Problem 37 - University Physics with Modern Physics

Question

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\(R\)?

Step-by-Step Solution

Answer

(a) Rate of change of flux: \( \pi r_1^2 \cdot \frac{dB}{dt} \). (b) Induced electric field inside: \( -\frac{r_1}{2} \cdot \frac{dB}{dt} \). (c) Outside field: \( -\frac{R^2}{2r_2} \cdot \frac{dB}{dt} \). (e) EMF for \( R/2 \): \( -\frac{1}{4}\pi R^2 \cdot \frac{dB}{dt} \). (f) EMF for \( R \): \( -\pi R^2 \cdot \frac{dB}{dt} \). (g) EMF for \( 2R \): \( -\pi R^2 \cdot \frac{dB}{dt} \).

Step 1: Determine the Flux Change Inside the Solenoid

The magnetic flux \( \Phi \) through a circle of radius \( r_1 \) in a magnetic field \( B \) is given by \( \Phi = B \cdot A \), where \( A = \pi r_1^2 \) is the area of the circle. Since \( B \) is increasing at a rate \( \frac{dB}{dt} \), the rate of change of flux is:\[ \frac{d\Phi}{dt} = \pi r_1^2 \cdot \frac{dB}{dt}. \]

Step 2: Calculate Induced Electric Field Inside the Solenoid

The induced electric field \( E \) at a distance \( r_1 \) from the solenoid axis can be found using Faraday's Law, which relates the EMF to the rate of change of flux: \( \text{EMF} = -\frac{d\Phi}{dt} \). The electric field is uniform along a circle of radius \( r_1 \), giving an EMF of \( E \cdot 2\pi r_1 = -\pi r_1^2 \cdot \frac{dB}{dt} \). Solving for \( E \), we get:\[ E = -\frac{r_1}{2} \cdot \frac{dB}{dt}. \] The direction of \( E \) is tangential to the circle, consistent with Lenz's Law.

Step 3: Calculate Induced Electric Field Outside the Solenoid

For a point outside the solenoid (\( r_2 > R \)), the total flux change within the solenoid is \( \Phi = \pi R^2 \cdot B \). From outside, the electric field \( E \) at \( r_2 \) is found noting that \( \text{EMF} = E \cdot 2\pi r_2 = -\pi R^2 \cdot \frac{dB}{dt} \). Solving this gives:\[ E = -\frac{R^2}{2r_2} \cdot \frac{dB}{dt}. \]

Step 4: Graph Electric Field as Function of Radius

The graph of the electric field \( E \) as a function of \( r \) will show a linear decrease from the maximum at \( r = 0 \) to \( r = R \), such that \( E = -\frac{r}{2} \cdot \frac{dB}{dt} \). For \( r > R \), \( E \) decreases as \( E = -\frac{R^2}{2r} \cdot \frac{dB}{dt} \). The transition at \( r = R \) will be smooth.

Step 5: Calculate Induced EMF for Radius R/2

For a circle with radius \( R/2 \) within the solenoid, the EMF is derived using \( \frac{d\Phi}{dt} = \pi (R/2)^2 \cdot \frac{dB}{dt} \) as \( \text{EMF} = -\frac{1}{4} \pi R^2 \cdot \frac{dB}{dt} \).

Step 6: Calculate Induced EMF for Radius R

For a circle with radius \( R \), the entire solenoid's cross-section is considered, giving \( \text{EMF} = -\pi R^2 \cdot \frac{dB}{dt} \).

Step 7: Calculate Induced EMF for Radius 2R

A circle with radius \( 2R \) encompasses and exceeds the solenoid area. However, since the added area outside the solenoid does not contribute magnetic flux (field is 0 outside), the induced EMF remains \( \text{EMF} = -\pi R^2 \cdot \frac{dB}{dt} \), same as for radius \( R \).

Key Concepts

Solenoid

A solenoid is a type of coil that creates a magnetic field when an electric current passes through it. If you've ever seen a tightly wound coil of wire, you've seen a solenoid. Solenoids have practical applications, particularly in electromagnets. They can generate controlled magnetic fields, which can be used in various machines, from MRI scanners to automotive components.

Inside the solenoid, the magnetic field is uniform and parallel to the axis of the coil, differing from the magnetic fields in other arrangements. This uniformity is because of how closely wound the wires are. If a solenoid has a circular cross-section, like the one described in the exercise, the field inside increases with the rate of change of the magnetic field, denoted as \( \frac{dB}{dt} \).

• The magnetic field strength inside is directly proportional to the number of turns and the current flowing through the wires.
• Solenoids are fundamental in implementing concepts from electromagnetism in real-world applications.

They have different utilizations based on how the magnetic field is harnessed. In physics problems involving solenoids, it often acts as the source of the magnetic field impacting other nearby objects.

Faraday's Law

Faraday's Law is a core principle in electromagnetism. It explains how electric currents can be induced by changing magnetic fields, a process crucial for harnessing electricity in generators and transformers. According to Faraday's Law, the induced electromotive force (EMF) in any closed loop is equal to the rate of change of the magnetic flux through the loop. Mathematically, this is represented as:

\[ \text{EMF} = -\frac{d\Phi}{dt} \]
Here, \( \Phi \) represents the magnetic flux, and the negative sign is due to Lenz's Law, indicating that the induced EMF creates a current that opposes the change in flux.

• The faster the change in the magnetic field, the greater the induced EMF.
• Faraday's Law is essential for understanding how electric transformers work, changing alternating current voltages from one level to another.

In exercises involving solenoids, like the one here, Faraday's Law helps us determine the induced electric field due to changing magnetic fields.

Induced Electric Field

An induced electric field is created as a direct response to a changing magnetic field, thanks to Faraday's Law. This electric field is not due to static charges but rather dynamic changes in the magnetic environment, such as those caused by a solenoid with an increasing magnetic field. In our exercise, the changing magnetic field inside the solenoid gives rise to an induced electric field.

Key characteristics of an induced electric field include:

• It is tangent to any loop or circumference around the solenoid.
• The strength depends on both the rate of change of the magnetic field (\( \frac{dB}{dt} \)) and the distance from the axis of the solenoid.
• Inside the solenoid, this field is proportional to the radius; outside, it varies inversely with the distance from the solenoid's center.

This induced field is what drives currents in the examples of electromagnetic induction systems, like electrical generators, showing the dynamic interplay between changing magnetic and electric fields.

Electromagnetic Induction

Electromagnetic induction is a phenomenon where a voltage or EMF is generated across a conductor, a result of changes in the magnetic environment. This principle, discovered by Michael Faraday, underpins many technologies, such as transformers and generators.

There are several ways this induction can occur:

• Moving a conductor through a stationary magnetic field.
• Changing the intensity of the magnetic field around a conductor.
• With rotating coils, like in many power generators.

In our scenario with a solenoid, electromagnetic induction is responsible for the induced voltage seen in the circular loops around the solenoid. As the magnetic field within the solenoid changes, different sections of the loop experience varying voltages. Understanding electromagnetic induction is crucial for grasping how energy can be transformed from mechanical to electrical forms, a foundational concept in both modern power generation and electronic applications.