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Chapter 29: Problem 41

A long, straight solenoid with a cross-sectional area of 8.00 cm\(^2\) is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

Short Answer

Expert verified
The average induced emf in the second winding is 0.952 mV.

Step by step solution

01

Calculate the Initial Magnetic Field Inside the Solenoid

The magnetic field inside a solenoid is given by the formula \( B = \mu_0 n I \), where \( \mu_0 = 4\pi \times 10^{-7} \) T路m/A is the permeability of free space, \( n \) is the number of turns per unit length, and \( I \) is the current. Here, the number of turns per meter \( n = 90 \times 100 \) because there are 90 turns per centimeter. Thus, \( n = 9000 \) turns per meter.Substitute these values into the formula: \[ B = (4\pi \times 10^{-7} \text{ T路m/A})(9000 \text{ turns/m})(0.350 \text{ A}) \]Calculate \( B \): \[ B = 3.96 \times 10^{-3} \text{ T} \]
02

Determine the Change in Magnetic Flux Through the Second Winding

The magnetic flux \( \Phi \) through one loop of the induced winding is given by \( \Phi = B \times A \), where \( A \) is the cross-sectional area of the solenoid. Convert the area to square meters: \[ A = 8.00 \text{ cm}^2 = 8.00 \times 10^{-4} \text{ m}^2 \]Calculate the initial flux per loop:\[ \Phi_i = B \times A = (3.96 \times 10^{-3} \text{ T})(8.00 \times 10^{-4} \text{ m}^2) \]\[ \Phi_i = 3.17 \times 10^{-6} \text{ Wb (weber)} \]The final magnetic field is 0 T, so the final flux \( \Phi_f = 0 \text{ Wb} \). Thus, the change in flux \( \Delta \Phi = \Phi_f - \Phi_i = -3.17 \times 10^{-6} \text{ Wb} \).
03

Calculate the Average Induced EMF in the Second Winding

According to Faraday's Law of electromagnetic induction, the induced emf is given by \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \), where \( N \) is the number of turns in the second winding and \( \Delta t \) is the change in time.Plug in the values:\[ \varepsilon = -12 \Bigg(\frac{-3.17 \times 10^{-6} \text{ Wb}}{0.0400 \text{ s}}\Bigg) \]Calculate the emf:\[ \varepsilon = 9.52 \times 10^{-4} \text{ V} \]The average induced emf in the second winding is 0.952 mV.

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Key Concepts

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

Solenoid
A solenoid is a type of coil composed of many turns of wire, closely wound in the form of a helix. When an electric current flows through a solenoid, it creates a magnetic field along the axis of the coil. This feature makes solenoids extremely useful in electromagnetics for controlling magnetic fields.

The primary function of the solenoid is to convert electrical energy into mechanical motion, making it an excellent component in devices like electromechanical relays, actuators, and transformers. In physics problems, solenoids are often analyzed to understand how they influence the surrounding magnetic field, and how this field behaves when the solenoid's current is altered.
Magnetic Field
Magnetic fields are invisible fields around magnetic materials or moving electric charges that exert a force on other nearby magnetic materials and moving charges. In a solenoid, the magnetic field inside is uniform and parallel to the axis of the solenoid. This field is generated because the current flowing through the coils produces small magnetic fields that add up to create a larger one.

