Problem 20 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The area of an elastic circular loop decreases at a constant rate, \(d A / d t=-3.50 \times 10^{-2} \mathrm{~m}^{2} / \mathrm{s}\). The loop is in a magnetic field \(B=0.28 \mathrm{~T}\) whose direction is perpendicular to the plane of the loop. At \(t=0,\) the loop has area \(A=0.285 \mathrm{~m}^{2} .\) Determine the induced \(\mathrm{emf}\) at \(t=0,\) and at \(t=2.00 \mathrm{~s}\).

Step-by-Step Solution

Answer

The induced emf is \(9.8 \times 10^{-3}\, \text{V} \) at both \(t=0\) and \(t=2.00\, \text{s}.\)

Step 1: Understand the concept

We need to calculate the induced electromotive force (emf) in the loop due to a changing magnetic flux. According to Faraday's Law of Electromagnetic Induction, the induced emf in the loop is equal to the negative rate of change of magnetic flux through the loop.

Step 2: Express magnetic flux

The magnetic flux \( \Phi \) through the loop is given by \( \Phi = B \cdot A \), where \( B \) is the magnetic field and \( A \) is the area of the loop. For this problem, \( B = 0.28 \, \text{T} \).

Step 3: Calculate rate of change of flux

The rate of change of magnetic flux \( \frac{d\Phi}{dt} \) is \( B \frac{dA}{dt} \), where \( \frac{dA}{dt} = -3.50 \times 10^{-2} \, \text{m}^2/\text{s} \). Plug the values into \( \frac{d\Phi}{dt} = 0.28 \times (-3.50 \times 10^{-2}) \).

Step 4: Apply Faraday's Law

According to Faraday's Law, the induced emf \( \mathcal{E} \) is \( \mathcal{E} = - \frac{d\Phi}{dt} \). So, \[ \mathcal{E} = - (0.28 \times (-3.50 \times 10^{-2})) = 9.8 \times 10^{-3} \text{ V} \].

Step 5: Confirm consistency over time

The rate of change of area is constant (given by \( \frac{dA}{dt} \)). Since \( B \) is constant, the induced emf stays constant regardless of time. Thus, \( \mathcal{E} = 9.8 \times 10^{-3} \text{ V} \) at both \( t=0 \) and \( t=2.00 \text{ s} \).

Key Concepts

Magnetic Flux

Magnetic flux is a foundational concept in electromagnetism that essentially measures the quantity of magnetic field lines passing through a given surface. When we talk about magnetic flux in a region, we are referring to the number of magnetic field lines intersecting a closed surface, like a loop. Mathematically, it's defined as the product of the magnetic field strength, denoted as \( B \), and the area \( A \) the field is passing through, i.e. \( \Phi = B \, \cdot \, A \).
This equation shows that the magnetic flux is directly proportional to both the magnetic field strength and the surface area it penetrates. If either of these values changes, the flux through the loop changes, which becomes important in understanding electromagnetic induction processes. For instance, if the area decreases, even with a constant magnetic field strength, the flux decreases, which in turn can induce an electromotive force (emf) as predicted by Faraday's Law.

Induced Electromotive Force (emf)

The induced electromotive force (emf) is the voltage generated around a closed loop when there is a change in magnetic flux through the loop. According to Faraday's Law of Electromagnetic Induction, the induced emf \( \mathcal{E} \) is equal to the negative of the rate of change of magnetic flux over time, expressed as \[ \mathcal{E} = - \frac{d\Phi}{dt} \].
The negative sign in this equation, known as Lenz's Law, indicates that the induced emf will generate a current with a magnetic field opposing the change in the original magnetic flux. This is a fundamental principle for devices like transformers and electric generators.

• Faraday's Law: Describes how a magnetic field interacts with an electric circuit to produce an emf.
• Lenz's Law: Dictates the direction of the induced emf and current.

Understanding how induced emf works is crucial in fields that rely on electric power and electromagnetism, emphasizing its role in generating electricity.

Rate of Change of Area

The rate of change of area is an important factor in calculating the variation of magnetic flux, particularly in scenarios where the geometry of the loop is changing over time. For a loop in a magnetic field, if its area is shrinking or expanding, the magnetic flux through it will change, which then can induce an emf.
In the context of Faraday's Law, if the area \( A \) changes at a constant rate, say \( \frac{dA}{dt} \), it simplifies the computation of the rate of change of magnetic flux:\[ \frac{d\Phi}{dt} = B \cdot \frac{dA}{dt} \]where \( B \) is a constant magnetic field.

• Constant Rate: In this exercise, the area changes at \( -3.50 \times 10^{-2} \text{ m}^2/\text{s} \), leading to a stable induced emf.
• Implication in Time: With constant magnetic field and rate of area change, the emf is steady over time.

This understanding highlights the direct correlation between physical changes in a loop and electromagnetic phenomena, showcasing Faraday's Law in a practical, observable way.