Question 209: Magnetism and Matter

Question

An insulating thin rod of length $$l$$ has a linear charge density $$\rho \left( x \right)$$ = $${\rho _0}{x \over l}$$ on it. The rod is rotated about an axis passing through the origin (x = 0) and perpendicular to the rod. If the rod makes n rotations per second, then the time averaged magnetic moment of the rod is -

Accepted Answer

$$ \because $$   M = NIA
dq = $$\lambda $$dx  &   A = $$\pi $$x
2
$$\int {dm}  = \int {\left( x \right){{{\rho _0}x} \over \ell }} \,dx.\pi {x^2}$$
M = $${{n{\rho _0}\pi } \over \ell }.\int\limits_0^\ell  {{x^3}.dx}  = {{n{\rho _0}\pi } \over \ell }.\left[ {{{{L^4}} \over 4}} \right]$$
M = $${{n{\rho _0}\pi {\ell ^3}} \over 4}$$  or  $${\pi  \over 4}n\rho {\ell ^3}$$

Answer

$${\pi  \over 3}n\rho {l^3}$$

Answer

$${\pi  \over 4}n\rho {l^3}$$

Answer

$$n\rho {l^3}$$

Answer

$$\pi n\rho {l^3}$$

Magnetism and Matter

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