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ShayanJ

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## Main Question or Discussion Point

One of the problems in my textbook of electromagnetism is about proving that the work done by the force [itex] \vec{F}=I \vec{dl} \times \vec{B} [/itex],is [itex] \delta W=I \delta \phi [/itex] where the circuit isn't rigid and the displacement vector of the element of interest is [itex] \vec{\delta r} [/itex] with a constant current and [itex] \delta \phi [/itex] is the change in magnetic flux.My calculation is as follows:

[itex]

\delta W=\vec{F}\cdot \vec{\delta r}=I (\vec{dl}\times \vec{B})\cdot \vec{\delta r}=I[ \delta x (dy B_z-dz B_y)+...]=I[(\delta x dy-\delta y dx)B_z+..]

[/itex]

To complete the proof,I should be able to set [itex] \delta A_z=\delta x dy-\delta y dx [/itex],etc.([itex]\delta A_z[/itex] being the change in area caused by [itex] B_z [/itex]).My problem is,I don't know how to justify it!

Any ideas?

Thanks

[itex]

\delta W=\vec{F}\cdot \vec{\delta r}=I (\vec{dl}\times \vec{B})\cdot \vec{\delta r}=I[ \delta x (dy B_z-dz B_y)+...]=I[(\delta x dy-\delta y dx)B_z+..]

[/itex]

To complete the proof,I should be able to set [itex] \delta A_z=\delta x dy-\delta y dx [/itex],etc.([itex]\delta A_z[/itex] being the change in area caused by [itex] B_z [/itex]).My problem is,I don't know how to justify it!

Any ideas?

Thanks