91Ó°ÊÓ

The magnetic field within a long, straight solenoid with a circular cross section and radius \(R\) is increasing at a rate of \(d B / d t\) . (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 \(r=2 R\) (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 2R?

Step by step solution

01

Calculate Rate of Change of Magnetic Flux

The magnetic flux through the circle of radius \( r_1 \) within the solenoid is given by \( \Phi = B \cdot A = B \cdot \pi r_1^2 \), where \( A = \pi r_1^2 \) is the area of the circle. The rate of change of this flux is \( \frac{d\Phi}{dt} = \pi r_1^2 \cdot \frac{dB}{dt} \).

02

Induced Electric Field Inside the Solenoid

Using Faraday's Law, the magnitude of the induced electric field \( E \) at a distance \( r_1 \) from the solenoid's axis is given by \( E \cdot 2\pi r_1 = - \frac{d\Phi}{dt} \). Substituting the expression for \( \frac{d\Phi}{dt} \), we get \( E = \frac{\pi r_1^2 \frac{dB}{dt}}{2 \pi r_1} = \frac{r_1}{2} \frac{dB}{dt} \). The direction of the induced electric field is circumferential around the axis, using the right-hand rule.

03

Induced Electric Field Outside the Solenoid

For \( r_2 > R \), the enclosed magnetic flux by a circle of radius \( r_2 \) is still = \( B \pi R^2 \), as only the magnetic field inside the solenoid changes. From Faraday’s Law: \( E \cdot 2\pi r_2 = - \frac{d(B \pi R^2)}{dt} \), so \( E = \frac{R^2}{2r_2} \frac{dB}{dt} \).

04

Graph Magnitude of Induced Electric Field as a Function of Distance

For \( r \leq R \), the induced field \( E = \frac{r}{2} \frac{dB}{dt} \). For \( r > R \), the induced field \( E = \frac{R^2}{2r} \frac{dB}{dt} \). Plotting \( E \) against \( r \), we see a linear increase inside the solenoid up to \( R \), after which the field decreases with \( \frac{1}{r} \).

05

Calculate Induced EMF for Radius R/2

The induced emf \( \mathcal{E} \) in a loop of radius \( R/2 \) inside the solenoid is given by \( \mathcal{E} = E \cdot 2\pi (R/2) = \pi \frac{R^2}{4} \frac{dB}{dt} \).

06

Calculate Induced EMF for Radius R

The induced emf for a loop of radius \( R \) (inside the solenoid) is \( \mathcal{E} = \pi \frac{R^2}{2} \frac{dB}{dt} \).

07

Calculate Induced EMF for Radius 2R

For a loop with radius \( 2R \), which is partially outside the solenoid, the emf is \( \mathcal{E} = \left(\frac{R^2}{2} + R^2 \right) \pi \frac{dB}{dt} = \frac{3\pi R^2}{2} \frac{dB}{dt} \), summing contributions inside and outside.

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Key Concepts

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

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental principle that describes how electric fields can be generated by changing magnetic fields. This law is the cornerstone of understanding how electromagnetic fields interact. It states that the electromotive force (EMF) induced in a closed loop is proportional to the rate of change of magnetic flux through the loop. Mathematically, it's expressed as \( \mathcal{E} = -\frac{d\Phi}{dt} \), where \( \mathcal{E} \) is the induced EMF and \( \Phi \) is the magnetic flux through the loop.

When the magnetic field around a loop changes—either by varying the field intensity, moving the magnet, or changing the loop's orientation—an electric current is induced. This principle forms the basis of many electrical devices, such as transformers, electric generators, and inductors.

• Induced EMF depends on the rate of change of flux, not the flux itself.
• The negative sign in Faraday’s law equation signifies Lenz's law, indicating that the induced EMF creates a current and magnetic field that opposes the initial change.
• This law can be applied to loops of various shapes and in different magnetic fields.

Magnetic Flux

Magnetic flux is essentially a measure of the number of magnetic field lines passing through a given area. It's a concept that helps us understand the field's influence over different surfaces. The magnetic flux \( \Phi \) is calculated as \( \Phi = B \cdot A \cdot \cos(\theta) \), where \( B \) is the magnetic field strength, \( A \) is the area through which the lines pass, and \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface.

Changes in magnetic flux are integral to the production of electric current, using the principle of electromagnetic induction.

• Flux is maximum when the field lines are perpendicular to the surface (i.e., \( \theta = 0\)).
• If the loop is tilted such that field lines are parallel, flux is zero because \( \cos(90^\circ) = 0 \).
• Flux changes can occur by varying the field strength, the area, or the orientation of the surface.

Solenoid

A solenoid is a coil of wire designed to create a uniform magnetic field in its interior space when an electric current is passed through it. This device is central to many electromagnetic applications. The field lines in a solenoid are parallel and closely spaced, creating a strong, consistent field at its core while rapidly diminishing outside.

The strength of the magnetic field inside a solenoid is given by \( B = \mu n I \), where \( \mu \) is the magnetic permeability of the core, \( n \) is the number of turns per unit length, and \( I \) is the current passing through the wire.

• Fields are strong inside due to the alignment and density of field lines.
• Outside the solenoid, the field is much weaker and tends to cancel out due to opposite loops of current.
• Solenoids are used in applications like electromagnets, valves, and loops to induce specific field patterns.

Induced Electric Field

An induced electric field is created as a result of changing magnetic flux, as per Faraday's Law. This field acts tangentially along any closed loop surrounding the changing magnetic field. When considering a solenoid, the varying magnetic field within induces an electric field that forms concentric loops around the solenoid's axis.

The magnitude of this induced field inside a solenoid at a distance \( r \) from its axis is determined using the equation \( E = \frac{r}{2} \frac{dB}{dt} \). For a position outside the solenoid, where \( r > R \), it is calculated using \( E = \frac{R^2}{2r} \frac{dB}{dt} \).

• The field lines of the induced electric field form closed loops, encircling the changing magnetic field lines.
• This field is non-conservative, meaning the work done in moving a charge around the loop isn't zero as it would be in a static field.
• The induced current direction follows the right-hand rule, aligning with the field to always oppose changes in flux, as explained by Lenz's Law.