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Chapter 30: Q50P (page 899)

The current in an RL circuit builds up to one-third of its steady-state value in 5.00 s . Find the inductive time constant.

Short Answer

Expert verified

$蟿_{L} = 12.3 s$

Step by step solution

01

Given

02

Understanding the concept

$i_{m} - current at steady value$ If a constant emf $蔚$ is introduced into a single-loop circuit containing a resistance Rand an inductance L, the current rises to an equilibrium value of $蔚 / R$ as

$i = \frac{蔚}{R} (1 - e^{- \frac{t}{蟿_{L}}})$

Formulae:

$i = \frac{蔚}{R} (1 - e^{- \frac{t}{蟿_{L}}})$

Here, $蔚 - emf; R - resistance; t - time; 蟿_{L} - inductive time constant of circuit$

Given:

At $t = 5.0 s, i = i_{m} / 3$

03

Calculate the inductive time constant

We have current in the inductive circuit as\

$i = \frac{蔚}{R} (1 - e^{- \frac{t}{蟿_{L}}})$

We know that steady state current of RL circuit is

$i_{m} = 蔚 / R$

Now plug in this value in equation:

$\frac{i_{m}}{3} = \frac{蔚}{R} (1 - e^{- \frac{t}{蟿_{L}}}) \frac{蔚}{3 R} = \frac{蔚}{R} (1 - e^{- \frac{t}{蟿_{L}}}) e^{- \frac{t}{蟿_{L}}} = 1 - \frac{1}{3} = 2 / 3$

Hence,

$- \frac{t}{蟿_{L}} = I n (\frac{2}{3}) - \frac{- 0.5}{蟿_{L}} = - 0.4054$

Therefore,

$蟿_{L} = 12.3 s$

The inductive time constant is 12.3s

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Most popular questions from this chapter

A rectangular coil of N turns and of length a and width b is rotated at frequency f in a uniform magnetic field, as indicated in Figure. The coil is connected to co-rotating cylinders, against which metal brushes slide to make contact. (a) Show that the emf induced in the coil is given (as a function of time t) by $蔚 = 2 蟿蟿 f N a b s i n (2 蟿蟿 f t) = 蔚_{0} s i n (2 蟿蟿 f t)$ . This is the principle of the commercial alternating-current generator. (b) What value of Nabgives an emf with $蔚_{0} 150 V$ when the loop is rotated at $60.0 \frac{rev}{s}$ in a uniform magnetic field of 0.500 T?

:Inductors in parallel. Two inductors L₁ and L₂ are connected in parallel and separated by a large distance so that the magnetic field of one cannot affect the other. (a)Show that the equivalent inductance is given by

$\frac{1}{L_{eq}} = \frac{1}{L_{2}} + \frac{1}{L_{2}}$

(Hint: Review the derivations for resistors in parallel and capacitors in parallel. Which is similar here?) (b) What is the generalization of (a) for N inductors in parallel?

In Figure, a wire loop of lengths L = 40.0 cmand W = 25.0 cmlies in a magnetic field $\overset{鈫�}{B}$ .(a)What is the magnitude $蔚$ if $\overset{鈫�}{B} = (4.00 脳 10^{- 2} \frac{T}{m}) y \hat{k}$ ?(b)What is the direction (clockwise or counterclockwise鈥攐r 鈥渘one鈥� if 0) of the emf induced in the loop if $\overset{鈫�}{B} = (4.00 脳 10^{- 2} \frac{T}{m}) y \hat{k}$ ?(c)What is the $蔚$ if $\overset{鈫�}{B} = (6.00 脳 10^{- 2} \frac{T}{s}) t \hat{k}$ (d)what is the direction if $\overset{鈫�}{B} = (6.00 脳 10^{- 2} \frac{T}{s}) t \hat{k}$ (e)What is the $蔚$ if $\overset{鈫�}{B} = (8.00 脳 10^{- 2} \frac{T}{m . s}) y t \hat{k}$ (f)What is the direction if $\overset{鈫�}{B} = (6.00 脳 10^{- 2} \frac{T}{s}) t \hat{k}$ (g)What is the $蔚$ if $\overset{鈫�}{B} = (3.00 脳 10^{- 2} \frac{T}{m . s}) x t \hat{k}$ (h)What is the direction if $\overset{鈫�}{B} = (3.00 脳 10^{- 2} \frac{T}{m . s}) x t \hat{k}$ (i)What is the if $\overset{鈫�}{B} = (5.00 脳 10^{- 2} \frac{T}{m . s}) y t \hat{k}$ (j)What is the direction if $\overset{鈫�}{B} = (5.00 脳 10^{- 2} \frac{T}{m . s}) y t \hat{k}$

Coil 1 has $L_{1} = 25 mH$ and $N_{1} = 100 turns$ . Coil 2 has $L_{2} = 40 mH$ and $N_{2} = 200 turns$ . The coils are fixed in place; their mutual inductance M is $M = 3.0 mH$ . A 6.0mAcurrent in coil 1 is changing at the rate of 4.0 A/s. (a) What magnetic flux $蠁_{12}$ links coil 1? (b) what self-induced emf appears in that coil? (c) What magnetic flux $蠁_{21}$ links coil 2? (d) what mutually induced emf appears in that coil?

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?

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