Question

A circular coil of $ 100 $ turns and an effective diameter of $ 20cm $ carries a current of $ 0.5A $ . It is to be turned in a magnetic field $ 2.0\;T $ from a position in which the normal to the plane of the coil makes an angle $ \theta $ equal to zero to one in which $ \theta $ equals $ 180 $ . The work required in this process is:
(A) $ \pi J $
(B) $ 2\pi J $
(C) $ 4\pi J $
(D) $ 8\pi J $

Answer 1

Hint: We know the current is produced due to the flow of electrons. A current-carrying wire produces a magnetic field. Also, when a current-carrying wire is placed in a magnetic field it experiences a magnetic force. The direction of force is found by Fleming’s right-hand rule.

Complete answer:
Work done is stored as the potential energy of the system. So, we have the following formula.
 $ W = U = - \vec m.\vec B $
Here, $ U $ is potential energy, $ m $ is magnetic dipole and $ B $ is a magnetic field.
Again, the magnetic dipole is calculated as below.
 $ \vec m = IA\overset{\lower0.5em\hbox{ $ \smash{\scriptscriptstyle\frown} $ }}{n} $
Here, $ I $ is current, $ A $ is an area, and $ \hat n $ is the unit normal.
Let us first write the information given in the question.
 $ N = 100 $ , $ d = 20cm \Rightarrow r = 10cm $ , $ I = 0.5A $ , $ B = 2T $ , $ {\theta _1} = 0 $ , $ {\theta _2} = 180 $
Magnetic moment: $ m = \left( {0.5} \right)\left( {\pi {{\left( {.10} \right)}^2}} \right) = 0.005\pi $
Work done will be a change in the potential energies.
 $ W = mB\cos {\theta _1} - mB\cos {\theta _2} $
We can rewrite this expression as below.
 $ W = mB\left( {\cos {\theta _1} - \cos {\theta _2}} \right) $
Let us substitute the values.
 $ W = 0.005\pi \times 2\left( {\cos 0 - \cos 180} \right) $
Let us simplify the above expression.
 $ W = 0.01\pi \left( {1 - \left( { - 1} \right)} \right) = 0.02\pi J $
Now, it is given that there are $ 100 $ turns. So total work done will become.
 $ W = 100 \times 0.02\pi = 2\pi J $
Therefore, total work done is $ 2\pi J $ .
Hence, option (B) $ 2\pi J $ is correct.

Note:
The magnetic dipole moment of an object is defined as the torque experienced by the object in a magnetic field. Unit of a magnetic dipole is $ A - {m^2} $ .