Problem 41 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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·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: \[ B = (4\pi \times 10^{-7} \text{ T·m/A})(9000 \text{ turns/m})(0.350 \text{ A}) \]Calculate \( B \): \[ B = 3.96 \times 10^{-3} \text{ T} \]

Step 2: 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 \( \Delta \Phi = \Phi_f - \Phi_i = -3.17 \times 10^{-6} \text{ Wb} \).

Step 3: 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.

Key Concepts

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·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.