform in nature. Thus, the net force acting on it will be zero. However, couple of forces will act on either charges of the dipole in opposite direction that would constitute torque. Due to this torque the dipole would rotate.
As, torque on the dipole is given as $\tau= p \times E$
Given, $p_{0}=P_{0} \cdot \hat{ k }$
and $E =E_{0}(2 \hat{ i }-3 \hat{ j }+4 \hat{ k }),$ then
$\tau =\left(P_{0} \cdot \hat{ k }\right) \times E_{0}(2 \hat{ i }-3 \hat{ j }+4 \hat{ k })$
$=E_{0}\left[2 P_{0}(\hat{ k } \times \hat{ i })-3 p_{0}(\hat{ k } \times \hat{ j })\right][\because \hat{ k } \times \hat{ k }=0]$
$=E_{0}\left[2 P_{0} \hat{ j }-3 P_{0}(-\hat{ i })\right]$
$[\because \hat{ k } \times \hat{ j }=\hat{ j }$ and $\hat{ k } \times \hat{ j }=-\hat{ i }]=P_{0} E_{0}(3 \hat{ i }+2 \hat{ j })$
So, it implies that torque is acting on the dipole rotating it on $x$ -y plane. However, the interaktion energy between the dipole and electric field
$E= P E = p _{0} \hat{ k } \cdot E _{0}(2 \hat{ i }-3 \hat{ j }+4 \hat{ k })=P_{0} E_{0} 4 \neq 0$