Problem 4 - College Physics

Question

A single loop of wire with an area of 0.0900 \(\mathrm{m}^{2}\) is in a uniform magnetic field that has an initial value of 3.80 \(\mathrm{T}\) , is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 \(\mathrm{T} / \mathrm{s}\) . (a) What emf is induced in this loop? (b) If the loop has a resistance of \(0.600 \Omega,\) find the current induced in the loop.

Step-by-Step Solution

Answer

(a) Emf induced is 0.0171 V. (b) Induced current is 0.0285 A.

Step 1: Understanding the Problem

We need to calculate the induced electromotive force (emf) in a circular loop when the magnetic field through it is changing. According to Faraday's Law of Electromagnetic Induction, the induced emf is equal to the negative change in magnetic flux divided by the change in time.

Step 2: Faraday's Law Equation

The induced emf can be calculated using Faraday's Law: \( \text{emf} = - \frac{d\Phi_B}{dt} \), where \( \Phi_B = B \times A \) is the magnetic flux, \( B \) is the magnetic field, and \( A \) is the area of the loop. Since \( B \) is decreasing at 0.190 T/s, we use \( \frac{dB}{dt} = -0.190 \ \mathrm{T/s} \).

Step 3: Calculate Change in Magnetic Flux

The change in magnetic flux \( \frac{d\Phi_B}{dt} \) is the product of the area \( A = 0.0900 \ \mathrm{m}^2 \) and the rate of change of the magnetic field \( \frac{dB}{dt} \). Thus, \( \frac{d\Phi_B}{dt} = A \times \frac{dB}{dt} = 0.0900 \ \mathrm{m}^2 \times -0.190 \ \mathrm{T/s} = -0.0171 \ \mathrm{T \cdot m^2/s} \).

Step 4: Calculate Induced emf

Using Faraday's Law, \( \text{emf} = - \frac{d\Phi_B}{dt} = - (-0.0171) = 0.0171 \ \mathrm{V} \). The negative sign indicates direction according to Lenz's Law, but here we are interested in magnitude.

Step 5: Calculate Induced Current Using Ohm's Law

Ohm's Law (\( V = I \cdot R \)) relates voltage, current, and resistance. Here, \( V \) is the induced emf (0.0171 V), \( R = 0.600 \ \Omega \). Solving for current \( I \), \( I = \frac{V}{R} = \frac{0.0171 \ \mathrm{V}}{0.600 \ \Omega} = 0.0285 \ \mathrm{A} \).

Key Concepts

Faraday's Law

The foundation of electromagnetic induction is explained by Faraday's Law. It states that the electromotive force (emf) induced in a circuit is directly proportional to the rate of change of magnetic flux through the circuit. An emf can be thought of as the voltage generated to oppose the change in the magnetic environment of a loop.

Faraday's Law is mathematically expressed as: \[ \text{emf} = - \frac{d\Phi_B}{dt} \], where \( \Phi_B \) is the magnetic flux through a loop and \( t \) represents time. The negative sign in the equation is a nod to Lenz's Law, indicating the direction of the induced emf opposes the change in flux.

To simplify, if the magnetic field through a loop changes consistent with time, this will induce an emf according to the above relationship. This principle is widely applied in generators and transformers, playing a critical role in electrical engineering.

magnetic flux

Magnetic flux helps in understanding how much magnetic field passes through a certain area. It's like counting how many magnetic lines penetrate a given surface. The magnetic flux \( \Phi_B \) through a loop of area \( A \) located in a magnetic field \( B \) is calculated as:
\[\Phi_B = B \times A \times \cos(\theta)\]
where \( \theta \) is the angle between the magnetic field and the normal to the surface.

For simplicity, when the magnetic field is perpendicular to the plane of the loop, \( \theta = 0 \) degrees and \( \cos(\theta) = 1 \), making our equation \( \Phi_B = B \times A \). Changing magnetic flux through a loop is the essence of electromagnetic induction, leading to the creation of an emf.

In the context of a changing magnetic environment, understanding how flux varies helps predict the behavior of induced emfs.

Ohm's Law

Once the voltage or emf across a circuit element is known, Ohm's Law comes into play to find the induced current. This basic law relates voltage (V), current (I), and resistance (R) in a simple linear relationship: \[V = I \times R\]
In our case, the voltage \( V \) is the induced emf we calculated using Faraday's Law. With the loop's resistance \( R \), Ohm's Law can be rearranged to find the current: \[I = \frac{V}{R}\]

Ohm's Law is an essential building block in understanding how electrical circuits respond to various forms of voltage inputs, whether from a battery or an induced emf.