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Chapter 23: Problem 51

A solenoid with a cross-sectional area of \(1.81 \times 10^{-3} \mathrm{m}^{2}\) is \(0.750 \mathrm{m}\) long and has 455 turns per meter. Find the induced emf in this solenoid if the current in it is increased from 0 to \(2.00 \mathrm{A}\) in \(45.5 \mathrm{ms}.\)

Short Answer

Expert verified
The induced emf is approximately -0.031 V.

Step by step solution

01

Understand the Problem

We need to find the induced electromotive force (emf) in the solenoid when the current changes from 0 to 2.00 A over 45.5 ms. Given values include the cross-sectional area, length, turns per meter, and the time duration of current change.
02

Calculate the Total Number of Turns

The solenoid is 0.750 m long with 455 turns per meter. The total number of turns (N) is given by:\[ N = n \times l = 455 \times 0.750 \]}
03

Use Faraday's Law of Induction

Faraday's Law states that the emf \( \mathcal{E} \) induced in the solenoid is given by:\[ \mathcal{E} = -N \frac{d\Phi_B}{dt} \]where \(d\Phi_B\) is the change in magnetic flux, and \(d\Phi_B/dt\) is the rate of change of magnetic flux.
04

Calculate the Change in Magnetic Flux

The magnetic flux \( \Phi_B \) through each turn is given by:\[ \Phi_B = B \cdot A \]where \(B\) is the magnetic field and \(A\) is the cross-sectional area. In this solenoid, \( B = \mu_0 n I \), where \(I\) is the current. Therefore, the change in magnetic flux, \( \Delta \Phi_B \), as current changes from 0 to 2.00 A is:\[ \Delta \Phi_B = A \cdot \Delta B = A \cdot (\mu_0 n (2.00) - \mu_0 n (0) ) = \mu_0 n A \cdot 2.00 \]
05

Substitute Values to Find Rate of Change of Magnetic Flux

Given \( \mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A} \), \(A = 1.81 \times 10^{-3} \text{ m}^2\), and \(t = 45.5 \text{ ms} = 45.5 \times 10^{-3} \text{ s}\):\[ \frac{d\Phi_B}{dt} = \mu_0 n A \cdot \frac{2.00}{45.5 \times 10^{-3}} \]
06

Solve for the Induced EMF

Substitute values from previous steps to calculate the induced emf:\[ N = 455 \times 0.750 = 341.25 \]\[ \frac{d\Phi_B}{dt} = \left(4\pi \times 10^{-7} \right) \cdot 455 \cdot 1.81 \times 10^{-3} \cdot \frac{2.00}{45.5 \times 10^{-3}} \]Then, multiply by the total number of turns:\[ \mathcal{E} = -341.25 \times \left(4\pi \times 10^{-7} \right) \cdot 455 \cdot 1.81 \times 10^{-3} \cdot \frac{2.00}{45.5 \times 10^{-3}} \]
07

Calculate and Interpret the Result

Upon computation, the calculation gives:\[ \mathcal{E} \approx -0.031 \, \text{V} \]The negative sign indicates the direction of the induced emf opposes the change in current according to Lenz's law.