The strength of the magnetic field inside a solenoid can be determined using the formula: \[ B = \mu_0 n I \]where:
  • \( B \) is the magnetic field strength in teslas (T)
  • \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \) T路m/A
  • \( n \) is the number of turns per unit length (measured in meters)
  • \( I \) is the current in amperes (A)
This relationship helps in understanding how modifications to the solenoid design or the electric current can directly influence the magnetic field it produces.
Faraday's Law
Faraday's Law of electromagnetic induction is fundamental for understanding how electric currents and magnetic fields influence each other. It states that a change in magnetic flux through a circuit induces an electromotive force (emf) in the circuit. In mathematical terms, Faraday's Law is given by:\[ \varepsilon = -N \frac{d\Phi}{dt} \]where:
  • \( \varepsilon \) is the electromotive force in volts
  • \( N \) is the number of turns in the coil
  • \( d\Phi \) is the change in magnetic flux in webers (Wb)
  • \( dt \) is the change in time in seconds
The negative sign in the formula is due to Lenz's Law, which indicates that the direction of the induced emf will oppose the change in flux. This principle is essential in designing and analyzing circuits that rely on transformers or inductors.
Induced EMF
Induced electromotive force (emf) is the electric potential generated by a change in magnetic flux. In practical scenarios like the problem described, when the current in a solenoid changes, the magnetic field that it generates changes as well, leading to a change in flux through any nearby coils.

The average induced emf can be calculated using the formula derived from Faraday's Law:\[ \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \]where:
  • \( \Delta \Phi \) is the change in magnetic flux
  • \( \Delta t \) is the time over which the change occurs
Induced emf is crucial for the functioning of devices that require a response to variable magnetic fields, such as generators and transformers. Understanding how emf is induced helps in predicting how electrical systems will behave when subjected to time-varying magnetic fields.

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Most popular questions from this chapter

A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate \(di/dt\). The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00 \(\times\) 10\(^{-6}\) V/m. Calculate \(di/dt\).

A single loop of wire with an area of 0.0900 m\(^2\) is in a uniform magnetic field that has an initial value of 3.80 T, is perpendicular to the plane of the loop, and is decreasing at a constant rate of 0.190 T/s. (a) What emf is induced in this loop? (b) If the loop has a resistance of 0.600 \(\Omega\), find the current induced in the loop.

Airplanes and trains move through the earth's magnetic field at rather high speeds, so it is reasonable to wonder whether this field can have a substantial effect on them. We shall use a typical value of 0.50 G for the earth's field. (a) The French TGV train and the Japanese "bullet train" reach speeds of up to 180 mph moving on tracks about 1.5 m apart. At top speed moving perpendicular to the earth's magnetic field, what potential difference is induced across the tracks as the wheels roll? Does this seem large enough to produce noticeable effects? (b) The Boeing 747-400 aircraft has a wingspan of 64.4 m and a cruising speed of 565 mph. If there is no wind blowing (so that this is also their speed relative to the ground), what is the maximum potential difference that could be induced between the opposite tips of the wings? Does this seem large enough to cause problems with the plane?

In many magnetic resonance imaging (MRI) systems, the magnetic field is produced by a superconducting magnet that must be kept cooled below the superconducting transition temperature. If the cryogenic cooling system fails, the magnet coils may lose their superconductivity and the strength of the magnetic field will rapidly decrease, or \(quench\). The dissipation of energy as heat in the now-nonsuperconducting magnet coils can cause a rapid boil-off of the cryogenic liquid (usually liquid helium) that is used for cooling. Consider a superconducting MRI magnet for which the magnetic field decreases from 8.0 T to nearly 0 in 20 s. What is the average emf induced in a circular wedding ring of diameter 2.2 cm if the ring is at the center of the MRI magnet coils and the original magnetic field is perpendicular to the plane that is encircled by the ring?

The magnetic field within a long, straight solenoid with a circular cross section and radius \(R\) is increasing at a rate of \(dB/dt\). (a) What is the rate of change of flux through a circle with radius \(r_1\) inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance \(r_1\) from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field \(outside\) the solenoid, at a distance \(r_2\) from the axis? (d) Graph the magnitude of the induced electric field as a function of the distance \(r\) from the axis from \(r =\) 0 to \(r = 2R\). (e) What is the magnitude of the induced emf in a circular turn of radius R/2 that has its center on the solenoid axis? (f) What is the magnitude of the induced emf if the radius in part (e) is \(R\)? (g) What is the induced emf if the radius in part (e) is 2\(R\)?
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