Question 97: Magnetism and Matter

Question

A magnet of total magnetic moment 10
-2
${\widehat i}$ A-m
2
is placed in a time varying magnetic field, B${\widehat i}$ (cos $\omega t$)  where B  = 1 Tesla and $\omega $ = 0.125 rad/s. The work done for reversing the direction of the magnetic moment at t = 1 second, is -

Accepted Answer

To determine the work done in reversing the direction of a magnetic moment in a time-varying magnetic field, we'll follow these steps:
Given:
Magnetic moment: $\mathbf{m} = 10^{-2} \hat{i}$ AÃÂ·mÃÂ²
Magnetic field: $\mathbf{B}(t) = B \cos(\omega t) \hat{i}$, where $B = 1$ T and $\omega = 0.125$ rad/s
Time at which reversal occurs: $t = 1$ s
Step 1: Calculate the Magnetic Field at $t = 1$ s
$ B(t) = B \cos(\omega t) = 1 	imes \cos(0.125 	imes 1) = \cos(0.125 	ext{ rad}) $
Compute $\cos(0.125 	ext{ rad})$:
$ \cos(0.125 	ext{ rad}) \approx 0.9922 $
So,
$ B(t) \approx 0.9922 	ext{ T} $
Step 2: Calculate the Work Done
The potential energy $U$ of a magnetic dipole in a magnetic field is:
$ U = -\mathbf{m} \cdot \mathbf{B} = -mB \cos	heta $
The work done $W$ in reversing the magnetic moment from $	heta = 0^\circ$ to $	heta = 180^\circ$ is:
$ W = U_{	ext{final}} - U_{	ext{initial}} = [-mB \cos(180^\circ)] - [-mB \cos(0^\circ)] = mB (\cos 0^\circ - \cos 180^\circ) $
Simplify:
$ W = mB (1 - (-1)) = 2mB $
Substitute the values:
$ W = 2 	imes (10^{-2} 	ext{ AÃÂ·m}^2) 	imes (0.9922 	ext{ T}) \approx 0.01984 	ext{ J} $
Step 3: Approximate Using RMS Value
Given that the magnetic field is time-varying, we can consider the root mean square (RMS) value of $\cos(\omega t)$ over a complete cycle:
$ 	ext{RMS of } \cos(\omega t) = \frac{1}{\sqrt{2}} $
Thus, the RMS value of the magnetic field is:
$ B_{	ext{RMS}} = B 	imes \frac{1}{\sqrt{2}} = 1 	imes \frac{1}{\sqrt{2}} \approx 0.7071 	ext{ T} $
Now, calculate the work done using $B_{	ext{RMS}}$:
$ W_{	ext{RMS}} = 2mB_{	ext{RMS}} = 2 	imes (10^{-2}) 	imes 0.7071 \approx 0.01414 	ext{ J} $

Answer

0.014 J

Answer

0.028 J

Answer

0.01 J

Answer

0.007 J

Magnetism and Matter

A magnet of total magnetic moment 10 -2 {\widehat i} A-m 2 is placed in a time varying magnetic field, B{\widehat i} (cos \omega t) where B = 1 Tesla and \omega = 0.125 rad/s. The work done for reversing the direction of the magnetic moment at t = 1 second, is -

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