A) \[\frac{\pi }{3}\,n\,\,\rho {{l}^{3}}\]
B) \[\frac{\pi }{4}\,n\,\,\rho {{l}^{3}}\]
C) \[n\,\,\rho {{l}^{3}}\]
D) \[\pi n\,\,\rho {{l}^{3}}\]
Correct Answer: B
Solution :
Charge of element \[=dq=\rho dx=\frac{{{\rho }_{0}}x}{l}dx\] Magnetic moment of this charge element \[=\,\,\frac{dqvr}{2}\] \[dM=\,\frac{{{\rho }_{0}}xdx}{l}\,\times \omega x\,\times \,\frac{x}{2}\] \[dM=\,\frac{{{\rho }_{0}}\omega {{x}^{3}}dx}{2l}\,\] \[M=\int\limits_{0}^{\ell }{\frac{{{\rho }_{0}}\omega {{x}^{3}}dx}{2l}}\,=\,\frac{{{\rho }_{0}}\omega }{2l}\,\,\int\limits_{0}^{\ell }{{{x}^{3}}dx}\] \[=\frac{\rho \omega }{2l}\,\,\frac{{{l}^{4}}}{4}\,=\,\frac{{{\rho }_{0}}\omega {{l}^{3}}}{8}\,\,(Put\,\omega =\,2\pi n)\] \[=\,\,\frac{2\pi n{{\rho }_{0}}\omega {{l}^{3}}}{8}\,=\,\frac{\pi }{4}\,n\,\rho \,{{l}^{3}}\]You need to login to perform this action.
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