Question

Two opposite and equal charges \[4 \times {10^{ - 8}}\]coulomb when placed \[2 \times {10^{ - 2}}\]cm ways form a dipole . If the dipole is placed in an external electric field \[4 \times {10^8}\]newton / coulomb , the value of maximum torque and the work done in rotating it through \[{180^0}\] will be
A.\[32 \times {10^{ - 4}}{\rm{ Nm and 64}} \times {\rm{1}}{{\rm{0}}^{ - 4}}J\]
B. \[64 \times {10^{ - 4}}{\rm{ Nm and 32}} \times {\rm{1}}{{\rm{0}}^{ - 4}}J\]
C. \[32 \times {10^{ - 4}}{\rm{ Nm and 32}} \times {\rm{1}}{{\rm{0}}^{ - 4}}J\]
D. \[64 \times {10^{ - 4}}{\rm{ Nm and 64}} \times {\rm{1}}{{\rm{0}}^{ - 4}}J\]

Answer 1

Hint:Electric dipole moment measures the separation of the positive and negative electrical charges within a system and is defined as the product of the magnitude of the charge and distance between them. Torque can be defined as the cross product of the dipole moment and the electric field.

Formula used:
Dipole moment is given as,
\[\overrightarrow p = q \times d\]
Where q is the magnitude of charge and d is the separating distance.
Torque is given as,
\[\overrightarrow \tau = \overrightarrow p \times \overrightarrow E = pE\sin \theta \]
Where p is the dipole moment, E is the electric field and \[\theta \] is the angle formed by the dipole axis with the electric field.
work done is given as,
\[W = pE(1 - \cos \theta )\]

Complete step by step solution:
Given charge, q=\[4 \times {10^{ - 8}}{\rm{ C}}\]
Electric field E=\[4 \times {10^8}{\rm{ N/C}}\]
Distance, \[d = 2 \times {10^{ - 2{\rm{ }}}}cm\]
As we know Dipole moment,\[\overrightarrow p = q \times d\]
By substituting the values,
\[\begin{array}{l}\overrightarrow p = 4 \times {10^{ - 8}} \times 2 \times {10^{ - 2{\rm{ }}}}\\{\rm{ = 8}} \times {\rm{1}}{{\rm{0}}^{ - 11}}{\rm{ C - m}}\end{array}\]

Also as we know torque, \[\overrightarrow \tau = \overrightarrow p \times \overrightarrow E = pE\sin \theta \]
By substituting the values,
\[\overrightarrow \tau = pE{\rm{sin9}}{{\rm{0}}^0}{\rm{ = 1}}\]
\[\begin{array}{l} \Rightarrow \overrightarrow \tau {\rm{ = 8}} \times {\rm{1}}{{\rm{0}}^{ - 11}} \times 4 \times {10^8}\\ \Rightarrow \overrightarrow \tau {\rm{ = 32}} \times {\rm{1}}{{\rm{0}}^{ - 4}}Nm\end{array}\]
Therefore the value of maximum torque will be \[{\rm{32}} \times {\rm{1}}{{\rm{0}}^{ - 4}}Nm\].

Now to find the work done in rotating it through \[{180^0}\]:
As we know work done,\[W = pE(1 - \cos \theta )\]
By substituting the values,
\[\begin{array}{l}W = 32 \times {10^{ - 4}}(1 - \cos {180^0})\\ \Rightarrow W {\rm{ = }}32 \times {10^{ - 4}}(1 - ( - 1))\\ \Rightarrow W {\rm{ = }}32 \times {10^{ - 4}} \times 2\\ \therefore W {\rm{ = 64}} \times {\rm{1}}{{\rm{0}}^{ - 4}}J\end{array}\]
Therefore the work done in rotating it through \[{180^0}\] will be \[{\rm{64}} \times {\rm{1}}{{\rm{0}}^{ - 4}}J\].

Hence option A is the correct answer.

Note: Torque is the rotational analogue of linear force and also termed as rotational force or turning effect. The torque is in clockwise direction if the direction of an electric field is positive. An electric field is an electric property associated with each point in the space where charge is present either at rest or in motion.