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Chapter 29: Problem 62

A wedding ring (of diameter \(2.0 \mathrm{~cm}\) ) is tossed into the air and given a spin, resulting in an angular velocity of 17 rotations per second. The rotation axis is a diameter of the ring. Taking the magnitude of the Earth's magnetic field to be \(4.0 \cdot 10^{-5} \mathrm{~T}\), what is the maximum induced potential difference in the ring?

Short Answer

Expert verified
Answer: The maximum induced potential difference in the wedding ring is approximately \(5.32 \times 10^{-7}\,\text{V}\).

Step by step solution

01

Convert the angular velocity to radians per second

First, we need to convert the given angular velocity from rotations per second (17 rotations/s) to radians per second. Since there are \(2\pi\) radians in one full rotation, we can multiply the given angular velocity by \(2\pi\): \(\omega = 17\,\text{rotations/s} \times 2\pi\,\text{radians/rotation} = 34\pi\,\text{radians/s}\)
02

Calculate the linear velocity of the ring

Next, we need to calculate the linear velocity of the ring. The linear velocity (\(v\)) is related to the angular velocity (\(\omega\)) and the radius (\(r\)) of the ring by the formula: \(v = \omega r\) The radius of the ring can be calculated as half the diameter, which is 1 cm or 0.01 m. Therefore, the linear velocity is: \(v = (34\pi\,\text{radians/s})(0.01\,\text{m}) = 0.34\pi\,\text{m/s}\)
03

Calculate the area of the ring

To find the magnetic flux through the ring, we need to calculate the area of the circular loop. The area of a circle is given by the formula: \(A = \pi r^2\) With the radius (\(r\)) being 0.01 m, the area is: \(A = \pi (0.01\,\text{m})^2 = 3.14 \times 10^{-4}\,\text{m}^2\)
04

Calculate the change in magnetic flux

The magnetic flux (\(\Phi_{B}\)) is given by the product of the magnetic field (\(B\)), the area (\(A\)), and the cosine of the angle between the magnetic field and the area vector (\(\cos\theta\)). In this problem, the angle between the magnetic field and the area vector changes as the ring rotates. At the maximum change in magnetic flux, the angle between the magnetic field and area vector changes by \(\pi\) radians as the ring rotates from a position where it is parallel to the magnetic field to a position where it is antiparallel. The change in magnetic flux is: \(\Delta \Phi_{B} = BA(\cos (0) - \cos (\pi)) = BA(1 - (-1)) = 2BA\) The given magnetic field is \(4.0 \cdot 10^{-5}\,\text{T}\), so the change in magnetic flux is: \(\Delta \Phi_{B} = 2(4.0 \cdot 10^{-5}\,\text{T})(3.14 \times 10^{-4}\,\text{m}^2) = 2.5136 \times 10^{-8}\,\text{Wb}\)
05

Calculate the maximum induced potential difference using Faraday's law

Now we can use Faraday's law to calculate the maximum induced potential difference (EMF) in the ring: \(EMF = -\frac{d\Phi_{B}}{dt}\) In this case, we can approximate the maximum induced EMF by dividing the change in magnetic flux by the time it takes for the ring to rotate from parallel to antiparallel to the magnetic field. The time period for one complete rotation is given by: \(T = \frac{1}{\text{angular velocity}} = \frac{1}{34\pi\,\text{radians/s}}\) The time it takes for the ring to rotate from parallel to antiparallel is half the time period: \(t = \frac{T}{2} = \frac{1}{68\pi\,\text{radians/s}}\) Finally, the maximum induced EMF is: \(EMF = \frac{2.5136 \times 10^{-8}\,\text{Wb}}{\frac{1}{68\pi\,\text{radians/s}}} = 5.32 \times 10^{-7}\,\text{V}\) Thus, the maximum induced potential difference in the wedding ring is approximately \(5.32 \times 10^{-7}\,\text{V}\).

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Key Concepts

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

Faraday's Law
If you've ever wondered how electricity is generated through motion, Faraday's Law is the fundamental principle to understand. Named after the renowned scientist Michael Faraday, the law deals with the concept of electromagnetic induction. Simply put, it states that a change in magnetic environment of a coil of wire will cause a voltage, or emf (electromotive force), to be induced in the coil.

