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Chapter 22: Problem 83

A long solenoid of length \(8.0 \times 10^{-2} \mathrm{m}\) and cross-sectional area \(5.0 \times 10^{-5} \mathrm{m}^{2}\) contains 6500 turns per meter of length. Determine the emf induced in the solenoid when the current in the solenoid changes from 0 to 1.5 A during the time interval from 0 to 0.20 s.

Short Answer

Expert verified
The induced emf is approximately \(-0.159\, \text{V}\).

Step by step solution

01

Identify the Given Values

We have the following data given in the problem:- Length of solenoid, \( L = 8.0 \times 10^{-2} \text{ m} \).- Cross-sectional area, \( A = 5.0 \times 10^{-5} \text{ m}^2 \).- Number of turns per unit length, \( n = 6500 \text{ turns/m} \).- Initial current, \( I_0 = 0 \text{ A} \).- Final current, \( I_f = 1.5 \text{ A} \).- Time interval, \( \Delta t = 0.20 \text{ s} \).
02

Calculate the Change in Current

The change in current \( \Delta I \) is given by the difference between the final and initial currents:\[\Delta I = I_f - I_0 = 1.5 \text{ A} - 0 \text{ A} = 1.5 \text{ A}\]
03

Compute the Total Number of Turns in the Solenoid

The total number of turns \( N \) in the solenoid can be calculated using the length of the solenoid:\[N = n \times L = 6500 \text{ turns/m} \times 8.0 \times 10^{-2} \text{ m} = 520 \text{ turns}\]
04

Determine the Induced emf Using Faraday's Law

Faraday's Law of Induction states that the induced emf \( \mathcal{E} \) in a solenoid is given by:\[\mathcal{E} = - N \cdot \frac{d(BA)}{dt}\]Where \( B = \mu_0 n I \) is the magnetic field inside the solenoid, and \( \mu_0 \) is the magnetic constant. Therefore, the change in magnetic flux is:\[\frac{d(BA)}{dt} = A \cdot \mu_0 n \cdot \frac{dI}{dt}\]Substituting this into Faraday's Law:\[\mathcal{E} = -N \cdot A \cdot \mu_0 n \cdot \frac{\Delta I}{\Delta t}\]
05

Substitute Numerical Values to Find the Induced emf

Use the values and the known constant \( \mu_0 = 4\pi \times 10^{-7} \text{ T} \cdot \text{m/A} \):\[\mathcal{E} = - (520 \text{ turns}) \times (5.0 \times 10^{-5} \text{ m}^2) \times (4\pi \times 10^{-7} \text{ T} \cdot \text{m/A}) \times (6500 \text{ turns/m}) \times \left( \frac{1.5 \text{ A}}{0.20 \text{ s}} \right)\]Calculating:\[\mathcal{E} = - 520 \times 5.0 \times 10^{-5} \times 4\pi \times 10^{-7} \times 6500 \times 7.5\]\[\mathcal{E} \approx - 1.59 \times 10^{-1} \text{ V} \approx -0.159 \text{ V}\]The negative sign indicates the direction of the induced emf.

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Key Concepts

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

Understanding a Solenoid
A solenoid is a helical coil of wire designed to produce a uniform magnetic field when an electric current passes through it. The strength of the magnetic field inside a solenoid depends on:
  • The number of turns per unit length of the coil, represented by \( n \).
  • The current \( I \) flowing through it.
  • The permeability of the core material, often air or vacuum, provided by \( \mu_0 \), the magnetic constant.
A solenoid's core can also be filled with magnetic materials to further enhance the magnetic field. In practical applications, solenoids are used in devices such as electromagnetic valves, actuators, and inductors in circuits. Their efficiency in creating a concentrated magnetic field is highly beneficial in many technological devices.
Induced Electromotive Force (emf)
The concept of induced electromotive force (emf) is based on Faraday's Law of Electromagnetic Induction. This law states that a change in magnetic flux through a circuit induces an emf in the circuit. The key elements that affect the induced emf include:
  • The rate of change of magnetic flux, \( \frac{d\Phi}{dt} \).
  • The number of turns \( N \) in the coil through which the change occurs.
This is mathematically expressed as:\[\mathcal{E} = - N \cdot \frac{d\Phi}{dt}\]where the negative sign indicates the direction of the induced emf as governed by Lenz's Law 鈥� the induced emf always opposes the change in magnetic flux. Induced emf is used in various applications like power generation in alternators and transformers, showcasing its importance in efficiently managing energy.
Understanding Magnetic Flux Change
Magnetic flux change plays a crucial role in inducing an emf in a solenoid. Magnetic flux \( \Phi \) through a surface is defined as the product of the magnetic field \( B \) and the perpendicular area \( A \) it penetrates:\[\Phi = B \times A\]When the magnetic field, the area, or their orientation changes, the magnetic flux also changes, potentially inducing an emf in circuits around the flux.To compute the change in magnetic flux for a solenoid whose current changes, we use the relationship of magnetic field with current:\[B = \mu_0 n I\]Thus, the change in magnetic flux per unit time, \( \frac{d(BA)}{dt} \), involves:
  • The cross-sectional area \( A \) of the solenoid.
  • The rate of change of current \( \frac{dI}{dt} \).
This understanding of magnetic flux is vital for devising efficient systems to harness electromagnetic energy, which is integral in technologies like wireless power transmission.

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Most popular questions from this chapter

A long, current-carrying solenoid with an air core has 1750 turns per meter of length and a radius of \(0.0180 \mathrm{m} .\) A coil of 125 turns is wrapped tightly around the outside of the solenoid, so it has virtually the same radius as the solenoid. What is the mutual inductance of this system?

Two coils of wire are placed close together. Initially, a current of 2.5 A exists in one of the coils, but there is no current in the other. The current is then switched off in a time of \(3.7 \times 10^{-2}\) s. During this time, the average emf induced in the other coil is \(1.7 \mathrm{V}\). What is the mutual inductance of the two-coil system?

Two flat surfaces are exposed to a uniform, horizontal magnetic field of magnitude 0.47 T. When viewed edge-on, the first surface is tilted at an angle of \(12^{\circ}\) from the horizontal, and a net magnetic flux of \(8.4 \times 10^{-3} \mathrm{Wb}\) passes through it. The same net magnetic flux passes through the second surface. (a) Determine the area of the first surface. (b) Find the smallest possible value for the area of the second surface.

The resistances of the primary and secondary coils of a transformer are 56 and \(14 \Omega,\) respectively. Both coils are made from lengths of the same copper wire. The circular turns of each coil have the same diameter. Find the turns ratio \(N_{J} / N_{\mathrm{p}}\).

Mutual induction can be used as the basis for a metal detector. A typical setup uses two large coils that are parallel to each other and have a common axis. Because of mutual induction, the ac generator connected to the primary coil causes an emf of \(0.46 \mathrm{V}\) to be induced in the secondary coil. When someone without metal objects walks through the coils, the mutual inductance and, thus, the induced emf do not change much. But when a person carrying a handgun walks through, the mutual inductance increases. The change in emf can be used to trigger an alarm. If the mutual inductance increases by a factor of three, find the new value of the induced emf.
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