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Chapter 23: Problem 19

A magnetic field increases from 0 to \(0.25 \mathrm{T}\) in \(1.8 \mathrm{s}\). How many turns of wire are needed in a circular coil \(12 \mathrm{cm}\) in diameter to produce an induced emf of \(6.0 \mathrm{V} ?\)

Short Answer

Expert verified
Approximately 3821 turns are needed.

Step by step solution

01

Understand the Problem

We need to determine the number of turns in a coil necessary to induce an electromotive force (emf) of 6.0 V when the magnetic field changes. Given the change in the magnetic field, we will apply Faraday's law of electromagnetic induction.
02

Recall Faraday's Law

Faraday's law states that the induced emf (\( \varepsilon \)) is equal to the negative rate of change of the magnetic flux through a coil of wire:\[ \varepsilon = -N \frac{d\Phi}{dt} \]where \( N \) is the number of turns and \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux.
03

Calculate Magnetic Flux Change

Magnetic flux (\( \Phi \)) is given by \( B \times A \), where \( B \) is the magnetic field and \( A \) is the area of the loop. For a circular coil of diameter 12 cm:\[ A = \pi \left( \frac{0.12}{2} \right)^2 \approx 1.13 \times 10^{-2} \mathrm{m}^2 \]The change in magnetic field is from 0 to 0.25 T, so \( \Delta B = 0.25 \mathrm{T}\). Hence, the change in magnetic flux is:\[ \Delta \Phi = A \times \Delta B = 1.13 \times 10^{-2} \times 0.25 = 2.825 \times 10^{-3} \mathrm{T} \cdot \mathrm{m}^2 \]
04

Calculate the Rate of Change of Flux

The change in magnetic flux occurs over 1.8 seconds, so we find the rate of change of flux:\[ \frac{d\Phi}{dt} = \frac{\Delta \Phi}{\Delta t} = \frac{2.825 \times 10^{-3}}{1.8} \approx 1.57 \times 10^{-3} \mathrm{T} \cdot \mathrm{m}^2/\mathrm{s} \]
05

Solve for Number of Turns

Using Faraday's law of induction, rearrange to find the number of turns:\[ \varepsilon = N \cdot \frac{d\Phi}{dt} \]\[ N = \frac{\varepsilon}{\frac{d\Phi}{dt}} = \frac{6.0}{1.57 \times 10^{-3}} \approx 3821 \]
06

Conclusion

We determine that approximately 3821 turns of wire are needed to induce an emf of 6.0 V under the given conditions.

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Key Concepts

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

Magnetic Flux
Magnetic flux is a core component in understanding electromagnetic induction. It quantifies the amount of magnetic field passing through a given area, making it crucial to concepts like Faraday's law. Imagine magnetic flux as invisible lines that permeate a surface, like arrows passing through a hoop.
The flux depends on two primary factors:
  • The strength of the magnetic field (\( B \)).
  • The area of the surface the field penetrates (\( A \)).
For a circular coil, the area (\( A \)) is calculated using the formula for the area of a circle,\( A = \pi \left( \frac{d}{2} \right)^2 \), where \( d \) is the diameter.
This concept helps us understand how changes in the magnetic field or the physical dimensions of a coil affect the magnetic flux, leading to induced currents as per Faraday's law.
Induced Electromotive Force (emf)
Induced electromotive force, or emf, is a critical outcome of a changing magnetic environment. It's essentially the electrical voltage generated by a change in magnetic flux. Faraday's law of electromagnetic induction provides the framework to calculate this induced emf.
As per Faraday's law, the relationship is expressed as:
  • \( \varepsilon = -N \frac{d\Phi}{dt} \),
where \( \varepsilon \) represents the induced emf, \( N \) is the number of turns in the coil, and \( \frac{d\Phi}{dt} \) symbolizes the rate of change of magnetic flux through the coil.
The negative sign is a nod to Lenz's law, which posits that the direction of induced emf and current is such that it opposes the change in the original magnetic flux, maintaining equilibrium in the system.
Rate of Change of Flux
The rate of change of flux is a pivotal factor that influences the magnitude of the induced emf in a coil. By understanding how flux changes over time, we can predict and even control the emf generated in devices.
The formula,\( \frac{d\Phi}{dt} \), represents this rate of change, dividing the change in magnetic flux (\( \Delta \Phi \)) by the time duration (\( \Delta t \)) over which the change occurs.
This concept portrays why the speed at which the magnetic environment alters can drastically impact the induced emf. For instance, in our original exercise, the change in magnetic flux takes place over 1.8 seconds, leading us to calculate:
  • \( \frac{d\Phi}{dt} \approx 1.57 \times 10^{-3} \mathrm{T} \cdot \mathrm{m}^2/\mathrm{s} \).
By manipulating this rate, one can effectively manage how devices like generators and transformers operate, optimizing them for desired electrical outputs.

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Most popular questions from this chapter

A \(1.6-\mathrm{m}\) wire is wound into a coil with a radius of \(3.2 \mathrm{cm}\). If this coil is rotated at 85 rpm in a \(0.075-\) T magnetic field, what is its maximum emf?

A step-up transformer has 25 turns on the primary coil and 750 turns on the secondary coil. If this transformer is to produce an output of \(4800 \mathrm{V}\) with a \(12-\mathrm{mA}\) current, what input current and voltage a re needed?

Predict/Explain A metal ring is dropped into a localized region of constant magnetic field, as indicated in Figure \(23-30\). The magnetic field is zero above and below the region where it is finite. (a) For each of the three indicated locations \((1,2,\) and 3 ), is the induced current clockwise, counterclockwise, or zero? (b) Choose the best explanation from among the following: Clockwise at 1 to oppose the field; zero at 2 because the field is uniform; counterclockwise at 3 to try to maintain the field. II. Counterclockwise at 1 to oppose the field; zero at 2 because the field is uniform; clockwise at 3 to try to maintain the field. III. Clockwise at 1 to oppose the field; clockwise at 2 to maintain the field; clockwise at 3 to oppose the field.

You hold a circular loop of wire at the equator. Consider the magnetic flux through this loop due to the Earth's magnetic field. Is the flux when the normal to the loop points north greater than, less than, or equal to the flux when the normal points vertically upward? Explain.

At a certain location, the Earth's magnetic field has a magnitude of \(5.9 \times 10^{-5} \mathrm{T}\) and points in a direction that is \(72^{\circ}\) below the horizontal. Find the magnitude of the magnetic flux through the top of a desk at this location that measures \(130 \mathrm{cm}\) by \(82 \mathrm{cm}.\)
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