Animated Solution for Physics - Electromagnetic Induction: A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it. The correct statement(s) is(are)

When a Changing Current Does Nothing

Electromagnetic induction is one of the most fascinating phenomena in physics. Usually, if you have a changing current near a wire loop, you can bet your bottom dollar that an electromotive force (emf) will be induced. But physics loves to throw a curveball, and this problem is a perfect example of a situation where our intuition might lead us astray.

Let's dive into the geometry of the setup to understand why a changing current in this specific scenario does absolutely nothing to the loop.

Imagine a circular wire loop resting on a table. Now, place an infinitely long straight wire exactly along its diameter. The wire carries a current $i$. Crucially, the wire and the loop are electrically insulated from each other—they don't touch.

To understand what happens, we first need to visualize the magnetic field produced by the straight wire. We use the Right-Hand Thumb Rule: if you point your right thumb in the direction of the current, your fingers curl in the direction of the magnetic field lines.

Applying this rule, we see that on one side of the wire (let's say the right half of the loop), the magnetic field lines plunge into the plane of the loop. On the other side (the left half), the magnetic field lines emerge out of the plane.

Here is where the magic happens. Because the straight wire lies exactly along the diameter, it cuts the circular loop into two perfectly identical semicircles.

The magnetic field strength at any distance $r$ from the wire is given by $B=\frac{{\mu }_{0}i}{2\pi r}$. This means the field strength is symmetric on both sides of the wire.

Now, let's calculate the magnetic flux, $\varphi =\int \vec{B}\cdot d\vec{A}$.

For the right semicircle, the flux is directed inwards. Let's call this $-{\varphi }_0$. For the left semicircle, the flux is directed outwards. Because of the perfect symmetry in both area and magnetic field strength, this flux is exactly $+{\varphi }_0$.

When we add them up to find the net magnetic flux through the entire circular loop, we get:

The net magnetic flux through the loop is zero. But does it stay zero? Yes! Because the geometry doesn't change, the inward flux will always perfectly cancel the outward flux, regardless of the magnitude of the current $i$.

According to Faraday's Law of Electromagnetic Induction, the induced emf $e$ is the negative rate of change of magnetic flux:

Since ${\varphi }_{\text{net}}$ is permanently $0$, its derivative with respect to time is also $0$.

This mathematical absolute tells us a profound physical truth: No matter what the current does, no emf will be induced.

If the current is constant, the emf is zero. If the current decreases at a steady rate, the emf is still zero. The symmetry of the system completely kills any chance of electromagnetic induction. Therefore, the correct statements are that the emf induced in the loop is zero if the current is constant, and it remains zero even if the current decreases at a steady rate.