Question

Two similar bar Magnets $P$ and $Q$ each of magnetic moment $M$ are taken. If $P$ is cut along its axial line and $Q$ along its equatorial line all the four pieces obtained have each of
A. Equal pole strength
B. Magnetic Moment $\dfrac{M}{4}$
C. Magnetic Moment $\dfrac{M}{2}$
D. Magnetic Moment $M$

Answer 1

Hint:
When a bar magnet is cut into two equal halves, each tiny magnet behaves like a magnet and has both the north and south pole. It is found that the magnetic field due to a tiny bar magnet at any point along its axial line is twice the magnetic field at a point along the equatorial line with the same distance.


Formula used:
The formula used to solve this question is given as follows: -
Magnetic dipole moment, $M = ml$




Complete step by step solution:
Initially, the magnetic moment of $P$ and $Q$ is $M$ (given)
We know that $M = ml$ is the magnetic dipole moment of a bar magnet.
Where m = pole strength and l = length of a bar magnet
Now, for magnet $P$, when it is cut along its axial line length remains the same but pole strength becomes halved as: -


Therefore, in this case, magnetic moment will be: ${M_P} = \left( {\dfrac{m}{2}} \right)l = \dfrac{{ml}}{2} = \dfrac{M}{2}$ $\left( {\therefore M = ml} \right)$
Now, for magnet $Q$, when it is cut along its equatorial line pole strength remains the same but length becomes halved as: -


Therefore, in this case, magnetic moment will be: ${M_Q} = m\left( {\dfrac{l}{2}} \right) = \dfrac{{ml}}{2} = \dfrac{M}{2}$

Thus, when the two magnets $P$ and $Q$ is cut according to the question, the resulting magnetic moment of both the magnets will become ${M_P} = {M_Q} = \dfrac{M}{2}$.
Hence, the correct option is (C) Magnetic Moment $\dfrac{M}{2}$.



Therefore, the correct option is C.




Note:
Since this is a problem related to a uniform magnetic field and magnetic dipole moment hence, we should always remember that the direction of magnetic field is the same direction as that of magnetic dipole moment vector due to tiny bar magnet.