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Griffiths 5.4 Force on a square loop

1 vote

Before doing any calculation, I see the net force being zero;

$$ F_{AB} = -F_{CD}$$

$$ F_{BC} = -F_{DA}$$

But apparently this is not the case. The provided solution is:

asked Mar 9, 2019 in Physics Problems by Jorge Daniel (696 points)
Your assumption that $F_{AB}=-F_{CD}$ is incorrect.

Your mistake is that the magnetic field is proportional to $z$, which is +ve above the xy plane and -ve below it.  So the $B$ field points in the +x direction at AB and in the -x direction at CD. The current is also in the opposite directions in AB and CD. So because both magnetic field and current are in opposite directions the force on AB and CD is in the same direction : (-) x (-) = (+).

If the magnetic field had been in the same direction at AB and CD, as is the case for a uniform magnetic field, or if eg $B(z)=k|z|\hat{x}$, then your assumption would be correct.

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