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Chapter 20: Problem 22

Verify that, in SI units, \(\Delta \Phi_{\mathrm{B}} / \Delta t\) can be measured in volts - in other words, that $1 \mathrm{Wb} / \mathrm{s}=1 \mathrm{V}$

Short Answer

Expert verified
Question: Verify that the rate of change of magnetic flux can be measured in volts, specifically that 1 Wb/s = 1 V. Answer: Following Faraday's law of electromagnetic induction and comparing the units of voltage with magnetic flux and time, we have verified that the rate of change of magnetic flux (1 Wb/s) is equivalent to 1 V in SI units.

Step by step solution

01

Write the definition of a volt in terms of electromotive force (EMF)

According to Faraday's law of electromagnetic induction, the electromotive force (EMF) induced in a circuit is directly proportional to the rate of change of magnetic flux through the circuit. Mathematically, it is represented by: $$\varepsilon = -\frac{\Delta \Phi_{\mathrm{B}}}{\Delta t}$$ Where \(\varepsilon\) is EMF, \(\Delta \Phi_{\mathrm{B}}\) is the change in magnetic flux, \(\Delta t\) is the change in time, and the negative sign indicates the direction of the induced EMF according to Lenz's law.
02

Define the units of measurements involved in the formula

We are given that magnetic flux \(\Phi_{\mathrm{B}}\) is measured in Weber (Wb) and time \(t\) is measured in seconds (s). The unit for EMF or voltage (V) is volts.
03

Compare the units of the formula with SI units of volts

From the formula derived in Step 1, we can write the units of voltage in terms of magnetic flux and time: $$1\mathrm{V} = \frac{1\mathrm{Wb}}{1\mathrm{s}}$$ This relation shows that the rate of change of magnetic flux \(1\mathrm{Wb} / \mathrm{s}\) is equivalent to \(1\mathrm{V}\) in SI units.
04

Conclusion

We have successfully verified that, in SI units, the rate of change of magnetic flux \(\Delta \Phi_{\mathrm{B}} / \Delta t\) can be measured in volts, or in other words, that \(1 \mathrm{Wb} / \mathrm{s} = 1 \mathrm{V}\).

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Most popular questions from this chapter

A transformer with a primary coil of 1000 turns is used to step up the standard \(170-\mathrm{V}\) amplitude line voltage to a \(220-\mathrm{V}\) amplitude. How many turns are required in the secondary coil?
The windings of an electromagnet have inductance \(L=8.0 \mathrm{H}\) and resistance \(R=2.0 \Omega . \quad A \quad 100.0-V \quad d c\) power supply is connected to the windings by closing switch \(S_{2} .\) (a) What is the current in the windings? (b) The electromagnet is to be shut off. Before disconnecting the power supply by opening switch \(S_{2},\) a shunt resistor with resistance \(20.0 \Omega\) is connected in parallel across the windings. Why is the shunt resistor needed? Why must it be connected before the power supply is disconnected? (c) What is the maximum power dissipated in the shunt resistor? The shunt resistor must be chosen so that it can handle at least this much power without damage. (d) When the power supply is disconnected by opening switch \(S_{2},\) how long does it take for the current in the windings to drop to \(0.10 \mathrm{A} ?\) (e) Would a larger shunt resistor dissipate the energy stored in the electromagnet faster? Explain.
A transformer for an answering machine takes an ac voltage of amplitude $170 \mathrm{V}\( as its input and supplies a \)7.8-\mathrm{V}$ amplitude to the answering machine. The primary has 300 turns. (a) How many turns does the secondary have? (b) When idle, the answering machine uses a maximum power of \(5.0 \mathrm{W}\). What is the amplitude of the current drawn from the 170 -V line?
An ideal solenoid ( \(N_{1}\) turns, length \(L_{1},\) radius \(r_{1}\) ) is placed inside another ideal solenoid ( \(N_{2}\) turns, length \(L_{2}>L_{1}\), radius \(r_{2}>r_{1}\) ) such that the axes of the two coincide. (a) What is the mutual inductance? (b) If the current in the outer solenoid is changing at a rate \(\Delta I_{2} / \Delta t,\) what is the magnitude of the induced emf in the inner solenoid?
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?
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