KVPY Sample Paper KVPY Stream-SX Model Paper-29

  • question_answer
    Wire bent as ABOCD as shown, carries current I entering at A and leaving at D. Three uniform magnetic fields each \[{{B}_{0}}\] exist in the region as shown. The force on the wire is

    A) \[\sqrt{3}I\,R\,{{B}_{0}}\]

    B) \[\sqrt{5}I\,R\,{{B}_{0}}\]

    C) \[\sqrt{8}I\,R\,{{B}_{0}}\]

    D) \[\sqrt{6}I\,R\,{{B}_{0}}\]

    Correct Answer: D

    Solution :

    \[\vec{F}=\vec{F}=I\vec{\ell }\times \vec{B}\]
    \[\vec{\ell }=\overrightarrow{AD}=R(\vec{i}-\vec{j})\]
    \[\therefore \vec{F}=IR{{B}_{0}}(\hat{i}-\hat{j})\times (\hat{i}+\hat{j}-\hat{k})\]
    \[=IR{{B}_{0}}\left| \begin{matrix}    {\hat{i}} & {\hat{j}} & {\hat{k}}  \\    1 & -1 & 0  \\    1 & 1 & -1  \\ \end{matrix} \right|=IR{{B}_{0}}(\hat{i}+\hat{j}+2\hat{k})\]
    \[\vec{B}={{B}_{0}}(\hat{i}+\hat{j}-\hat{k})\,\,:\,\,\,\,\,\,\vec{\ell }=R(\hat{i}-\hat{j})\]
    \[\vec{B}\,\vec{\ell }=0\,\,\,\,\,\,\Rightarrow Angle\,\,=90{}^\circ \,\,\,\,\,\,\Rightarrow \,\,\,\,\,\,\,F=BI\ell \]

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