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(A) The force per unit length on wire $a$ due to wire $b$ is $f_{ab} = \frac{\mu_0 i_a i_b}{2 \pi d_1}$. Since currents are in the same direction, thi

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$\frac{\mu_0}{6 \pi d_1} i_a^2 l$

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(A) The force per unit length on wire $a$ due to wire $b$ is $f_{ab} = \frac{\mu_0 i_a i_b}{2 \pi d_1}$. Since currents are in the same direction, this force is attractive.The force per unit length on wire $a$ due to wire $c$ is $f_{ac} = \frac{\mu_0 i_a i_c}{2 \pi (d_1 + d_2)}$. Since currents are in opposite directions, this force is repulsive.Given $d_2 = 2 d_1$, $i_b = i_a$, and $i_c = 4 i_a$, the net force per unit length on wire $a$ is:$f_{net} = f_{ab} - f_{ac} = \frac{\mu_0 i_a^2}{2 \pi d_1} - \frac{\mu_0 i_a (4 i_a)}{2 \pi (d_1 + 2 d_1)}$$f_{net} = \frac{\mu_0 i_a^2}{2 \pi d_1} - \frac{4 \mu_0 i_a^2}{2 \pi (3 d_1)} = \frac{\mu_0 i_a^2}{2 \pi d_1} \left( 1 - \frac{4}{3} ight) = \frac{\mu_0 i_a^2}{2 \pi d_1} \left( -\frac{1}{3} ight)$The magnitude of the force on length $l$ is $F = |f_{net}| 	imes l = \frac{\mu_0 i_a^2 l}{6 \pi d_1}$.

Three parallel wires $a$, $b$, and $c$ carrying currents $i_a$, $i_b$, and $i_c$ as shown in the figure are placed next to each other. The magnitude of the force on a length $l$ of the wire $a$, if $d_2 = 2 d_1$, $i_b = i_a$, and $i_c = 4 i_a$ is:

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