Question

An electric dipole of dipole moment $\overrightarrow{p}=(2.0\hat{i}+3.0\hat{j})\mu \text{C-m}$ is placed in a uniform electric field $\overrightarrow{E}=(3.0\hat{i}+2.0\hat{k})\times {10}^5\text{N/C}$

• A. The potential energy of the dipole is $+0.6\text{J}$
• B. The torque that $\overrightarrow{E}$ exertes on $\overrightarrow{p}$ is $(0.6\hat{i}-0.4\hat{j}-0.9\hat{k})\text{N-m}$
• C. The potential energy of the dipole is $-0.6\text{J}$
• D. If the dipole is rotated in the electric field , the maximum potential energy of the dipole is $1.3\text{J}$

Answer 1

Answer: (B), (C), (D)

If the dipole is rotated in the electric field , the maximum potential energy of the dipole is $1.3\text{J}$

Given,

Dipole moment of electric dipole $\overrightarrow{p}=(2.0\hat{i}+3.0\hat{j})\mu \text{C-m}$

Electric field $\overrightarrow{E}=(3.0\hat{i}+2.0\hat{k})\times {10}^5\text{N/C}$

Torque acting on electric dipole w.r.t origin is given by $\overrightarrow{\tau }=\overrightarrow{p}\times \overrightarrow{E}$

$\therefore \overrightarrow{\tau }=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2.0\times {10}^{-6}& 3.0\times {10}^{-6}& 0.0\times {10}^{-6}\\ 3.0\times {10}^5& 0.0\times {10}^5& 2.0\times {10}^5\end{array}\right|$

$\Rightarrow \overrightarrow{\tau }=(0.6\hat{i}-0.4\hat{j}-0.9\hat{k})\text{N-m}$

Potential energy of dipole , $U=-\overrightarrow{p}\cdot \overrightarrow{E}$

Substituting the data given we get,

$U=-(2.0\hat{i}+3.0\hat{j})\times {10}^{-6}\cdot (3.0\hat{i}+2.0\hat{k})\times {10}^5$

$\Rightarrow U=-0.6\text{J}$

Potential energy of dipole is minimum when the dipole moment direction is along the direction of electric field and the potential energy is maximum when the dipole moment direction is anti-parallel to the direction of electric field.

$\therefore U_i=-0.6\text{J}$

After rotation, Potential energy of dipole , $U_f=|\overrightarrow{p}||\overrightarrow{E}|$

$\Rightarrow U_f=2\times 0.1\times \sqrt{{\left(2.0\right)}^2+{\left(3.0\right)}^2}$

$\Rightarrow U_f=0.2\times \sqrt{13}=0.7211\text{J}$

The work done in rotating the dipole is stored as potential energy, Maximum potential energy stored is given by

$\Delta U_{max}=|U_f-U_i|$

$\Rightarrow \Delta U_{max}=|0.7211-(-0.6)|=1.3211\text{J}\approx 1.3\text{J}$

Hence, options (a) ,(b) and (d) are the correct answers.