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Chapter 30: Q46P (page 899)

The current i through a 4.6 Hinductor varies with time t as shown by the graph of Figure, where the vertical axis scale is set by $i_{s} = 8.0 A$ and the horizontal axis scale is set by $t_{s} = 6.0 m s$ . The inductor has a resistance of $12 惟$ .(a) Find the magnitude of the induced emf $蔚$ during time intervals 0 to 2 ms. (b) Find the magnitude of the induced emf $蔚$ during time intervals 2 ms to 5 ms. (c) Find the magnitude of the induced emf $蔚$ during time intervals 5 ms to 6 ms. (Ignore the behavior at the ends of the intervals.)

Short Answer

Expert verified

$蔚 = 1.6 x 10^{4} V$
$蔚 = 3.1 x 10^{3} V$
$蔚 = 2.3 x 10^{4} V$

Step by step solution

01

Given

$i_{s} = 8 A t_{s} = 6 m s$

02

Understanding the concept

According to Faraday鈥檚 law, the emf induced in a circuit is proportional to the time rate of change of the magnetic flux linking that circuit.

Formulae:

Induced emfis given by

$蔚 = L \frac{d i}{d t}$

03

(a) Calculate the induced emf for a time interval 0 to 2 ms

Induced emf $蔚$ is given by

$\overset{,}{o} = L \frac{d i}{d t} i = 0 t o 7 A 蔚 = L \frac{d i}{d t} = \frac{4.6 脳 (7 - 0)}{(2 - 0) 脳 10^{- 3}} 蔚 = 1.6 脳 10^{4} V$

04

(b) Calculate the induced emf for time interval 2ms to5ms

For time interval 2ms to 5ms

$1 = 7 t 0 5 A 蔚 = L \frac{d i}{d t} = \frac{4.6 脳 (5 - 7)}{(5 - 2) 脳 10^{- 3}} 蔚 = 3.1 脳 10^{3} 鈥� V$

05

(c) Calculate the induced emf for time interval 5ms to 6ms

For time interval 5ms to 6ms

$1 = 0 t o 5 A 蔚 = L \frac{d i}{d t} = \frac{4.6 脳 (5 - 0)}{(6 - 5) 脳 10^{- 3}} 蔚 = 2.3 脳 10^{4} V$

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Most popular questions from this chapter

A uniform magnetic field $\overset{鈫�}{B}$ is perpendicular to the plane of a circular loop of diameter 10 cmformed from wire of diameter2.5 mm and resistivity $1.69 脳 10^{8} 惟 - m$ . At what rate must the magnitude of $\overset{鈫�}{B}$ change to induce a 10Acurrent in the loop?

A solenoid having an inductance of $6.30 渭贬$ is connected in series with a $1.20 k 惟$ resistor. (a) If a 14.0Vbattery is connected across the pair, how long will it take for the current through the resistor to reach 80.0% of its final value?

(b) What is the current through the resistor at time $t = 1.0 蟿_{L}$ ?

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 $\overset{鈫�}{B} = (5.00 脳 10^{- 2} \frac{T}{m . s}) y t \hat{k}$

A square wire loop with 2.00msides is perpendicular to a uniform magnetic field, with half the area of the loop in the field as shown in Figure. The loop contains an ideal battery with emf $蔚 = 20.0 V$ . If the magnitude of the field varies with time according to $B = 0.0420 - 0.870 t$ , with B in Tesla and t in seconds, (a)what is the net emf in the circuit?(b)what is the direction of the (net) current around the loop?

A small loop of area 6.8 ${mm}^{2}$ is placed inside a long solenoid that hasand carries a sinusoidally varying current i of amplitude1.28 A and angular frequency rad/s.The central axes of the loop and solenoid coincide. What is the amplitude of the emf induced in the loop?

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