Two parallel wires carry currents i_1 and i_2 | vedclass

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(C) Let the distance from each wire to the midpoint be $x$. The magnetic field due to a long straight wire at distance $x$ is given by $B = \frac{\mu_

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$2$

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(C) Let the distance from each wire to the midpoint be $x$. The magnetic field due to a long straight wire at distance $x$ is given by $B = \frac{\mu_0 i}{2\pi x}$.Case $(i)$: Currents are in the same direction. The magnetic fields at the midpoint due to the two wires are in opposite directions. Since $i_1 > i_2$,the net magnetic field is:$B_{net} = B_1 - B_2 = \frac{\mu_0}{2\pi x} (i_1 - i_2) = 10 \, \mu T \quad .....(1)$Case $(ii)$: The direction of $i_2$ is reversed. Now,the magnetic fields due to both wires at the midpoint are in the same direction. The net magnetic field is:$B_{net}' = B_1 + B_2 = \frac{\mu_0}{2\pi x} (i_1 + i_2) = 30 \, \mu T \quad .....(2)$Dividing equation $(2)$ by equation $(1)$:$\frac{i_1 + i_2}{i_1 - i_2} = \frac{30}{10} = 3$$i_1 + i_2 = 3(i_1 - i_2)$$i_1 + i_2 = 3i_1 - 3i_2$$4i_2 = 2i_1$$\frac{i_1}{i_2} = \frac{4}{2} = 2$Thus,the ratio $\frac{i_1}{i_2}$ is $2$.

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