Question

A rectangular loop of wire containing a high-resistance voltmeter has corners initially at \((a / 2, b / 2,0),(-a / 2, b / 2,0),(-a / 2,-b / 2,0)\), and \((a / 2,-b / 2,0)\). The loop begins to rotate about the \(x\) axis at constant angular velocity \(\omega\), with the first-named corner moving in the \(\mathbf{a}_{z}\) direction at \(t=0\). Assume a uniform magnetic flux density \(\mathbf{B}=B_{0} \mathbf{a}_{z} .\) Determine the induced emf in the rotating loop and specify the direction of the current.

Short answer

Question: Determine the induced emf in a rectangular loop of wire rotating about the x-axis with a constant angular velocity, given a uniform magnetic flux density. Specify the direction of the induced current. Answer: The induced emf in the rotating loop is given by \(\epsilon(t) = -abB_0\omega\cos(\omega t)\), and the current flows in the clockwise direction when observed from the positive x-axis as the first-named corner moves in the positive z-direction.

Step-by-step solution

Determine the area vector of the loop

We are given the position of each corner of the rectangular loop of wire. We can compute the length and width of the rectangle while considering its orientation. The length of the rectangle is 'a' (parallel to the x-axis), and the width is 'b' (parallel to the y-axis). Hence, the loop's area, A, can be calculated as \(A = a \times b\). The magnitude of the area vector is equal to A, and it lies perpendicular to the loop. Since the loop initially rotates about the x-axis, the area vector lies along the z-axis. So, the area vector \(\textbf{A} = a \times b \; \textbf{a}_z\)

Determine the magnetic field vector

The uniform magnetic flux density \(\textbf{B}=B_{0}\; \textbf{a}_{z}\) is given.

Compute the magnetic flux enclosed by the loop

Now, we will calculate the magnetic flux enclosed by the loop. This is given by the dot product of the area vector of the loop and the magnetic field vector: \(\Phi_{B} = \textbf{A} \cdot \textbf{B} = ab \; \textbf{a}_z \cdot B_0 \; \textbf{a}_z = abB_0\)

Compute the rate of change of the magnetic flux

To compute the induced emf, we need the rate of change of the magnetic flux with respect to time. As the loop rotates about the x-axis, the angle between the z-axis and the area vector changes. Let \(\theta(t)\) be the angle between \(\textbf{a}_z\) and \(\textbf{A}(t)\). When the loop rotates a quarter of a cycle, the area vector will become anti-parallel to the magnetic field vector, making the angle between them \(\pi\) radians. Also, from the problem statement, when \(t=0\), the position of the first-named corner moves in the positive z-direction. Therefore, the function \(\theta(t)\) can be defined as: \(\theta(t) = \omega t\) Now the area vector at time t is given by \(\textbf{A}(t) = ab \sin(\omega t) \;\textbf{a}_z\) and hence, the magnetic flux enclosed by the loop at time t is \(\Phi_{B}(t) = abB_0\sin(\omega t)\) Now, let's calculate the rate of change of magnetic flux: \(\frac{d \Phi_{B}(t)}{dt} = abB_0\omega\cos(\omega t)\) This is the induced emf in the rotating loop: \(\epsilon(t) = -\frac{d \Phi_{B}(t)}{dt} = -abB_0\omega\cos(\omega t)\)

Specify the direction of the induced current

To determine the direction of the induced current, we can use Lenz's law, which states that the induced current will produce a magnetic field that opposes the change in the magnetic flux. When the area vector is parallel to the magnetic field vector (as the loop is initially), the induced current will be in the clockwise direction from the perspective of the positive x-axis. So, when the first-named corner moves in the positive z-direction, the current flows in the clockwise direction. In summary, the induced emf in the rotating loop is given by \(\epsilon(t) = -abB_0\omega\cos(\omega t)\), and the current flows in the clockwise direction when observed from the positive x-axis as the first-named corner moves in the positive z-direction.

Key concepts

Electromagnetic Induction

Electromagnetic induction is the process by which a change in magnetic flux generates an electromotive force (emf) across a conductor. The concept was discovered by Michael Faraday, who noted that a voltage would be induced in a conductor when it is exposed to a changing magnetic field. This principle is the working foundation of many electrical generators and transformers.

Essentially, in the case of our rotating loop problem, as the loop spins in a magnetic field, the magnetic flux through the loop changes over time, inducing a voltage (or emf) across the wire. The induced emf can be used to drive current through a circuit, which is the basic principle behind the generation of electricity in hydroelectric dams, wind turbines, and other energy conversion devices.

Magnetic Flux

Magnetic flux, often denoted as \( \Phi_B \), refers to the measure of the quantity of magnetism, considering the force and the extent to which a magnetic field influences the motion of charged particles around it. It is calculated as the product of the magnetic field strength and the area perpendicular to the field through which it passes. Mathematically, it's defined as \( \Phi_B = \textbf{B} \cdot \textbf{A} \), where \( \textbf{B} \) is the magnetic field vector, and \( \textbf{A} \) is the area vector.

In our rectangular loop example, as the loop rotates, the effective area through which the magnetic field lines pass changes over time, contributing to a changing magnetic flux. Computing the flux is critical for predicting the induced emf in the wire loop, hence a fundamental concept in analyzing electromagnetics.

Lenz's Law

Lenz's law is paramount when determining the direction of induced current resulting from electromagnetic induction. It states that the direction of the current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial change in magnetic flux through the loop.

In practical terms, Lenz’s law is like a physical manifestation of the adage 'every action has an equal and opposite reaction.' This means that when the loop in our problem starts rotating and inducing an emf, the direction of the resulting current will be such that it tries to maintain the original magnetic state of the system. This law is integral to the design of electric motors, generators, and transformers to ensure that they operate correctly in their intended applications.

Angular Velocity

Angular velocity, symbolized as \( \omega \), represents how fast an object rotates or revolves relative to another point, which, in our case, is the x-axis. It tells us not only the speed but also the direction of rotation. Angular velocity is a vector quantity, possessing both magnitude and direction. For a body rotating about an axis, like our rotating rectangular loop, the angular velocity defines how quickly it rotates expressed in radians per second.

Understanding angular velocity is crucial in problems involving rotational motion, such as our rotating loop. It dictates how quickly the magnetic flux through the loop changes, which directly affects the magnitude of the induced emf. By adopting a constant angular velocity, we can create a predictable and calculable scenario for the induced emf in such a system.