Question

For a dipole \[q = 2 \times {10^{ - 6}}C\], \[d = 0.01m\], find the maximum torque on the dipole in \[E = 5 \times {10^5}N/C\]
A. \[1 \times {10^{ - 3}}N{m^{ - 1}}\]
B. \[10 \times {10^{ - 3}}N{m^{ - 1}}\]
C. \[10 \times {10^{ - 3}}Nm\]
D. \[1 \times {10^{ - 3}}Nm\]

Answer 1

Hint:A dipole is the combination of two charges of equal in magnitude and opposite in nature. When a charge is kept in a uniform electric field then it experiences the electric force. When a force has a perpendicular component then it applies torque.

Formula used:
\[p = qd\], here p is the magnitude of the dipole moment, q is the magnitude of charge separated at distance d.
\[\overrightarrow \tau = \overrightarrow p \times \overrightarrow E \], here \[\overrightarrow \tau \]is the torque vector, \[\overrightarrow p \]is the dipole moment vector and \[\overrightarrow E \]is the electric field vector.

Complete step by step solution:
The magnitude of the charge is given as \[q = 2 \times {10^{ - 6}}C\] and the electric field is given as \[5 \times {10^5}N/C\]. The charges are separated by the distance \[d = 0.01\,m\]. The magnitude of the dipole moment is,
\[p = \left( {2 \times {{10}^{ - 6}}C} \right) \times \left( {0.01} \right)Cm\]
\[\Rightarrow p = 2 \times {10^{ - 8}}Cm\]

Let the angle between the dipole moment vector and the electric field vector is \[\theta \]
Then the magnitude of the torque on the dipole will be,
\[\tau = pE\sin \theta \]
Putting the values, we get
\[\tau = \left[ {\left( {2 \times {{10}^{ - 8}}} \right) \times \left( {5 \times {{10}^5}} \right)\sin \theta } \right]Nm\]
\[\Rightarrow \tau = \left[ {1 \times {{10}^{ - 2}}\sin \theta } \right]Nm\]
For maximum value of torque, the value of \[\sin \theta \]should be maximum.

As we know that the range of the sine function is -1 to 1.
\[ - 1 \le \sin \theta \le 1\]
The maximum value of the sine function is 1 when the angle between the electric field vector and the dipole moment vector is 90°, i.e. the electric field vector and the dipole moment are perpendicular to each other. So, the maximum value of the torque will be,
\[{\tau _{\max }} = \left[ {1 \times {{10}^{ - 2}}{{\left( {\sin \theta } \right)}_{\max }}} \right]Nm\]
\[\Rightarrow {\tau _{\max }} = \left[ {1 \times {{10}^{ - 2}} \times 1} \right]Nm\]
\[\Rightarrow {\tau _{\max }} = \left[ {1 \times {{10}^{ - 2}}} \right]Nm\]
\[\therefore {\tau _{\max }} = 10 \times {10^{ - 3}}Nm\]
Hence, the maximum torque on the dipole is \[10 \times {10^{ - 3}}Nm\] due to the given electric field.

Therefore, the correct option is C.

Note: The torque is maximum when the dipole moment is perpendicular to the electric field. In other words, the value of \[\theta\] will be \[90^{\circ}\] for maximum torque.Thus the value of \[\sin\theta\] will be 1 for maximum torque.