Magnetic moment of a thin bar magnet is ‘M’. If it is bent into a semicircular form, its new magnetic moment will be________. Fill in the blank with the correct answer from the options given below.
To find the new magnetic moment of a thin bar magnet when bent into a semicircular shape, we need to understand how the magnetic moment is affected by the change in shape. The magnetic moment \(M\) of a straight bar magnet is given by the product of its pole strength \(m\) and the length \(l\) of the bar magnet: \(M = m \cdot l\).
When the magnet is bent into a semicircle, its pole strength remains unchanged, but the effective length of the magnet is now the chord length of the semicircle. The semicircular arc length is the original length \(l\). The radius \(r\) of the semicircle can be determined using \(r = \frac{l}{\pi}\), as the arc length (original straight length) is \(l = \pi r\).
Now, the straight-line distance between the two ends of the magnet (the chord length) becomes the diameter of the semicircle, which is \(2r = \frac{2l}{\pi}\). The new magnetic moment \(M'\) can be calculated as:
\(M' = m \cdot \text{chord length} = m \cdot \frac{2l}{\pi}\).
Since the original magnetic moment \(M\) was \(m \cdot l\), the new magnetic moment in terms of \(M\) is:
\(M' = \frac{2M}{\pi}\).
Hence, the magnetic moment of the magnet when bent into a semicircular form is \(\frac{2M}{\pi}\).
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