Problem 5 - College Physics

Question

A coil of wire with 200 circular turns of radius 3.00 \(\mathrm{cm}\) is in a uniform magnetic field along the axis of the coil. The coil has \(R=40.0 \Omega\) . At what rate, in teslas per second, must the magnetic field be changing to induce a current of 0.150 \(\mathrm{A}\) in the coil?

Step-by-Step Solution

Answer

The magnetic field must be changing at approximately -10.61 T/s.

Step 1: Understand Faraday's Law of Induction

Faraday's Law of Induction states that the electromotive force (emf) induced in a coil is equal to the rate of change of magnetic flux through the coil. Mathematically, it can be expressed as \( \varepsilon = -N \frac{d\Phi}{dt} \), where \( \varepsilon \) is the induced emf, \( N \) is the number of turns, and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux.

Step 2: Relate EMF to Induced Current

The induced current \( I \) in the coil is related to the emf by Ohm's Law: \( I = \frac{\varepsilon}{R} \), where \( R \) is the resistance. Rearranging, we get \( \varepsilon = I \cdot R \). Given that \( I = 0.150 \ \mathrm{A} \) and \( R = 40.0 \, \Omega \), substitute these values to calculate \( \varepsilon \): \( \varepsilon = 0.150 \times 40.0 = 6.0 \, \mathrm{V} \).

Step 3: Calculate Rate of Change of Magnetic Field

Substitute the expression for \( \varepsilon \) from Step 2 into Faraday's Law: \( 6.0 = -200 \cdot \frac{d\Phi}{dt} \). Simplify to find \( \frac{d\Phi}{dt} = -\frac{6.0}{200} = -0.03 \, \mathrm{Wb/s} \).

Step 4: Find Magnetic Flux Formula and Rate

The magnetic flux \( \Phi \) through one turn of the coil is given by \( \Phi = B \cdot A \), where \( A = \pi r^2 \) is the area and \( r = 0.03 \, \mathrm{m} \) (converted from cm). Calculate \( A \): \( A = \pi \times (0.03)^2 = 2.827 \times 10^{-3} \, \mathrm{m^2} \). The rate of change of \( B \) is \( \frac{dB}{dt} = \frac{1}{A} \cdot \frac{d\Phi}{dt} \). Substitute the values to find \( \frac{dB}{dt} = \frac{-0.03}{2.827 \times 10^{-3}} \approx -10.61 \, \mathrm{T/s} \).

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental principle of electromagnetism discovered by Michael Faraday. It refers to the process where an electromotive force (emf) is generated in a coil due to a change in the magnetic field surrounding it. This effect can be summarized by Faraday's Law of Induction, which states that the induced emf is proportional to the rate of change of magnetic flux through the coil.
The equation that represents this concept is \[ \varepsilon = -N \frac{d\Phi}{dt} \] where:

• \( \varepsilon \) is the induced emf.
• \( N \) is the number of turns in the coil.
• \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux.

This negative sign indicates the direction of the induced emf follows Lenz's Law, meaning it opposes the change in magnetic flux. Electromagnetic induction is the basic principle behind many electrical generators and transformers.

Ohm's Law

Ohm's Law is a simple yet powerful principle in electrical circuits that relates voltage, current, and resistance. It states that the current (\( I \)) flowing through a conductor between two points is directly proportional to the voltage (\( V \)) across the two points and inversely proportional to the resistance (\( R \)) of the conductor. It is mathematically expressed as: \[ V = I \cdot R \] In the context of electromagnetic induction, the induced emf \( (\varepsilon )\) acts as the voltage in Ohm's Law. Therefore, the relationship between the induced current and the emf in a circuit is given by: \[ I = \frac{\varepsilon}{R} \] This formula allows us to calculate the required emf if the current and resistance are known. By rearranging, we found that the induced emf for our solution was \( 6.0 \ \mathrm{V} \), given a current of \( 0.150 \ \mathrm{A} \) and resistance of \( 40.0 \ \Omega \).
Understanding Ohm's Law allows us to figure out how changes in voltage, current, and resistance affect each other in electrical circuits.

Magnetic Flux

Magnetic flux (\( \Phi \)) is a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It is expressed as the dot product of the magnetic field \( B \) and the area \( A \) the field penetrates, at a perpendicular angle. Mathematically, magnetic flux is represented by: \[ \Phi = B \cdot A \] where:

• \( B \) is the magnetic field measured in teslas (T).
• \( A \) is the area through which the field lines pass, calculated by \( \pi r^2 \) if circular.

In our exercise, we calculated the area of a single turn of the coil as \( 2.827 \times 10^{-3} \ \mathrm{m^2} \).
Magnetic flux is pivotal to understanding electromagnetic induction, as it is the parameter that changes to create emf. A greater change in magnetic flux over time results in a higher induced emf.

Coil Resistance

Coil resistance refers to the opposition that a coil presents to the flow of electric current. Resistance is influenced by factors such as the coil's material, length, cross-sectional area, and temperature. It is measured in ohms (\( \Omega \)).
The resistance of a coil plays a critical role when calculating the induced current using Ohm's Law, where the current is the ratio of the induced emf to the resistance of the coil. In the given exercise, the coil's resistance was \( 40.0 \ \Omega \).
When dealing with electromagnetic induction, resistance determines how easily current may flow through the coil. Lower resistance will allow for more current to flow for a given emf, while higher resistance will limit the current. Understanding coil resistance is important for designing electrical circuits and ensuring they operate efficiently and safely.