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Chapter 31: Problem 65

The magnetic flux through a metal ring varies with time \(t\) according to \(\Phi_{B}=3\left(a t^{3}-b t^{2}\right) \mathrm{T} \cdot \mathrm{m}^{2},\) with \(a=2.00 \mathrm{s}^{-3}\) and \(b=6.00 \mathrm{s}^{-2}\) . The resistance of the ring is 3.00\(\Omega\) Determine the maximum current induced in the ring during the interval from \(t=0\) to \(t=2.00 \mathrm{s}\) .

Short Answer

Expert verified
The maximum induced current \(I_{max}\) in the ring is the maximum value of \(I\) obtained when substituting \(\varepsilon_{max}\) into Ohm's Law.

Step by step solution

01

Write down the magnetic flux

The magnetic flux \(\Phi_B\) given by the problem is \(\Phi_B = 3(a t^3 - b t^2)\) where \(a = 2.00\ s^{-3}\) and \(b = 6.00\ s^{-2}\).
02

Apply Faraday's Law of Induction

According to Faraday's Law of Induction, the induced electromotive force (EMF) in the ring is equal to the negative rate of change of the magnetic flux through the ring. This can be expressed as \(\varepsilon = -\frac{d\Phi_B}{dt}\).
03

Differentiate the magnetic flux with respect to time

To find the rate of change of \(\Phi_B\), differentiate it with respect to \(t\): \(\frac{d\Phi_B}{dt} = \frac{d}{dt}[3(a t^3 - b t^2)] = 3[3a t^2 - 2b t]\). Substituting the values of \(a\) and \(b\), we get \(\frac{d\Phi_B}{dt} = 3[3(2.00) t^2 - 2(6.00) t]\).
04

Find the maximum rate of change of \(\Phi_B\)

Evaluate the derivative between \(t=0\) and \(t=2.00\ s\) to determine the maximum rate of change of the magnetic flux: \(\frac{d\Phi_B}{dt}|_{max} = 3[3(2.00)(2.00)^2 - 2(6.00)(2.00)]\).
05

Calculate the maximal induced EMF

The maximal induced EMF \(\varepsilon_{max}\) is the negative of the maximum rate of change of the magnetic flux: \(\varepsilon_{max} = -\frac{d\Phi_B}{dt}|_{max}\).
06

Calculate the maximum current

Using Ohm's Law \(I = \frac{\varepsilon}{R}\), the maximum induced current \(I_{max}\) can be calculated using the maximal induced EMF and the resistance \(R = 3.00\ \Omega\): \(I_{max} = \frac{\varepsilon_{max}}{3.00}\).
07

Substitute the values and solve for \(I_{max}\)

Substitute \(\varepsilon_{max}\) into the equation for \(I_{max}\) to find the maximum current. Make sure to calculate \(\varepsilon_{max}\) from the previous steps and use the negative sign as per Faraday's law.

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Key Concepts

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

Magnetic Flux
Understanding magnetic flux is crucial when delving into the principles of electromagnetism. Magnetic flux, represented by the Greek letter \( \Phi_B \), is a measure of the quantity of the magnetic field (or magnetic lines of force) passing through a given surface area. Imagine holding a hula hoop inside a magnetic field; the amount of field lines that penetrate the hoop relates to the magnetic flux through it.

The formula \( \Phi_B = 3(a t^3 - b t^2) \) describes the magnetic flux through the metal ring, with \( a \) and \( b \) being constants shaping its dependency on time \( t \). As time changes, the flux varies, leading us to the cornerstone of electromagnetic induction, which is pivotal for generators and transformers.
Induced Electromotive Force (EMF)
When the magnetic flux through a conductor changes, an electromotive force (EMF) is induced. This is known as Faraday's Law of Induction. In simple terms, it means that when a magnetic field around a wire shifts, electricity is produced.

