91Ó°ÊÓ

A rectangular loop of area \(A\) is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of the field is allowed to vary in time according to \(B=B_{\max } e^{-l / \tau},\) where \(B_{\max }\) and \(\tau\) are constants. The field has the constant value \(B_{\max }\) for \(t<0\) (a) Use Faraday's law to show that the emf induced in the loop is given by $$\mathcal{E}=\frac{A B_{\max }}{\tau} e^{-t / \tau}$$ (b) Obtain a numerical value for \(\boldsymbol{\varepsilon}\) at \(t=4.00 \mathrm{s}\) when \(A=0.160 \mathrm{m}^{2}, B_{\max }=0.350 \mathrm{T},\) and \(\tau=2.00 \mathrm{s} .\) (c) For the values of \(A, B_{\max },\) and \(\tau\) given in (b), what is the maximum value of \(\mathcal{E} ?\)

Step by step solution

01

Apply Faraday's Law

Faraday's law states that the induced electromotive force \( \epsilon \) in any closed loop is equal to the negative of the time rate of change of the magnetic flux \( \Phi_B \) through the loop. So, we use the equation for induced emf:\( \epsilon = - \frac{d \Phi_B}{dt}\) Where the magnetic flux \( \Phi_B = BA \), B is the magnetic field and A is the area of the loop. Given B varies with time, we have \( B = B_{max} e^{- \frac{t}{\tau}} \). Therefore, \( \Phi_B = AB_{max} e^{- \frac{t}{\tau}} \).

02

Differentiate the Magnetic Flux

Find the time derivative of the magnetic flux: \( \frac{d\Phi_B}{dt} = - \frac{A B_{max}}{\tau} e^{- \frac{t}{\tau}}\). Since the induced emf is equal to the negative rate of change of the magnetic flux, we get \( \epsilon = \frac{A B_{max}}{\tau} e^{- \frac{t}{\tau}}\). This completes the proof for part (a).

03

Calculate the Value of Induced EMF at Specific Time

For part (b), we plug in the given values into the formula we just derived: \( \epsilon = \frac{A B_{max}}{\tau} e^{- \frac{t}{\tau}}\). Thus, \( \epsilon = \frac{0.160m^2 \cdot 0.350 T}{2.00s} e^{-\frac{4.00s}{2.00s}} = 0.022T \).

04

Compute Maximum Value of Induced EMF

For part (c), the maximum value of \( \epsilon \) occurs when \( e^{-\frac{t}{\tau}} = 1 \), which occurs when \( t = 0s \). Substituting these values into the equation from Step 3, we find \( \epsilon_{max} = \frac{A B_{max}}{\tau} = \frac{0.160m^2 \cdot 0.350T}{2.00s} = 0.028T \).

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Key Concepts

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

Magnetic Flux

Magnetic flux represents the quantity that describes the magnetic field lines passing through a given area. Imagine the area as the surface of a pond, and the magnetic field lines as straws passing vertically through it. The more straws (or field lines) pass through the surface, the larger the magnetic flux. Mathematically, magnetic flux \( \Phi_B \) is given by the product of the magnetic field \( B \) and the area \( A \) of the loop it penetrates:

• \( \Phi_B = BA \cos (\theta) \)
• Where \( \theta \) is the angle between the magnetic field and the normal to the surface
• For our exercise, the field is perpendicular to the surface so \( \cos(\theta) = 1 \)

Thus, \( \Phi_B = BA \). In the problem, the magnetic field \( B \) is allowed to change with time, which directly affects the magnetic flux. As \( B \) decreases, so does \( \Phi_B \).

Induced EMF

According to Faraday's Law of electromagnetic induction, a change in magnetic flux over time induces an electromotive force (EMF) in a loop. This is the fundamental principle behind many electrical generators and transformers. The induced EMF \( \mathcal{E} \) is given by:

• \( \mathcal{E} = - \frac{d \Phi_B}{dt} \)

The negative sign reflects Lenz's Law, which indicates that the induced EMF works to oppose the change in flux. In our exercise, calculating this change requires differentiation:

• As \( \Phi_B = AB_{max} e^{- \frac{t}{\tau}} \), its derivative is \( -\frac{A B_{max}}{\tau} e^{- \frac{t}{\tau}} \)
• Hence, \( \mathcal{E} = \frac{A B_{max}}{\tau} e^{- \frac{t}{\tau}} \)

Magnetic Field Variation

The magnetic field in our problem is not constant—it varies with time according to an exponential function. An exponential decay is characterized by a rapid decrease that tapers off over time. The given formula is:

• \( B = B_{max} e^{- \frac{t}{\tau}} \)
• "\( B_{max} \)" is the initial magnetic field strength when \( t = 0 \)
• "\( \tau \)" is the time constant which dictates how quickly \( B \) decreases

This specific form for \( B \) directly impacts how the magnetic flux changes over time. As \( t \) increases, \( B \) decreases, leading to a decrease in magnetic flux and hence inducing an EMF, until eventually, \( B \) becomes negligible.

Rectangular Loop

In this exercise, we use a rectangular loop of area \( A \), which is essential for examining the interaction between the loop and the changing magnetic field. The shape and area of the loop play a critical role:

• The area \( A = 0.160 \text{ m}^2 \) sets the scale for the magnetic flux \( \Phi_B \).
• Because the magnetic field is perpendicular to the loop, all field lines pass through it efficiently, maximizing the flux.
• The loop is positioned in such a way that changes in \( B \) lead directly to changes in \( \Phi_B \).

This configuration allows us to effectively use Faraday's Law to determine the induced EMF. As \( A \) is a constant, it amplifies the changes in \( B \), and consequently, the EMF according to the derived formula.