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Chapter 29: Problem 23

When a magnet in an MRI is abruptly shut down, the magnet is said to be quenched. Quenching can occur in as little as \(20.0 \mathrm{~s}\). Suppose a magnet with an initial field of \(1.20 \mathrm{~T}\) is quenched in \(20.0 \mathrm{~s},\) and the final field is approximately zero. Under these conditions, what is the average induced potential difference around a conducting loop of radius \(1.00 \mathrm{~cm}\) (about the size of a wedding ring) oriented perpendicular to the field?

Short Answer

Expert verified
Answer: The average induced potential difference around the conducting loop is approximately \(1.89 \times 10^{-5}\) volts.

Step by step solution

01

Define the given variables and constants

We are given: Initial magnetic field, \(B_1 = 1.20\) T Final magnetic field, \(B_2 \approx 0\) T Time to quench the magnet, \(t = 20.0\) s Radius of the conducting loop, \(r = 1.00\) cm \(= 0.01\) m
02

Define the formula for Faraday's Law

Faraday's Law states that the induced electromotive force (EMF) is equal to the rate of change of magnetic flux through a closed loop. $$ \varepsilon = - \frac{\Delta \Phi}{\Delta t} $$
03

Calculate the change in magnetic flux

The magnetic flux \(\Phi\) through a loop of area A perpendicular to the magnetic field is given by: $$ \Phi = B\cdot A $$ Since the loop is a circle, its area can be calculated using $$ A = \pi r^2 $$ The change in magnetic flux \(\Delta \Phi\) is given by the difference between the initial and final fluxes: $$ \Delta \Phi = \Phi_2 - \Phi_1 = B_2A - B_1A $$ As \(B_2 \approx 0\), $$ \Delta \Phi = -B_1A = -1.20 T \cdot \pi (0.01 m)^2 $$
04

Calculate the average induced potential difference

Now, we can calculate the average potential difference (induced EMF) using Faraday's Law: $$ \varepsilon = - \frac{\Delta \Phi}{\Delta t} = \frac{[1.20 \mathrm{T} \cdot \pi (0.01 \mathrm{m})^2]}{20.0 \mathrm{s}} $$ Calculating the value, $$ \varepsilon \approx 1.89 \times 10^{-5} V $$ So, the average induced potential difference around the conducting loop is approximately \(1.89 \times 10^{-5}\) volts.

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Most popular questions from this chapter

A motor has a single loop inside a magnetic field of magnitude \(0.87 \mathrm{~T}\). If the area of the loop is \(300 \mathrm{~cm}^{2}\), find the maximum angular speed possible for this motor when connected to a source of emf providing \(170 \mathrm{~V}\).

A solenoid with 200 turns and a cross-sectional area of \(60 \mathrm{~cm}^{2}\) has a magnetic field of \(0.60 \mathrm{~T}\) along its axis. If the field is confined within the solenoid and changes at a rate of \(0.20 \mathrm{~T} / \mathrm{s}\), the magnitude of the induced potential difference in the solenoid will be a) \(0.0020 \mathrm{~V}\). b) \(0.02 \mathrm{~V}\). c) \(0.001 \mathrm{~V}\). d) \(0.24 \mathrm{~V}\).

A magnetar (magnetic neutron star) has a magnetic field near its surface of magnitude \(4.0 \cdot 10^{10} \mathrm{~T}\) a) Calculate the energy density of this magnetic field. b) The Special Theory of Relativity associates energy with any mass \(m\) at rest according to \(E_{0}=m c^{2}(\) more on this in Chapter 35 ). Find the rest mass density associated with the energy density of part (a).

A rectangular wire loop (dimensions of \(h=15.0 \mathrm{~cm}\) and \(w=8.00 \mathrm{~cm}\) ) with resistance \(R=5.00 \Omega\) is mounted on a door. The Earth's magnetic field, \(B_{\mathrm{E}}=2.6 \cdot 10^{-5} \mathrm{~T}\), is uniform and perpendicular to the surface of the closed door (the surface is in the \(x z\) -plane). At time \(t=0,\) the door is opened (right edge moves toward the \(y\) -axis) at a constant rate, with an opening angle of \(\theta(t)=\omega t,\) where \(\omega=3.5 \mathrm{rad} / \mathrm{s}\) Calculate the direction and the magnitude of the current induced in the loop, \(i(t=0.200 \mathrm{~s})\).

A steel cylinder with radius \(2.5 \mathrm{~cm}\) and length \(10.0 \mathrm{~cm}\) rolls without slipping down a ramp that is inclined at \(15^{\circ}\) above the horizontal and has a length (along the ramp) of \(3.0 \mathrm{~m} .\) What is the induced potential difference between the ends of the cylinder at the bottom of the ramp, if the surface of the ramp points along magnetic north?
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