Problem 33 - University Physics with Modern Physics

Question

A long, straight solenoid with a cross-sectional area of 8.00 \(\mathrm{cm}^{2}\) is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 \(\mathrm{A}\) . A second winding of 12 tums 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 indnced emf in the second winding?

Step-by-Step Solution

Answer

The average induced emf is 0.949 mV.

Step 1: Understand the Problem

We need to find the average induced electromotive force (emf) in a secondary winding when the magnetic field inside a solenoid changes to zero.

Step 2: Calculate the Initial Magnetic Field

The magnetic field inside the solenoid is given by the formula: \[ B = \mu_0 n I \]where \( \mu_0 = 4\pi \times 10^{-7} \ \text{T} \cdot \text{m/A} \) is the permeability of free space, \( n = 90 \ \text{turns/cm} = 9000 \ \text{turns/m} \) is the turn density, and \( I = 0.350 \ \text{A} \) is the current. Substitute in the values:\[ B = (4\pi \times 10^{-7})(9000)(0.350) \approx 3.955 \times 10^{-3} \ \text{T} \]

Step 3: Calculate the Change in Magnetic Flux

The magnetic flux through each loop of the second winding is given by:\[ \Phi = B \cdot A \]where \( A = 8.00 \ \text{cm}^2 = 8.00 \times 10^{-4} \ \text{m}^2 \) is the area. Initially, \( \Phi = 3.955 \times 10^{-3} \times 8.00 \times 10^{-4} = 3.164 \times 10^{-6} \ \text{Wb} \). When the current is turned off, \( B \) becomes 0 and thus \( \Phi = 0 \ \text{Wb} \). Therefore, the change in magnetic flux \( \Delta \Phi = 0 - 3.164 \times 10^{-6} = -3.164 \times 10^{-6} \ \text{Wb} \).

Step 4: Calculate the Average Induced EMF

According to Faraday's Law, the induced emf is given by:\[ \text{emf} = -N \frac{\Delta \Phi}{\Delta t} \]where \( N = 12 \) is the number of turns in the second winding, and \( \Delta t = 0.0400 \ \text{s} \) is the time interval. Substituting the values:\[ \text{emf} = -12 \frac{-3.164 \times 10^{-6}}{0.0400} \approx 9.49 \times 10^{-4} \ \text{V} \]

Step 5: Finalize the Result

The induced emf magnitude is positive, which means the direction of the induced emf is such that it opposes the change in current, according to Lenz's Law. The average induced emf in the second winding is 0.949 mV.

Key Concepts

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental principle that describes how electric currents are induced in conductors by changing magnetic fields. It forms the basis for many technological applications, such as transformers and electric generators. The key idea is simple and elegant: a change in magnetic flux through a circuit induces an electromotive force (emf) in the circuit.
Faraday's law is mathematically expressed as:

• \( \text{emf} = -N \frac{\Delta \Phi}{\Delta t} \)

Where:

• \( \text{emf} \) is the induced electromotive force in volts.
• \( N \) is the number of turns in the coil.
• \( \Delta \Phi \) is the change in magnetic flux, given in Weber (Wb).
• \( \Delta t \) is the time during which the change occurs, given in seconds.

The negative sign in Faraday's Law indicates the direction of the induced emf and is a reminder of Lenz's Law: the induced emf will always oppose the change in magnetic flux that produced it.

Solenoid

A solenoid is a long coil of wire that creates a uniform magnetic field inside its loops when an electric current passes through it. It acts like a magnet when carrying a current and is instrumental in applications that require a controllable magnetic field, such as electromagnets, inductors, and magnetic valves. The strength and properties of the magnetic field inside the solenoid depend on:

• The number of turns per unit length of the solenoid (turn density).
• The current passing through the solenoid.

To calculate the magnetic field inside a long solenoid, we use:\[ B = \mu_0 n I \]Where:

• \( B \) is the magnetic field inside the solenoid, measured in Teslas (T).
• \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \).
• \( n \) is the number of turns per meter (turn density).
• \( I \) is the current in amperes.

The uniform magnetic field inside a solenoid makes it an ideal environment for studying electromagnetic induction and other magnetic phenomena.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. Think of it as the number of magnetic field lines passing through a given area. It is a critical factor in electromagnetic induction, as changes in magnetic flux induce an electromotive force (emf) according to Faraday's Law. Magnetic flux is given by:

• \( \Phi = B \times A \)

Where:

• \( \Phi \) is the magnetic flux, measured in Webers (Wb).
• \( B \) is the magnetic field strength (in Tesla).
• \( A \) is the area through which the field lines pass (in square meters).

When the magnetic field or the area changes, the magnetic flux changes, leading to induction of emf. Understanding magnetic flux helps in visualizing how and why electromotive forces are generated in circuits.

Induced EMF

Induced electromotive force (emf) refers to the voltage generated in a circuit due to a change in magnetic flux. Faraday's Law gives us the framework to quantify this induced emf, and its direction is always such that it opposes the change causing it, as per Lenz's Law.
The importance of induced emf cannot be understated, as it's a critical concept in various technologies:

• In electric generators, mechanical energy is converted into electrical energy through induction.
• In transformers, the voltage levels are altered due to the induction occurring between primary and secondary coils.
• Induced emf is a key principle in the functioning of induction cooktops, electric guitars, and more.

The typical application of induced emf involves coiling wires and moving them through magnetic fields or changing the magnetic field around the coils. By closely understanding and controlling these changes, engineers and scientists can develop powerful electromagnetic devices.