Question

Two long, parallel wires separated by a distance \(d\) carry currents in opposite directions. If the left-hand wire carries a current \(i / 2\) and the righthand wire carries a current \(i,\) determine where the magnetic field is zero.

Short answer

Answer: The position where the magnetic field is zero will be at a distance of r = (d/3) away from the left-hand wire, towards the right-hand wire.

Step-by-step solution

Understand the magnetic field created by the wires

First, we need to understand the principle of the magnetic field created by the current-carrying wires. Each wire will create a magnetic field around itself, and since they carry the current in opposite directions, their respective magnetic fields will have opposite directions as well. According to Biot-Savart's Law, the formula for the magnetic field (\(B\)) created by a current-carrying wire is: \(B = \dfrac{\mu_0 i}{2\pi r}\) where \(\mu_0\) is the permeability of vacuum (around \(4\pi \times 10^{-7} Tm/A\)), \(i\) is the current in the wire, and \(r\) is the perpendicular distance between the wire and the point where the magnetic field is measured. In this exercise, we have two wires with currents, \(i/2\) and \(i\).

Set up the equation to find the position where the magnetic field is zero

Next, we want to find the position, distance \(r\) from the left-hand wire, where the magnetic fields created by the two wires will cancel each other, i.e., their magnitudes will be equal but their directions will be opposite. We can set up the equation for the magnetic fields created by both wires: \(\dfrac{\mu_0 i}{4\pi r} = \dfrac{\mu_0 i}{2\pi (d - r)}\) Notice that \(\mu_0 i\) can be cancelled from both sides.

Solve for \(r\)

Now, we can solve the equation for the position \(r\): \(2r = d - r\) Adding \(r\) to both sides, we get: \(3r = d\) Now, divide by 3: \(r = \dfrac{d}{3}\)

Find the position where the magnetic field is zero

The distance \(r\) from the left-hand wire where the magnetic field is zero is: \(r = \dfrac{d}{3}\) So, to find the position where the magnetic field is zero, we need to move \(\dfrac{d}{3}\) away from the left-hand wire, towards the right-hand wire.