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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
Faraday's Law of Induction is a fundamental principle that explains how voltage, or electromotive force (emf), is generated by a change in magnetic flux. Imagine a loop of wire. When the magnetic field passing through the loop changes, a voltage is induced. This happens because the changing magnetic field interacts with the charges in the wire, setting them in motion.
In mathematical terms, Faraday's Law states:
  • The induced emf \( \mathcal{E} \) is equal to the negative of the rate of change of magnetic flux through the loop.
  • This is expressed by the formula: \( \mathcal{E} = - \frac{d\Phi_B}{dt} \).
Where:
  1. \( \mathcal{E} \) is the induced electromotive force in volts (V).
  2. \( \Phi_B \) represents the magnetic flux.
  3. The negative sign reflects Lenz's Law, indicating that the induced emf opposes the change in magnetic flux.
This principle is used in various applications, such as transformers and electric generators, to produce electric current.
Solenoid
A solenoid is a coil of wire designed to create a magnetic field when an electric current passes through it. Think of it as a tightly wrapped tube of wire. The solenoid structure allows it to produce a uniform magnetic field in its interior, making it highly useful in electromagnetism and electromechanical devices.
Here's how a solenoid works:
  • When current flows through the wires, it generates a magnetic field.
  • The field strengthens as more turns are added or as the current increases.
  • Inside a long solenoid, the field is parallel to the axis and virtually uniform.
Key characteristics of solenoids include:
  1. The magnetic field inside is determined by the number of turns per unit length \( n \), the current \( I \), and the magnetic constant \( \mu_0 \).
  2. This field can be calculated using the formula: \( B = \mu_0 n I \).
The solenoid's ability to create a controlled and sizable magnetic field makes it essential in applications like MRI machines, relays, and solenoid valves.
Magnetic Flux
Magnetic flux represents the amount of magnetic field passing through a given area. You can think of it as the 'web' created by magnetic field lines passing through a surface, influencing how they interact with electrical conductors.
In simple terms:
  • Magnetic flux is represented by \( \Phi_B \).
  • It's calculated as the product of the magnetic field \( B \) and the area \( A \) through which it passes, given by \( \Phi_B = B \cdot A \cos \theta \).
Where:
  1. \( B \) is the magnetic field, usually in teslas (T).
  2. \( A \) is the area through which the field lines pass (m虏).
  3. \( \theta \) is the angle between the magnetic field and the normal (perpendicular) to the surface.
A change in magnetic flux is crucial for the induction of voltage as described by Faraday's Law. Thus, measuring and manipulating magnetic flux is fundamental in the study and application of electromagnetism.
Lenz's Law
Lenz's Law provides insight into the direction of the induced current or emf. It asserts that the induced current will flow in such a way that its magnetic field opposes the initial change in flux that produced it.
This means:
  • If the magnetic flux through a loop is increasing, the induced emf will act to decrease it.
  • If the magnetic flux is decreasing, the induced emf will work to increase it.
Here's why Lenz's Law is vital:
  1. It is essentially a manifestation of the conservation of energy, ensuring that energy created by the change in magnetic flux doesn't come out of nowhere.
  2. The negative sign in Faraday's Law formula \( \mathcal{E} = - \frac{d\Phi_B}{dt} \) is due to Lenz's Law.
Understanding Lenz's Law is crucial for predicting the behavior of circuits and components subjected to changing magnetic fields. It's important in practical applications such as designing electrical circuits and improving the efficiency of electric motors and generators.

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Most popular questions from this chapter

IP A disk drive plugged into a 120 -V outlet operates on a voltage of \(9.0 \mathrm{V}\). The transformer that powers the disk drive has 125 turns on its primary coil. (a) Should the number of turns on the secondary coil be greater than or less than 125 ? Explain. (b) Find the number of turns on the secondary coil.

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?

A rectangular coil \(25 \mathrm{cm}\) by \(35 \mathrm{cm}\) has 120 turns. This coil produces a maximum emf of \(65 \mathrm{V}\) when it rotates with an angular speed of \(190 \mathrm{rad} / \mathrm{s}\) in a magnetic field of strength \(B .\) Find the value of \(B.\)

"Smart" traffic lights are controlled by loops of wire embedded in the road (Figure \(23-49\) ). These "loop detectors" sense the change in magnetic field as a large metal object-such as a car or a truck-moves over the loop. Once the object is detected, electric circuits in the controller check for cross traffic, and then turn the light from red to green. A typical loop detector consists of three or four loops of 14 gauge wire buried 3 in. below the pavement. You can see the marks on the road where the pavement has been cut to allow for installation of the wires. There may be more than one loop detector at a given intersection; this allows the system to recognize that an object is moving as it activates first one detector and then another over a short period of time. If the system determines that a car has entered the intersection while the light is red, it can activate one camera to take a picture of the car from the front-to see the driver's face-and then a second camera to take a picture of the car and its license plate from behind. This red-light camera system was used to good effect during an exciting chase scene through the streets of London in the movie National Treasure: Book of Secrets. Motorcycles are small enough that they often fail to activate the detectors, leaving the cyclist waiting and waiting for a green light. Some companies have begun selling powerful neodymium magnets to mount on the bottom of a motorcycle to ensure that they are "seen" by the detectors. Suppose the downward vertical component of the magnetic field increases as a car drives over a loop detector. As viewed from above, is the induced current in the loop clockwise, counterclockwise, or zero?

A single-turn square loop of side \(L\) is centered on the axis of a long solenoid. In addition, the plane of the square loop is perpendicular to the axis of the solenoid. The solenoid has 1250 turns per meter and a diameter of \(6.00 \mathrm{cm},\) and carries a current of \(2.50 \mathrm{A}\). Find the magnetic flux through the loop when (b) \(L=6.00 \mathrm{cm},\) and (c) \(L=12.0 \mathrm{cm}\) (a) \(L=3.00 \mathrm{cm}.\)
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