Question

A short bar magnet placed with its axis at \(45^\circ\) with an external field of \(400 \times 10^{-4}\,\text{T}\) experiences a torque of \(0.024\,\text{Nm}\). If a solenoid of cross-sectional area \(10^{-4}\,\text{m}^2\) and \(500\) turns replaces the short bar magnet such that they have the same magnetic moment, then the current flowing through the solenoid is:

• A. \(2\sqrt{2}\,\text{A}\)
• B. \(5\sqrt{2}\,\text{A}\)
• C. \(12\sqrt{2}\,\text{A}\)
• D. \(4\sqrt{2}\,\text{A}\)

Hint:

For a magnetic dipole in a uniform magnetic field, use \(\tau = MB\sin\theta\). For a solenoid, magnetic moment is \(M = NIA\).

Answer 1

The Correct Option is C

Step 1: Use torque formula for a magnetic dipole.
When a magnetic dipole of magnetic moment \(M\) is placed in a magnetic field \(B\) at an angle \(\theta\), the torque acting on it is given by:
\[ \tau = MB\sin\theta \]
Here,
\[ \tau = 0.024\,\text{Nm} \]
\[ B = 400 \times 10^{-4}\,\text{T} \]
\[ \theta = 45^\circ \]

Step 2: Simplify the magnetic field.
\[ B = 400 \times 10^{-4} \]
\[ B = 4 \times 10^{-2}\,\text{T} \]

Step 3: Substitute values in the torque formula.
\[ 0.024 = M \times 4 \times 10^{-2} \times \sin 45^\circ \]
Since,
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]
Therefore,
\[ 0.024 = M \times 4 \times 10^{-2} \times \frac{1}{\sqrt{2}} \]

Step 4: Find the magnetic moment of the bar magnet.
\[ M = \frac{0.024 \sqrt{2}}{4 \times 10^{-2}} \]
\[ M = \frac{0.024 \sqrt{2}}{0.04} \]
\[ M = 0.6\sqrt{2}\,\text{A m}^2 \]

Step 5: Use magnetic moment formula for solenoid.
The magnetic moment of a current carrying solenoid is given by:
\[ M = NIA \]
where \(N\) is number of turns, \(I\) is current, and \(A\) is cross-sectional area.
Given,
\[ N = 500 \]
\[ A = 10^{-4}\,\text{m}^2 \]

Step 6: Substitute the magnetic moment value.
\[ 0.6\sqrt{2} = 500 \times I \times 10^{-4} \]
\[ 0.6\sqrt{2} = 5 \times 10^{-2} I \]

Step 7: Find the current.
\[ I = \frac{0.6\sqrt{2}}{5 \times 10^{-2}} \]
\[ I = \frac{0.6\sqrt{2}}{0.05} \]
\[ I = 12\sqrt{2}\,\text{A} \]
\[ \boxed{12\sqrt{2}\,\text{A}} \]