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Chapter 20: Problem 74

Suppose you wanted to use Earth's magnetic field to make an ac generator at a location where the magnitude of the field is \(0.50 \mathrm{mT}\). Your coil has 1000.0 turns and a radius of \(5.0 \mathrm{cm} .\) At what angular velocity would you have to rotate it in order to generate an emf of amplitude $1.0 \mathrm{V} ?$

Short Answer

Expert verified
Answer: The coil must be rotated with an angular velocity of approximately 1273 s⁻¹ to generate an emf of amplitude 1.0 V at the given location.

Step by step solution

01

Convert the units to MKS system

Convert the radius from centimeters to meters and Earth's magnetic field from millitesla (mT) to tesla (T). - Radius: \(5.0\,cm = 0.05\,m\) - Magnetic field: \(0.50\,mT = 0.0005\,T\)
02

Write Faraday's Law of electromagnetic induction

Faraday's Law states that the induced emf in a coil is equal to the rate of change of magnetic flux through the coil. The formula for the emf induced in a coil with N turns is: \(\epsilon = N \frac{d\Phi_{B}}{dt}\)
03

Calculate the magnetic flux through the coil

The magnetic flux through a circular loop of area A and magnetic field B in the direction perpendicular to the loop is given by: \(\Phi_{B} = B A\) Here, the area of the circular loop can be calculated as \(A = \pi r^2\).
04

Differentiate the magnetic flux with respect to time

By differentiating the magnetic flux with respect to time, we will obtain the rate of change of magnetic flux through the coil: \(\frac{d\Phi_{B}}{dt} = \frac{d(B A )}{dt}\) Keeping the value of B constant, the derivative of the area of the coil with respect to time becomes the variable part. As the coil is rotating, the area changes with respect to time, so we should find the area as a function of time, i.e., \(A(t) = A\sin(\omega t)\). Differentiating this with respect to time, we get: \(\frac{dA(t)}{dt} = A\omega\cos(\omega t)\)
05

Calculate angular velocity

To find the required angular velocity, plug the values of N, B, A, and the amplitude of the emf (1.0 V) back into the Faraday's Law equation: \(1.0\,V = 1000\,turns \times 0.0005\,T \times (\pi (0.05\,m)^2)\omega\cos(\omega t)\) For the peak emf, we have \(\cos(\omega t) = 1\). After simplifying the equation, we can solve for the angular velocity \(\omega\): \(\omega = \frac{1.0\,V}{1000\,turns \times 0.0005\,T \times (\pi (0.05\,m)^2)}\) After calculations, we get:
06

Final Answer

\(\omega \approx 1273\,\mathrm{s}^{-1}\) Hence, the coil must be rotated with an angular velocity of approximately 1273 s⁻¹ to generate an emf of amplitude 1.0 V at the given location.

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Most popular questions from this chapter

How much energy due to Earth's magnetic field is present in $1.0 \mathrm{m}^{3}$ of space near Earth's surface, at a place where the field has magnitude \(0.45 \mathrm{mT} ?\)
A transformer with 1800 turns on the primary and 300 turns on the secondary is used in an electric slot car racing set to reduce the input voltage amplitude of \(170 \mathrm{V}\) from the wall output. The current in the secondary coil is of amplitude \(3.2 \mathrm{A}\). What is the voltage amplitude across the secondary coil and the current amplitude in the primary coil?
The primary coil of a transformer has 250 turns; the secondary coil has 1000 turns. An alternating current is sent through the primary coil. The emf in the primary is of amplitude \(16 \mathrm{V}\). What is the emf amplitude in the secondary? (tutorial: transformer)
The component of the external magnetic field along the central axis of a 50 -turn coil of radius \(5.0 \mathrm{cm}\) increases from 0 to 1.8 T in 3.6 s. (a) If the resistance of the coil is \(2.8 \Omega,\) what is the magnitude of the induced current in the coil? (b) What is the direction of the current if the axial component of the field points away from the viewer?
Two solenoids, of \(N_{1}\) and \(N_{2}\) turns respectively, are wound on the same form. They have the same length \(L\) and radius \(r\) (a) What is the mutual inductance of these two solenoids? (b) If an ac current $$I_{1}(t)=I_{\mathrm{m}} \sin \omega t$$ flows in solenoid $$1\left(N_{1} \text { turns }\right)$$ write an expression for the total flux through solenoid \(2 .\) (c) What is the maximum induced emf in solenoid \(2 ?[\text { Hint: Refer to Eq. }(20-7) .]\)
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