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Chapter 30: Q. 69 (page 874)

69. II The current through inductance $L$ is given by $I = I_{0} \sin 蝇 t$ .
CALC a. Find an expression for the potential difference $螖 V_{L}$ across the inductor.
b. The maximum voltage across the inductor is $0.20 V$ when $L = 50 渭 H$ and $f = 500 kHz$ . What is $I_{0}$ ?

Short Answer

Expert verified

Part (a) The expression for potential difference across the inductor is $螖 V_{L} = 鈭� L 蝇 I_{e} \cos 蝇 t$

Part (b) The maximum voltage across the inductor is $1.27 脳 10^{鈭� 3} A$

Step by step solution

01

Part (a)

The current in the inductor at any instant of time is $I = I_{e} \sin 蝇 t$

Here, $蝇$ is the angular frequency, $t$ is the time taken by the current in the inductor, and $I_{e}$ is the maximum current in the inductor.

Differentiate the equation with respect to time $t$ .

$\frac{d I}{d t} = \frac{d}{d t} (I_{e} \sin 蝇 t)$

$= 蝇 I_{0} \cos 蝇 t$

The potential difference across the inductance is

role="math"

螖 V_{L} = 鈭� L \frac{d I}{d t} 鈭� L 蝇 I_{o} \cos 蝇 t

Therefore, the expression for potential difference across the inductor is

$螖 V_{L} = 鈭� L 蝇 I_{e} \cos 蝇 t$

02

Part (b)

Convert the units for inductance from to .

Convert the units for frequency from to

The maximum cosine value varies from +1 to -1 . So the maximum potential difference across the inductor is,

Rewrite the equation for current

Therefore, the maximum voltage across the inductor is

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Most popular questions from this chapter

70. II The current through inductance $L$ is given by $I = I_{0} e^{鈭� t / 蟿}$ .

CALC a. Find an expression for the potential difference $螖 V_{L}$ across the inductor.

b. Evaluate $螖 V_{L}$ at $t = 0, 1.0$ , and $3.0 ms$ if $L = 20 mH, I_{0} = 50 mA$ , and $蟿 = 1.0 ms$ .

81. III In recent years it has been possible to buy a $1.0 F$ capacitor. This is an enormously large amount of capacitance. Suppose you want to build a $1.0 Hz$ oscillator with a $1.0 F$ capacitor. You have a spool of 0.25-mm-diameter wire and a $4.0 鈭� cm$ -diameter plastic cylinder. How long must your inductor be if you wrap it with 2 layers of closely spaced turns?

An $8.0 cm 脳 8.0 cm$ square loop is halfway into a magnetic CALC field perpendicular to the plane of the loop. The loop's mass is and its resistance is $0.010 惟$ . A switch is closed at $t = 0 s$ , causing the magnetic field to increase from $0$ to $1.0 T$ in $0.010 s$ .

a. What is the induced current in the square loop?

b. With what speed is the loop "kicked" away from the magnetic field?

Hint: What is the impulse on the loop?

What value of resistor $R$ gives the circuit in FIGURE $EX 30.34$ a time constant of $25 渭蝉$ $?$

FIGURE shows a U-shaped conducting rail that is oriented vertically in a horizontal magnetic field. The rail has no electric resistance and does not move. A slide wire with mass $m$ and resistance $R$ can slide up and down without friction while maintaining electrical contact with the rail. The slide wire is released from rest.

a. Show that the slide wire reaches a terminal speed $v_{term}$ , and find an expression for $v_{term}$ .

b. Determine the value of $v_{term}$ if $l = 20 cm, m = 10 g, R =$ $0.10 惟$ and $B = 0.50 T$

See all solutions

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