Question

A Bar magnet has a magnetic moment M and length l. If its length is reduced to half, find its new magnetic moment.

Hint:

Remember the two ways to cut a magnet:
1. Transversely (perpendicular to length): length halves ($l \rightarrow l/2$), pole strength is constant ($m \rightarrow m$). $M' = M/2$.
2. Longitudinally (along the length): length is constant ($l \rightarrow l$), pole strength halves ($m \rightarrow m/2$). $M' = M/2$.
In both typical cutting scenarios into two equal pieces, the magnetic moment halves.

Answer 1

Step 1: Understanding the Concept:
The magnetic moment ($M$) of a bar magnet is defined as the product of its pole strength ($m$) and its magnetic length ($l$).

Step 2: Key Formula or Approach:
\[ M = m \times l \]
When a bar magnet is cut to reduce its length, we must consider how the cut is made. "Length is reduced to half" typically implies cutting it transversely (perpendicular to its length) into two equal pieces.

Step 3: Detailed Explanation:
Let the initial magnetic moment be $M = m \times l$.
If the magnet is cut transversely into two equal halves, the length of each new piece is $l' = l/2$.
The pole strength ($m$) depends on the cross-sectional area of the magnet, which remains unchanged by a transverse cut. So, $m' = m$.
The new magnetic moment ($M'$) for one of the halves is:
\[ M' = m' \times l' \]
\[ M' = m \times \left(\frac{l}{2}\right) \]
\[ M' = \frac{m \times l}{2} \]
Substituting the original magnetic moment $M$:
\[ M' = \frac{M}{2} \]

Step 4: Final Answer:
The new magnetic moment is $M/2$.