Two long, parallel wires are separated by a distance of 2.50 cm. The force per unit length that each wire exerts on the other is $4.00 \times 10^{-5} N/m$, and the wires repel each other. The current in one wire is 0.600 A. (a) What is the current in the second wire? (b) Are the two currents in the same direction or in opposite directions?

Solution

VerifiedConsider two long, parallel wires which are separated by a distance of 2.50 cm. It is given that the force per unit length that each wire exerts on the other is $F/L=4.00 \times 10^{-5}$ N/m. The two wires repel each others, and the current in one wire is $I=0.600$ A.

a.

We need to find the current in the other wire. The magnitude of the force per unit length between two wires carrying currents of $I$ and $I^{\prime}$ is given by,

$\begin{align}\frac{F}{L}=\frac{\mu_{0} I^{\prime} I}{2 \pi r}\end{align}$

solve for $I^{\prime}$ and substitute to get,

$\begin{align*}I^{\prime}&=\frac{F}{L} \frac{2 \pi r}{\mu_{0} I}\\ &=\left(4.0 \times 10^{-5} \mathrm{~N/m}\right) \frac{2 \pi(0.0250 \mathrm{~m})}{(4 \pi \times 10^{-7} \mathrm{~T \cdot m/A})(0.60 \mathrm{~A})}\\ &=8.33 \mathrm{~A}\end{align*}$

$\boxed{I^{\prime}=8.33 \mathrm{~A}}$