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Chapter 22: Problem 62

In each of two coils the rate of change of the magnetic flux in a single loop is the same. The emf induced in coil 1 , which has 184 loops, is \(2.82 \mathrm{~V}\). The emf induced in coil 2 is \(4.23 \mathrm{~V}\). How many loops does coil 2 have?

Short Answer

Expert verified
Coil 2 has 276 loops.

Step by step solution

01

Understanding Electromagnetic Induction

When a magnetic flux changes through a loop, an electromotive force (emf) is induced according to Faraday's law of induction, which is expressed as \( \text{emf} = -N \frac{d\Phi}{dt} \), where \( N \) is the number of loops and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux. For both coils, \( \frac{d\Phi}{dt} \) is the same.
02

Setting up the Equation for Coils

For coil 1, the induced emf is given by \( \text{emf}_1 = -N_1 \frac{d\Phi}{dt} \) and for coil 2, it is \( \text{emf}_2 = -N_2 \frac{d\Phi}{dt} \). Given that \( N_1 = 184 \), \( \text{emf}_1 = 2.82 \mathrm{~V} \), and \( \text{emf}_2 = 4.23 \mathrm{~V} \), the expressions for the emfs are \( 2.82 = -184 \frac{d\Phi}{dt} \) and \( 4.23 = -N_2 \frac{d\Phi}{dt} \).
03

Expressing Relationships Between Coils

We know \( \frac{d\Phi}{dt} \) is the same for both coils. Thus, we can equate the expressions of \( \frac{d\Phi}{dt} \) from both coils: \( \frac{2.82}{184} = \frac{4.23}{N_2} \).
04

Solving for the Number of Loops in Coil 2

Solve the equation \( \frac{2.82}{184} = \frac{4.23}{N_2} \) to find \( N_2 \). First, simplify the left side: \( \frac{2.82}{184} \approx 0.0153 \). Now, solve for \( N_2 \): \( 0.0153 = \frac{4.23}{N_2} \), giving \( N_2 = \frac{4.23}{0.0153} \), which calculates to approximately \( 276.47 \). Since the number of loops must be an integer, round to the nearest whole number, resulting in \( N_2 = 276 \).

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Key Concepts

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

Faraday's Law
Faraday's Law of electromagnetic induction is a fundamental principle that describes how a changing magnetic field can create an electromotive force (emf) in a conductor. This principle is named after Michael Faraday, who discovered it in 1831. It is represented mathematically as:
  • \( \text{emf} = -N \frac{d\Phi}{dt} \)
  • Where \( N \) is the number of loops in a coil, and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux.
The negative sign in the equation is an application of Lenz's Law. This indicates that the direction of the induced emf will oppose the change in magnetic flux that produced it. It is a crucial aspect because it explains how devices such as transformers, electric generators, and induction motors function.
Faraday's discovery shows that only changes in magnetic fields can generate an emf. Therefore, if the magnetic field remains constant, no emf is induced regardless of the strength of the field.
Magnetic Flux
Magnetic flux, often denoted by \( \Phi \), is a measure of the amount of magnetic field passing through a given area. It quantifies the strength and extent of the magnetic field over an area. The formula to calculate magnetic flux is:
  • \( \Phi = B \cdot A \cdot \cos(\theta) \)
  • Where \( B \) is the magnetic field strength, \( A \) is the area the field passes through, and \( \theta \) is the angle between the magnetic field lines and the normal (perpendicular) to the surface.
Changes in magnetic flux are critical for the induction of emf, as seen in Faraday's Law. When the magnetic flux through a coil changes, it disturbs the equilibrium, which leads to the generation of emf. The change can occur due to...
  • Changing the strength of the magnetic field \( B \)
  • Changing the area \( A \) that the magnetic field lines pass through
  • Changing the angle \( \theta \)
Induced emf
Induced emf refers to the voltage generated in a loop of wire or a coil when it experiences a change in magnetic flux. This concept is a direct outcome of Faraday's Law. The key point to remember is that the induced emf is directly proportional to the rate of change of magnetic flux and the number of loops in the coil.
Think of induced emf as a result of nature's way to resist changes. When you try to change something like magnetic flux through a coil, the coil generates an emf to oppose that change. It's much like resistance to change in everyday life.
The greater the change of magnetic flux, or the faster the change occurs, the larger the induced emf. Increasing the number of loops in a coil also amplifies the emf, as seen in the exercise where two coils experience the same flux change, but the one with more loops (coil 2) has a higher induced emf.

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Most popular questions from this chapter

The drawing shows a copper wire (negligible resistance) bent into a circular shape with a radius of \(0.50 \mathrm{~m} .\) The radial section \(B C\) is fixed in place, while the copper bar \(A C\) sweeps around at an angular speed of \(15 \mathrm{rad} / \mathrm{s}\). The bar makes electrical contact with the wire at all times. The wire and the bar have negligible resistance. A uniform magnetic field exists everywhere, is perpendicular to the plane of the circle, and has a magnitude of \(3.8 \times 10^{-3} \mathrm{~T}\). Find the magnitude of the current induced in the \(\operatorname{loop} A B C .\)

A generator is connected across the primary coil \(\left(N_{\mathrm{p}}\right.\) turns) of a transformer, while a resistance \(R_{2}\) is connected across the secondary coil \(\left(N_{\mathrm{s}}\right.\) turns \() .\) This circuit is equivalent to a circuit in which a single resistance \(R_{1}\) is connected directly across the generator, without the transformer. Show that \(R_{1}=\left(N_{\mathrm{p}} / N_{\mathrm{s}}\right)^{2} R_{2},\) by starting with Ohm's law as applied to the secondary coil.

The drawing shows a type of flow meter that can be used to measure the speed of blood in situations when a blood vessel is sufficiently exposed (e.g., during surgery). Blood is conductive enough that it can be treated as a moving conductor. When it flows perpendicularly with respect to a magnetic field, as in the drawing, electrodes can be used to measure the small voltage that develops across the vessel. Suppose the speed of the blood is \(0.30 \mathrm{~m} / \mathrm{s}\) and the diameter of the vessel is \(5.6 \mathrm{~mm} .\) In a 0.60 -T magnetic field what is the magnitude of the voltage that is measured with the electrodes in the drawing?

The secondary coil of a step-up transformer provides the voltage that operates an electrostatic air filter. The turns ratio of the transformer is \(50: 1\). The primary coil is plugged into a standard \(120-V\) outlet. The current in the secondary coil is \(1.7 \times 10^{-3} \mathrm{~A}\). Find the power consumed by the air filter.

A magnetic field has a magnitude of \(0.078 \mathrm{~T}\) and is uniform over a circular surface whose radius is \(0.10 \mathrm{~m}\). The field is oriented at an angle \(\phi=25^{\circ}\) with respect to the normal to the surface. What is the magnetic flux through the surface?
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