Mathematically expressed, Faraday's Law is represented as:
\[EMF = -N\frac{d\text{桅}_B}{dt}\]
where EMF is the induced potential difference, N represents the number of turns of wire, and 诲桅叠/诲迟 denotes the rate of change of magnetic flux through the wire. The negative sign alludes to Lenz's Law, indicating the direction of the induced EMF opposes the change in magnetic flux causing it.

This phenomenon is exploited in various applications, from the generators producing electricity in power plants to the principle behind transformers used in power distribution.
Magnetic Flux
Imagine a loop of wire in a magnetic field: how much of this magnetic field passes through the loop? The answer lies in the concept of magnetic flux. It's quantified as the product of the magnetic field and the loop's area through which the field lines pass, and it's vital for understanding electromagnetic phenomena.

Mathematically, it's given by:
\[\text{桅}_B = B \times A \times \text{cos}(\theta)\]
where 桅叠 is the magnetic flux, B is the magnitude of the magnetic field, A is the area of the loop, and 胃 is the angle between the magnetic field and the normal to the loop's surface. A change in any of these factors鈥攆ield strength, area, or angle鈥攃an induce a potential difference as described by Faraday's Law.

For example, in the exercise mentioned, as the ring spins, there is a change in the angle 胃 affecting the magnetic flux, which, in turn, leads to the generation of an induced potential difference in the ring.
Angular Velocity
Think of angular velocity as the rotational equivalent of linear velocity. It represents the rate at which an object spins around a fixed axis. It's crucial for understanding motions in everything from car engines to the Earth's rotation.

It is denoted by the symbol 蝇 and is often measured in radians per second (rad/s). Since there are 2蟺 radians in a full circle (360掳), you can convert revolutions per second to radians per second by multiplying by 2蟺.

The formula is:
\[\omega = \frac{d\theta}{dt}\]
where 蝇 is the angular velocity, 诲胃 is the infinitesimal change in angle, and dt is the change in time. In our exercise, the angular velocity of the wedding ring determines how quickly the ring's position changes relative to the magnetic field, affecting the rate of change of magnetic flux and, by extension, the magnitude of the induced EMF.

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Most popular questions from this chapter

A wire of length \(\ell=10.0 \mathrm{~cm}\) is moving with constant velocity in the \(x y\) -plane; the wire is parallel to the \(y\) -axis and moving along the \(x\) -axis. If a magnetic field of magnitude \(1.00 \mathrm{~T}\) is pointing along the positive \(z\) -axis, what must the velocity of the wire be in order to induce a potential difference of \(2.00 \mathrm{~V}\) across it?

A \(100 .-V\) battery is connected in series with a \(500 .-\Omega\) resistor. According to Faraday's Law of Induction, current can never change instantaneously, so there is always some "stray" inductance. Suppose the stray inductance is \(0.200 \mu \mathrm{H}\). How long will it take the current to build up to within \(0.500 \%\) of its final value of \(0.200 \mathrm{~A}\) after the resistor is con- nected to the battery?

Faraday's Law of Induction states a) that a potential difference is induced in a loop when there is a change in the magnetic flux through the loop. b) that the current induced in a loop by a changing magnetic field produces a magnetic field that opposes this change in magnetic field. c) that a changing magnetic field induces an electric field. d) that the inductance of a device is a measure of its opposition to changes in current flowing through it. e) that magnetic flux is the product of the average magnetic field and the area perpendicular to it that it penetrates.

An 8 -turn coil has square loops measuring \(0.200 \mathrm{~m}\) along a side and a resistance of \(3.00 \Omega\). It is placed in a magnetic field that makes an angle of \(40.0^{\circ}\) with the plane of each loop. The magnitude of this field varies with time according to \(B=1.50 t^{3}\), where \(t\) is measured in seconds and \(B\) in teslas. What is the induced current in the coil at \(t=2.00 \mathrm{~s} ?\)

A circular conducting loop with radius \(a\) and resistance \(R_{2}\) is concentric with a circular conducting loop with radius \(b \gg a(b\) much greater than \(a\) ) and resistance \(R_{1}\). A time-dependent voltage is applied to the larger loop, having a slow sinusoidal variation in time given by \(V(t)=V_{0} \sin \omega t\) where \(V_{0}\) and \(\omega\) are constants with dimensions of voltage and inverse time, respectively. Assuming that the magnetic field throughout the inner loop is uniform (constant in space) and equal to the field at the center of the loop, derive expressions for the potential difference induced in the inner loop and the current \(i\) through that loop.
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