Question

A circular loop of wire moving in the \(x y\) -plane with a constant velocity in the negative \(x\) -direction enters a uniform magnetic field, which covers the region in which \(x<0,\) as shown in the figure. The surface normal vector of the loop points in the direction of the magnetic field. Which of the following statements is correct? a) The induced potential difference in the loop is at a maximum as the edge of the loop just enters the region with the magnetic field. b) The induced potential difference in the loop is at a maximum when one fourth of the loop is in the region with the magnetic field. c) The induced potential difference in the loop is at a maximum when the loop is halfway into the region with the magnetic field. d) The induced potential difference in the loop is constant from the instant the loop starts to enter the region with the magnetic field.

Short answer

Answer: The induced potential difference in the loop is at a maximum as the edge of the loop just enters the region with the magnetic field.

Step-by-step solution

1. Calculate the magnetic flux through the loop

To calculate the magnetic flux through the loop, we can use the formula: $$ \Phi_B = \int \vec{B} \cdot d\vec{A} $$ where \(\vec{B}\) is the magnetic field, and \(d\vec{A}\) is the infinitesimal vector area. As given in the problem, the magnetic field is uniform, and the surface normal vector of the loop points in the direction of the magnetic field. Therefore, the magnetic flux through the loop can be represented as: $$ \Phi_B = B \cdot A $$ where \(B\) is the magnitude of the magnetic field, and \(A\) is the area of the loop.

2. Calculate the area of the loop as it enters the magnetic field

As the loop enters the magnetic field, the area inside the magnetic field changes. Let's denote the ratio of the part of the loop that is inside the magnetic field as \(x\) (ranging from 0 to 1). Then, the area of the loop inside the magnetic field can be represented as: $$ A_x = x \cdot A $$

3. Calculate the rate of change of magnetic flux

Now that we have the area of the loop inside the magnetic field, we can find the rate of change of magnetic flux with respect to time. We first differentiate magnetic flux with respect to \(x\), considering that \(B\) is constant: $$ \frac{d\Phi_B}{dx} = B \cdot \frac{dA_x}{dx} = B \cdot A $$ Now we can find the rate of change of magnetic flux with respect to time using the chain rule: $$ \frac{d\Phi_B}{dt} = \frac{d\Phi_B}{dx} \cdot \frac{dx}{dt} = B \cdot A \cdot \frac{dx}{dt} $$

4. Calculate EMF and find the condition for maximum induced potential difference

Now, we can use Faraday's law to find the induced electromotive force (EMF), which is equal to the induced potential difference in the loop: $$ \varepsilon = -\frac{d \Phi_B}{dt} = -B \cdot A \cdot \frac{dx}{dt} $$ For the EMF to be maximum (induced potential difference to be maximum), the rate of change of the ratio of the part of the loop inside the magnetic field (\(\frac{dx}{dt}\)) should be maximum. This condition occurs when the edge of the loop just enters the region with the magnetic field, as the rate of change of the area inside the magnetic field is maximum at that point. As the loop further moves into the magnetic field, the rate of change of the area decreases. This means that the correct answer is option a) The induced potential difference in the loop is at a maximum as the edge of the loop just enters the region with the magnetic field.

Key concepts

Magnetic Flux

Magnetic flux, symbolized as \( \Phi_B \), is a measure of the amount of magnetic field passing through a given surface area. It's crucial in understanding how electromagnetic fields interact with physical objects.

The formula for calculating magnetic flux is \( \Phi_B = \vec{B} \cdot \vec{A} \), where \( \vec{B} \) is the magnetic field strength and \( \vec{A} \) is the vector area through which the field lines pass. If the field is uniform and the surface is perpendicular to the field lines, as in the original exercise, this simplifies to \( \Phi_B = B \cdot A \).

Understanding the concept of magnetic flux is essential for grasping other electromagnetic phenomena, such as the induction of electromotive force (EMF) in a conducting loop.

Faraday's Law

Faraday's Law is a fundamental principle in electromagnetism that explains how a change in magnetic flux induces an electromotive force (EMF) within a closed circuit.

Mathematically, it's expressed as \( \varepsilon = -\frac{d \Phi_B}{dt} \), where \( \varepsilon \) is the induced EMF, and the negative sign indicates the direction of the induced EMF opposes the change in flux, in accordance with Lenz's Law.

In practical terms, this law infers that an electric potential is generated in any circuit whenever there is a change in the magnetic environment of the circuit. For instance, as the loop from the exercise question moves into the uniform magnetic field, an EMF is induced due to the change in flux through the loop—a direct application of Faraday's Law.

Electromotive Force (EMF)

Electromotive force (EMF) is not an actual force but a potential difference that can cause a current to flow in a circuit. It's what 'pushes' electrons to move through a conductor.

EMF can be induced by a change in magnetic flux, as described by Faraday's Law. When we talk about an induced EMF, like in our exercise, it refers to the voltage generated across a loop when there is a change in magnetic flux. The induced EMF can be calculated using \( \varepsilon = -\frac{d \Phi_B}{dt} \) as derived in the steps of the solution.

A maximum induced EMF occurs when the rate of change of magnetic flux through the loop is greatest. This ties back to why, when the loop first enters the magnetic field, the induced potential difference is at its maximum—because the change in flux is initially highest.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic field strength (\(B\)) is constant at every point in space. It creates parallel, equally-spaced field lines, indicating the direction of the force that would be exerted on a north magnetic pole placed within the field.

In our exercise scenario, the loop enters a region with a uniform magnetic field. This simplifies calculations as we can predict the behavior of magnetic phenomena, such as magnetic flux, without having to account for variations in the strength or direction of the field. When a conductor, like the circular loop, moves through a uniform magnetic field, the induced EMF is directly proportional to the velocity at which the loop enters the field.