Two long parallel wires carry currents I_1 an | vedclass

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Question

(A) Let $d$ be the distance between the two wires. The distance of the midpoint from each wire is $r = d/2$.The magnetic field due to a long wire at d

Vedclass · Accepted Answer

$3 : 2$

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(A) Let $d$ be the distance between the two wires. The distance of the midpoint from each wire is $r = d/2$.The magnetic field due to a long wire at distance $r$ is $B = \frac{\mu_0 I}{2 \pi r}$.For the midpoint,$B = \frac{\mu_0 I}{2 \pi (d/2)} = \frac{\mu_0 I}{\pi d}$.Case $1$: Currents in the same direction. The magnetic fields at the midpoint are in opposite directions. The net field is $B_1 = \frac{\mu_0}{\pi d} (I_1 - I_2) = 6 	imes 10^{-6} \ T$.Case $2$: $I_2$ is reversed. The magnetic fields are now in the same direction. The net field is $B_2 = \frac{\mu_0}{\pi d} (I_1 + I_2) = 3 	imes 10^{-5} \ T$.Dividing the two equations: $\frac{I_1 + I_2}{I_1 - I_2} = \frac{3 	imes 10^{-5}}{6 	imes 10^{-6}} = \frac{30}{6} = 5$.$I_1 + I_2 = 5 I_1 - 5 I_2 \implies 4 I_1 = 6 I_2 \implies \frac{I_1}{I_2} = \frac{6}{4} = \frac{3}{2}$.Thus,the ratio $I_1 : I_2$ is $3 : 2$.

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