In our exercise, by differentiating the magnetic flux in respect to time \( t \), \( \frac{d\Phi_B}{dt} \) gives us the rate at which this flux changes, leading to an induced EMF in the metal ring. Consequently, \( \varepsilon = -\frac{d\Phi_B}{dt} \) defines the relationship between the changing magnetic environment and the electricity generated, and this induced EMF is what eventually drives the current through the ring, exemplifying the law's predictive power in electromechanical systems.
Ohm's Law
Ohm's Law is one of the fundamental principles in the field of electricity. It states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. The formula is expressed as \( I = \frac{V}{R} \), where \( I \) is the current, \( V \) is the voltage (often referred to as EMF when dealing with induced currents), and \( R \) is the resistance.

In the context of our exercise, once an EMF is induced in the ring, Ohm's Law allows us to calculate the maximum current by dividing the maximal induced EMF by the known resistance of the ring. It is a simple yet powerful equation that connects electrical properties in a reliable and predictable manner.
Differentiation in Physics
Differentiation is a mathematical process used to calculate the rate of change of a quantity. In physics, it helps to understand how a system evolves over time. When we derive one variable with respect to another, we are effectively seeking the rate at which these variables change with respect to each other.

In the exercise, we differentiate the magnetic flux \( \Phi_B \) with respect to time \( t \) to find the rate of change of the magnetic flux, which is crucial for calculating the induced EMF using Faraday's Law. This operation translates the physical phenomena into a language that can be quantified and manipulated mathematically to predict real-world outcomes, such as the induced current in the metal ring.

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Most popular questions from this chapter

A rectangular coil of 60 turns, dimensions 0.100 \(\mathrm{m}\) by 0.200 \(\mathrm{m}\) and total resistance \(10.0 \Omega,\) rotates with angular speed 30.0 \(\mathrm{rad} / \mathrm{s}\) about the \(y\) axis in a region where a 1.00 - \(\mathrm{T}\) magnetic field is directed along the \(x\) axis. The rotation is initiated so that the plane of the coil is perpendicular to the direction of \(\mathbf{B}\) at \(t=0 .\) Calculate (a) the maximum induced emf in the coil, (b) the maximum rate of change of magnetic flux through the coil, \((c)\) the induced emf at \(t=0.0500\) s, and (d) the torque exerted by the magnetic field on the coil at the instant when the emf is a maximum.

The rotating loop in an AC generator is a square 10.0 \(\mathrm{cm}\) on a side. It is rotated at 60.0 \(\mathrm{Hz}\) in a uniform field of 0.800 T. Calculate (a) the flux through the loop as a function of time, (b) the emf induced in the loop, (c) the current induced in the loop for a loop resistance of \(1.00 \Omega,(\mathrm{d})\) the power delivered to the loop, and (e) the torque that must be exerted to rotate the loop.

An automobile has a vertical radio antenna 1.20 \(\mathrm{m}\) long. The automobile travels at 65.0 \(\mathrm{km} / \mathrm{h}\) on a horizontal road where the Earth's magnetic field is 50.0\(\mu \mathrm{T}\) directed toward the north and downward at an angle of \(65.0^{\circ}\) below the horizontal. (a) Specify the direction that the automobile should move in order to generate the maximum motional emf in the antenna, with the top of the antenna positive relative to the bottom. (b) Calculate the magnitude of this induced emf.

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 \(B=B_{\max } e^{-t / \tau},\) where \(B_{\max }\) and \(\tau\) are constants. The field has the constant value \(B_{\text { max }}\) for \(t<0\) . (a) Use Faraday's law to show that the emf induced in the loop is given by $$ \boldsymbol{\varepsilon}=\frac{A B_{\max }}{\tau} e^{-t / \tau} $$ (b) Obtain a numerical value for \(\varepsilon\) 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 \(A, B_{\text { max }},\) and \(\tau\) given in (b), what is the maximum value of \(\mathcal{E}\) ?

A solenoid wound with 2000 turns/m is supplied with current that varies in time according to \(I=\) \((4 \mathrm{A}) \sin (120 \pi t),\) where \(t\) is in seconds. A small coaxial circular coil of 40 turns and radius \(r=5.00 \mathrm{cm}\) is located inside the solenoid near its center. (a) Derive an expression that describes the manner in which the emf in the small coil varies in time. (b) At what average rate is energy delivered to the small coil if the windings have a total resistance of 8.00\(\Omega\) ?
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