Question

Two identical circular loops of metal wire are lying on a table near to each other without touching. Loop A carries a current which increasing with time. In response the loop B......... (a) Is repelled by loop \(\mathrm{A}\) (b) Is attracted by loop \(\mathrm{A}\) (c) rotates about its centre of mass (d) remains stationary

Short answer

The correct answer is (b) Loop B is attracted by loop A. This is because the induced current in loop B creates a magnetic field that opposes the change in the magnetic field created by loop A's increasing current, resulting in an attractive force between the loops.

Step-by-step solution

Understand the Scenario and Apply Faraday's Law

Since loop A carries a current that increases with time, it will create a magnetic field around it. According to Faraday's law of electromagnetic induction, the change in this magnetic field induces an electromotive force (EMF) in loop B. The EMF is given by the equation: \(EMF = -\frac{d\Phi}{dt}\), where \(\Phi\) is the magnetic flux through loop B and \(t\) is time.

Understand and Apply Lenz's Law

Now, we can apply Lenz's law, which states that the direction of the induced current in loop B will be such that it opposes the change in the magnetic field created by loop A. In other words, loop B will generate a magnetic field to oppose the increase in magnetic field due to loop A's increasing current.

Determine the Force Acting on Loop B

As the magnetic field of loop A increases, loop B will experience a force due to the interaction between the magnetic fields of both loops. To determine the nature of this force (attractive or repulsive), we need to consider the direction of the induced current in loop B and its corresponding magnetic field. Since the current induced in loop B is such that it opposes the change in the magnetic field created by loop A, we can conclude that the force acting on loop B will be attractive, as it will try to minimize the rate of change of the magnetic flux. Hence, the correct answer is (b) Loop B is attracted by loop A. Option (a) is incorrect because the force is attractive, not repulsive. Option (c) is incorrect because there is no mention of any external force causing loop B to rotate about its center of mass. Option (d) is incorrect because loop B does not remain stationary; it experiences an attractive force due to the interaction with loop A's magnetic field.

Key concepts

Lenz's Law

Lenz's Law is an essential part of understanding electromagnetic induction. It tells us how induced currents develop in response to changing magnetic fields.
According to Lenz's Law, when an electromotive force (EMF) is induced by a change in magnetic flux, the direction of the induced current will be such that it opposes the change that caused it.
This is why it's described with a negative sign in Faraday's equation:

• This means if the magnetic field from a loop is increasing, the induced field will try to reduce this increase.
• If the field is decreasing, the induced field will try to increase it.

This opposition ensures that energy conservation is upheld in magnetic systems. So, in the problem scenario, as loop A's current increases, loop B's induced current creates a magnetic field opposing this change.
This results in a particular reaction, known to be attractive according to Lenz's Law.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, taking into account the strength and the extent of the magnetic field.
It is represented by the symbol \( \Phi \) and is calculated as follows: \[ \Phi = B \cdot A \cdot \cos(\theta) \]where \( B \) is the magnetic field, \( A \) is the area the field lines pass through, and \( \theta \) is the angle between the field lines and the perpendicular to the surface.

• When the magnetic field through a loop changes, the magnetic flux changes.
• This change in magnetic flux is key to electromagnetic induction, creating an EMF across the loop.

In our scenario, loop A's increasing current changes the magnetic flux through loop B. This change is what induces a current in loop B, which opposes the change in magnetic flux.

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a conductor induces a current in the conductor.
This phenomenon is elegantly encapsulated by Faraday's Law, which can be expressed as: \[ EMF = -\frac{d\Phi}{dt} \]where \( EMF \) is the induced electromotive force and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux.

• It's crucial in many technologies like electric generators and transformers.
• Faraday's Law is integral to understanding how energy is converted from mechanical to electrical forms.

In the given problem, loop A's changing magnetic field induces an EMF in loop B, showing how interconnected magnetic and electric fields can be.
This induced EMF eventually leads to a current in loop B, again demonstrating electromagnetic induction in action.