Animated Solution for Physics - Magnetic Effects of Current: Two thin long parallel wires separated by a distance $b$ are carrying a current $i$ ampere each. The magnitude of the force per unit length exerted by one wire on the other is

The interaction between two parallel current-carrying wires is a classic and fundamental concept in electromagnetism. It beautifully bridges two core ideas: how a current creates a magnetic field, and how a magnetic field exerts a force on a moving charge (or current).

Imagine two infinitely long, thin, parallel wires separated by a distance $b$. Both wires are carrying an identical current $i$. We want to find the force per unit length that one wire exerts on the other.

To solve this, we break the problem into two logical steps. First, we calculate the magnetic field produced by one wire at the location of the second wire. Then, we calculate the force that this magnetic field exerts on the second wire.

Let's focus on the first wire. According to Ampere's Law, a long straight wire carrying a current $i$ generates a magnetic field around it. At a perpendicular distance $b$ from the wire, the magnitude of this magnetic field $B_1$ is given by:

Using the right-hand grip rule, if we point our thumb in the direction of the current in the first wire, our fingers curl in the direction of the magnetic field. At the position of the second wire, this magnetic field points perpendicularly into (or out of, depending on the relative orientation) the plane containing the two wires.

Now, consider the second wire. It is carrying a current $i$ and is immersed in the external magnetic field $B_1$ created by the first wire. The magnetic force $\vec{F}$ on a straight segment of wire of length $L$ carrying current $i$ in a uniform magnetic field $\vec{B}$ is given by the Lorentz force equation for currents:

Since the second wire is parallel to the first wire, it is perpendicular to the magnetic field $B_1$ (which forms concentric circles around the first wire). Therefore, the angle between $\vec{L}$ and ${\vec{B}}_1$ is $90^{\circ }$, and $\sin (90^{\circ })=1$. The magnitude of the force is simply:

The question asks for the force per unit length, which is $\frac{F}{L}$. Rearranging our force equation gives:

Now, we substitute the expression for $B_1$ that we derived earlier:

Multiplying the terms together, we arrive at our final, elegant result:

This tells us that the force per unit length is directly proportional to the square of the current and inversely proportional to the distance between the wires. If the currents are in the same direction, the force is attractive; if they are in opposite directions, the force is repulsive. In this problem, we only needed the magnitude, which perfectly matches option (b).