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In the attached figure, a coil is wrapped around the central arm of a flat H-shaped thin plate. DC current flows through the coil and magnetizes the arm. At the same time, DC current also flows from another source through the plate as shown in the figure. Within the central arm, the directions of flow of the magnetic field and current are perpendicular to each other.

If B is the magnetic flux density, L is the length along which the current I flows through the arm, does the plate experience a force F according to the equation F = B I L ?

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does the plate experience a force F according to the equation F = B I L ?

In this formula, I and B are vectors, the operation is a cross-product and not a multiplication, and the resulting force is also a vector. You could also consider I to be a scalar and L to be a vector representing the length and direction of the wire.

$ \vec{F} = L \vec{I} \times \vec{B} $

Apply the usual finger rule to get the direction of the force.

The result is an approximation, as the current in the plate will not be uniform and in a straight line from top to bottom. Consider several different current paths (in blue):

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These paths have different lengths, thus different resistance, and they will not share current equally. In fact current density will be higher in the corners and along the edges (in yellow). This wouldn't matter for calculating the force if magnetic field was uniform, but the coil does not generates a uniform magnetic field...

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