question_answer

Figure shows an infinitely long wire carrying an outward current\[{{I}_{1}}\]. The current is along Z axis. There is a curved wire carrying current\[{{I}_{2}}\]. The magnetic force on this wire between \[({{x}_{1}},{{y}_{1}})\] and \[({{x}_{2}},{{y}_{2}})\] is

A) \[\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}{4\pi }\,\frac{(x_{2}^{2}+y_{2}^{2})}{(x_{1}^{2}+y_{1}^{2})}\]

B) \[\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}{4\pi }\,In\,\frac{(x_{2}^{2}+y_{2}^{2})}{(x_{1}^{2}+y_{1}^{2})}\]

C) \[\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}{4\pi }\,In\,\frac{({{x}_{2}}+{{y}_{2}})}{({{x}_{1}}+{{y}_{1}})}\]

D) \[\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}{2\pi }\,In\,\frac{(x_{2}^{2}+y_{2}^{2})}{(x_{1}^{2}+y_{1}^{2})}\]

Correct Answer: B

Solution :

[b] \[dF=\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}dr}{2\pi r}\Rightarrow F=\int\limits_{{{r}_{1}}}^{{{r}_{2}}}{\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}{2\pi r}}dr=\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}{2\pi }[\ell nr]_{{{r}_{1}}}^{{{r}_{2}}}\] \[=\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}{2\pi }\left[ \ell n{{r}_{2}}-\ell n{{r}_{1}} \right]=\frac{{{\mu }_{0}}{{I}_{1}}{{I}_{2}}}{4\pi }\ell n\left( \frac{x_{2}^{2}+y_{2}^{2}}{x_{1}^{2}+y_{1}^{2}} \right)\]