Q. A steel wire of length $l$ has a magnetic moment $M$ . It is then bent into a semicircular arc. The new magnetic moment is
NTA AbhyasNTA Abhyas 2022
Solution:
Let, the pole strength of wire $= m$
Magnetic moment $=$ Pole strength $\times $ Separation between poles.
For straight wire:
Magnetic moment, $M = m l$
When a wire of length $l$ is bent in semi-circular shape, then, $l=\frac{\pi d}{2}$ .
where, $d$ is the diameter of a semi-circle.
$\Rightarrow d=\frac{2 l}{\pi }$
Also in semi-circle, the separation between poles is its diameter.
Hence, the magnetic moment for semi-circle, $M^{'}=m\left(\frac{2 l}{\pi }\right)=\frac{2 M}{\pi }$
Magnetic moment $=$ Pole strength $\times $ Separation between poles.
For straight wire:
Magnetic moment, $M = m l$
When a wire of length $l$ is bent in semi-circular shape, then, $l=\frac{\pi d}{2}$ .
where, $d$ is the diameter of a semi-circle.
$\Rightarrow d=\frac{2 l}{\pi }$
Also in semi-circle, the separation between poles is its diameter.
Hence, the magnetic moment for semi-circle, $M^{'}=m\left(\frac{2 l}{\pi }\right)=\frac{2 M}{\pi }$