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Chapter 13: Problem 40

A coil of 1000 tums encloses an area of \(25 \mathrm{cm}^{2}\) It is rotated in 0.010 s from a position where its plane is perpendicular to Earth's magnetic field to one where its plane is parallel to the field. If the strength of the field is \(6.0 \times 10^{-5} \mathrm{T}, \quad\) what is the average emf induced in the coil?

Short Answer

Expert verified
The average induced emf in the coil is 15 mV.

Step by step solution

01

Given the area of the coil A = 25 cm虏, we need to convert it to square meters: \[ A = 25 \,\text{cm}^2 \cdot \left(\frac{1 \,\text{m}}{100 \,\text{cm}}\right)^2 = 0.0025 \,\text{m}^2 \] #step2# Step 2: Calculate the initial and final magnetic flux

Initially, when the plane of the coil is perpendicular to the Earth's magnetic field, the angle \(\theta_{1}\) between the normal to the plane of the coil and the magnetic field is 0掳, and the magnetic flux is: \[ \Phi_{1} = B \cdot A \cdot \cos{0^\circ} = B \cdot A \cdot 1 \] Finally, when the plane of the coil is parallel to the Earth's magnetic field, the angle \(\theta_{2}\) between the normal to the plane of the coil and the magnetic field is 90掳, and the magnetic flux is: \[ \Phi_{2} = B \cdot A \cdot \cos{90^\circ} = B \cdot A \cdot 0 = 0 \] Since the magnetic field strength B is constant, we can express the change in magnetic flux as: \[ \Delta \Phi = \Phi_{2} - \Phi_{1} = -B \cdot A \] #step3# Step 3: Calculate the average induced emf using Faraday's law
02

Now, we can use Faraday's law to find the average induced emf: \[ \text{average emf} = -N \frac{\Delta \Phi}{\Delta t} = -N \frac{-B \cdot A}{\Delta t} = N \frac{B \cdot A}{\Delta t} \] Substitute the given values: \[ \text{average emf} = 1000 \frac{6.0 \times 10^{-5} \,\text{T} \cdot 0.0025 \,\text{m}^2}{0.010 \,\text{s}} \] #step4# Step 4: Calculate the average induced emf

Calculate the average induced emf: \[ \text{average emf} = 1000 \frac{6.0 \times 10^{-5} \,\text{T} \cdot 0.0025 \,\text{m}^2}{0.010 \,\text{s}} = 15 \,\text{mV} \] The average induced emf in the coil is 15 mV.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Faraday's law of induction
Understanding how electrical currents can be generated in a coil of wire is fundamental in electromagnetism. This is where Faraday's law of induction comes into play. This law states that the induced electromotive force (emf) in any closed circuit is equal to the negative of the rate of change of the magnetic flux through the circuit. The formula representing this law is given as:
e = -N (螖桅/螖t)
where 'e' is the induced emf, 'N' is the number of turns in the coil, '螖桅' is the change in magnetic flux, and '螖t' is the change in time over which this change occurs.

Application in Exercises

In exercises like the one given, Faraday's law allows us to calculate the average induced emf when given the change in magnetic flux and the time interval over which the coil is rotated. By providing all the necessary changes in the coil's characteristics, such as the angle and time, Faraday's law helps students find the average emf induced, which is critical for understanding how electric generators operate.
Magnetic flux
A related concept that is vital to grasp is magnetic flux, denoted usually by the symbol '桅'. Think of magnetic flux as the 'amount' of the magnetic field passing through a given area. Mathematically, it is the product of the magnetic field strength 'B', the area 'A' the magnetic field is passing through, and the cosine of the angle '胃' between the magnetic field lines and the normal (perpendicular) to the area:
桅 = B * A * cos(胃)
This equation highlights that the maximum flux passes through an area when the field is perpendicular to it (cos(0) = 1) and no flux passes through when it is parallel (cos(90掳) = 0), as demonstrated in the step-by-step solution of our exercise. Remember: The unit of magnetic flux is the weber (Wb), and it helps quantify the effect of the magnetic field on the coil.
Electromagnetic induction
Electromagnetic induction is the process by which a change in magnetic flux induces an emf and consequently an electrical current, if a path is provided. This is at the heart of how electric generators and transformers work.
The phenomenon involves the interplay between electric and magnetic fields and is fundamental to much of our modern technology, including charging devices without direct electrical connections (inductive charging).
  • Understanding how the movement or rotation of a coil relative to a magnetic field induces an emf requires the grasp of this principle.
  • In the given exercise, electromagnetic induction occurs when the coil is rotated within the magnetic field of the Earth.
The change in flux that happens because of this rotation is what induces the emf in the wire.
Magnetic field strength
The magnetic field strength, often denoted as 'B', is a measure of the magnitude of the magnetic field at a point in space. It tells us how strong the magnetic field is, which directly impacts the magnetic flux and, subsequently, the amount of emf that can be induced.

Significance in Practice

In practice, strong magnetic fields can induce a more significant emf and are essential in applications ranging from electric motors to MRI machines.

Units and Measurements

The strength of the magnetic field is measured in teslas (T) in the International System of Units (SI). Earth's magnetic field is relatively weak, often in the range of microteslas (渭T), but even this small field strength can induce a measurable emf as seen in our exercise.

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Most popular questions from this chapter

The accompanying figure shows a single-turn rectangular coil that has a resistance of \(2.0 \Omega .\) The magnetic field at all points inside the coil varies according to \(B=B_{0} e^{-\alpha t}, \) where \(B_{0}=0.25 \mathrm{T}\) and \(\alpha=200 \mathrm{Hz}\) What is the current induced in the coil at (a) \(t=0.001 \mathrm{s}\) (b) 0.002 s, (c) 2.0 s?

A flip coil is a relatively simple device used to measure a magnetic field. It consists of a circular coil of \(N\) turns wound with fine conducting wire. The coil is attached to a ballistic galvanometer, a device that measures the total charge that passes through it. The coil is placed in a magnetic field \(\overrightarrow{\mathbf{B}}\) such that its face is perpendicular to the field. It is then flipped through \(180^{\circ},\) and the total charge \(Q\) that flows through the galvanometer is measured. (a) If the total resistance of the coil and galvanometer is \(R\) what is the relationship between \(B\) and \(Q\) ? Because the coil is very small, you can assume that \(\overrightarrow{\mathbf{B}}\) is uniform over it. (b) How can you determine whether or not the magnetic field is perpendicular to the face of the coil?

A flat, square coil of 20 turns that has sides of length \(15.0 \mathrm{cm}\) is rotating in a magnetic field of strength \(0.050 \mathrm{T}\) If the maximum emf produced in the coil is \(30.0 \mathrm{mV}\), what is the angular velocity of the coil?

A wire loop moves translationally (no rotation) in a uniform magnetic field. Is there an emf induced in the loop?

A metal bar of length 25 cm is placed perpendicular to a uniform magnetic field of strength 3 T. (a) Determine the induced emf between the ends of the rod when it is not moving. (b) Determine the emf when the rod is moving perpendicular to its length and magnetic field with a speed of \(50 \mathrm{cm} / \mathrm{s}\).
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