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I should add context to this question that what I know so far of the emf is that it is defined as ##\mathscr E=\oint f \cdot d\ell## where ##f## is the sum of the forces on the charges. In the textbook I'm reading it gives an example of where a person pulls a rectangular wire with a velocity ##\vec{v}## and the left side of that wire is in a uniform magnetic field.

After some calculations, the author states that calculating the emf from the current induced in line ##ab## at a particular instant is equivalent to the work done by the person pulling the wire even though the force of the magnetic field, ##\vec{v}\times \vec{B}##, has no effect on the work done by the person and the force exerted by the person, ##f_{pull}## has no effect on the emf. How is that we can interpret the emf as ##\begin{equation} \mathscr E=\int f_{pull} \cdot d\ell=\oint f_{mag} \cdot d\ell\end{equation}##, yet the force produced by a magnetic field does no work?

After some calculations, the author states that calculating the emf from the current induced in line ##ab## at a particular instant is equivalent to the work done by the person pulling the wire even though the force of the magnetic field, ##\vec{v}\times \vec{B}##, has no effect on the work done by the person and the force exerted by the person, ##f_{pull}## has no effect on the emf. How is that we can interpret the emf as ##\begin{equation} \mathscr E=\int f_{pull} \cdot d\ell=\oint f_{mag} \cdot d\ell\end{equation}##, yet the force produced by a magnetic field does no work?

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