Question

Two different loops are concentric $\mathrm{\&}$ lie in the same plane. The current in outer loop is clockwise \& increasing with time. The induced current in the inner loop then, is ....... (a) clockwise (b) zero (c) counter clockwise (d) direction depends on the ratio of loop radii

Short answer

The direction of the induced current in the inner loop is counter-clockwise (c).

Step-by-step solution

Recognize the concepts involved in the problem

To solve this problem, we will use Faraday's Law of Electromagnetic Induction and Lenz's Law. Faraday's Law states that a change in the magnetic field across a coil will induce an electromotive force (induced voltage) in the coil, which in turn causes current to flow if there is a closed path. Lenz's Law tells us the direction of the induced current, which is such that it opposes the change in the magnetic field that caused it.

Analyze the behavior of the magnetic field

Since the current in the outer loop is increasing and clockwise, the magnetic field generated by the outer loop at the position of the inner loop is also increasing. The magnetic field lines created by a circular loop will form concentric circles around it. The direction of the magnetic field can be determined by the right-hand rule: for a clockwise current, the field lines will point towards the center of the loop.

Apply Lenz's Law to the problem

Now we need to find the direction of the induced current in the inner loop. Lenz's Law states that the induced current will flow in such a way that its magnetic field opposes the change in the magnetic field that caused it. In our case, the magnetic field from the outer loop is increasing and pointing towards the center of the loops. So, the induced current in the inner loop will create a magnetic field that opposes the increase, i.e., it will have a direction such that its magnetic field points outward, away from the center of the loops.

Determine the direction of the induced current in the inner loop using the right-hand rule

We use the right-hand rule again, but this time to determine the direction of the induced current in the inner loop that generates a magnetic field pointing outward. To do this, curl your right-hand fingers in the direction of the desired magnetic field (outward), and your extended thumb will indicate the direction of the induced current. In this case, the induced current in the inner loop will be counter-clockwise. Solution: The direction of the induced current in the inner loop is counter-clockwise (c).

Key concepts

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental principle in electromagnetism. It explains how a changing magnetic field can induce an electric voltage in a closed circuit. This induced voltage is what we call the electromotive force (EMF). Faraday's Law is expressed mathematically as $\mathcal{E}=-\frac{d\mathrm{\Phi }}{dt}$, where $\mathcal{E}$ is the EMF and $\mathrm{\Phi }$ is the magnetic flux. The negative sign indicates the direction of the induced EMF, following Lenz's Law.

The concept of magnetic flux ($\mathrm{\Phi }$) is significant here. It represents the total magnetic field passing through an area, given by $\mathrm{\Phi }=B\cdot A\cdot \cos \theta$, where $B$ is the magnetic field strength, $A$ is the area, and $\theta$ is the angle between the magnetic field and the normal to the surface. When any of these variables change over time, an EMF is induced.

This law has many practical applications in everyday life. For example, it is the basis for the operation of transformers, electric generators, and induction stovetops. Whenever a magnetic field changes around a loop of wire, as described in the exercise, Faraday’s Law comes into play, creating an induced current in the loop.

Lenz's Law

Lenz's Law is closely tied to Faraday's Law, as it dictates the direction of the induced current and EMF. Simply put, Lenz's Law states that the induced current will flow in such a direction that its magnetic field opposes the change causing it. This is why the negative sign appears in the expression for Faraday's Law: $\mathcal{E}=-\frac{d\mathrm{\Phi }}{dt}$.

This principle is grounded in the conservation of energy. If the induced EMF were to enhance the change in magnetic flux, it would create energy from nothing, violating the law of conservation. Instead, the induced current works against the change, ensuring that energy is conserved.

In the context of the problem, when the current in the outer loop increases, it strengthens the magnetic field pointing towards the center. Lenz's Law tells us that the inner loop's induced current will flow in a direction that produces a magnetic field opposing this increase. This results in the induced current in the inner loop flowing counter-clockwise, as it attempts to generate a magnetic field in the outward direction.

Right-Hand Rule

The Right-Hand Rule is a mnemonic tool used in physics to determine the direction of the magnetic field relative to the flow of electric current. When a current flows through a wire, it creates a surrounding magnetic field. To predict the orientation of the field lines around a circular current loop, we use the Right-Hand Rule.

Here's how it works:

• Imagine holding the loop with your right hand.
• Curl your fingers in the direction of the current flow.
• Your thumb will point towards the direction of the magnetic field lines produced by the current.

In the original exercise, we apply the Right-Hand Rule twice. First, to determine the direction of the magnetic field due to the clockwise current in the outer loop—pointing inward toward the center.

Then, again, to establish the direction of the induced current in the inner loop required for the magnetic field opposing the increase. By curling your fingers in the direction of the desired magnetic field (outward), your thumb then points in the direction of the counter-clockwise induced current in the inner loop. This consistent application of the Right-Hand Rule provides clarity and predictability in solving electromagnetic problems.