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Chapter 30: Q. 35 (page 871)

At , the current in the $t = 0 s$ circuit in FIGURE $EX 30.35$ is $I_{0}$ . At what time is the current $\frac{1}{2} I_{0}$

Short Answer

Expert verified

Time for current $\frac{1}{2} I_{0}$ is $173 μs$ .

Step by step solution

01

Formula for current and time constant

Current decays across the inductor in a $LR$ circuit.

$I = I_{o} e^{- t / τ}$

localid="1648833013057" $τ$ is a constant in time.

Furthermore, the temporal constant is given by

$τ = \frac{L}{R}$

02

Calculation for time for current12I0

As it stands,

$R_{eq} = 300 Ω$

$L = 75 \times 10^{- 3} H$

all values should be substituted,

τ = \frac{L}{R} = \frac{75 \times 10^{- 3}}{300 Ω}

$I = \frac{I_{o}}{2}$ Then,

$I = I_{o} e^{- t / τ}$

$\frac{I_{o}}{2} = I_{o} e^{- t / 250 μs}$

$\ln 0.5 = \frac{- t}{250 μs}$

$t = 173 μs$

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Most popular questions from this chapter

A bar magnet is pushed toward a loop of wire as shown in FIGURE Q30.7. Is there a current in the loop? If so, in which direction? If not, why not?

The current in the solenoid of FIGURE EX30.12 is increasing. The solenoid is surrounded by a conducting loop. Is there a current in the loop? If so, is the loop current cw or ccw?

The earth's magnetic field strength is $5.0 \times 10^{- 5} T$ . How fast would you have to drive your car to create a $1.0 V$ motional emf along your $1.0 - m$ -tall radio antenna? Assume that the motion of the antenna is perpendicular to $\vec{B}$ .

56. II Your camping buddy has an idea for a light to go inside your CALC tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a $0.20$ T field between two pole tips $10 cm$ apart. His idea is to build the hand-cranked generator shown in FIGURE P30.56. He thinks you can make enough current to fully light a $1.0 Ω$ lightbulb rated at $4.0 W$ . That's not super bright, but it should be plenty of light for routine activities in the tent.

a. Find an expression for the induced current as a function of time if you turn the crank at frequency f Assume that the semicircle is at its highest point at $t = 0 s$ .

b. With what frequency will you have to turn the crank tor the maximum current to fully light the bulb? Is this feasible?

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

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

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

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