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Chapter 30: Q58P (page 900)

Suppose the emf of the battery in the circuit shown in Figure varies with time t so that the current is given by i(t) = 3.0 +5.0 t , where i is in amperes and t is in seconds. Take R=4.0惟andL=5.0H, and find an expression for the battery emf as a function of t. (Hint: Apply the loop rule.)

Short Answer

Expert verified

Expression for the battery emf as a function of t is蔚=20t+42

Step by step solution

01

Given

i) Current isit=3.0+5.0t

ii) Resistance isR=4.0惟

iii) Inductance isL=6.0H

iv) Figure 30-16 is the RL circuit.

02

Understanding the concept

We use the concept of loop rule. Applying loop rule to the circuit, we can write the equation. Plugging the current equation, we can solve for emf.

Formulae:

危V=0

03

Find an expression for the battery emf as a function of t.

By applying the loop rule, we can write

蔚-Ldidt-iR=0蔚=Ldidt+iR

Plugging the given equation of current, we get

蔚=Ld3.0+5.0tdt+3.0+5.0tR蔚=L5.0+3.0+5.0tR

Using the values of inductance and resistance,

蔚=6.05.0+3.0+5.0t4.0蔚=30+12+20t蔚=20t+42

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Most popular questions from this chapter

A coil with an inductance of2.0 H and a resistance of 10惟is suddenly connected to an ideal battery with 蔚=100V. At 0.10 safter the connection is made, (a) what is the rate at which energy is being stored in the magnetic field? (b) what is the rate at which thermal energy is appearing in the resistance? (c) what is the rate at which energy is being delivered by the battery?

Figure 30-31 shows three situations in which a wire loop lies partially in a magnetic field.The magnitude of the field is either increasing or decreasing, as indicated. In each situation, a battery is part of the loop. In which situations are the induced emf and the battery emf in the same direction along the loop?

Two solenoids are part of the spark coil of an automobile. When the current in one solenoid falls from 6.0Ato zero in 2.5ms, an emf of 30 kVis induced in the other solenoid. What is the mutual inductance M of the solenoids?

The switch in the circuit of Fig. 30-15 has been closed on a for a very long time when it is then thrown to b. The resulting current through the inductor is indicated in Fig. 30-28 for four sets of values for the resistance R and inductance L: (1) R=R0, (2) 2R0=R, (3) R0and2L0 , (4) 2R0and2L0. Which set goes with which curve?

In Fig. 30-26, a wire loop has been bent so that it has three segments: segment bc(a quarter-circle), ac(a square corner), and ab(straight). Here are three choices for a magnetic field through the loop:

(1)B1鈫�=3i^+7j^-5tk^,(2)B2鈫�=5ti^-4j^-15k^,(3)B3鈫�=2i^-5tj^-12k^,

where Bis in milliteslas and tis in seconds. Without written calculation, rank the choices according to (a) the work done per unit charge in setting up the induced current and (b) that induced current, greatest first. (c) For each choice, what is the direction of the induced current in the figure?
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