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Chapter 30: Q. 74 (page 874)

74. II The inductor in FIGURE P30.74 is a $9.0 鈭� cm$ -long, $2.0 鈭� cm 鈭�$ diameter solenoid wrapped with 300 turns. What is the current in the circuit $10 渭 s$ after the switch is moved from a to b ?

Short Answer

Expert verified

Hence, the required current is $0.093 A$

Step by step solution

01

Step 1. Given information

The number of turns if 300, and length of inductor is 9 cm

The radius of the coil is,

$\begin{array}{l} r = \frac{d}{2} \\ r = \frac{2 cm (\frac{1 m}{100 cm})}{2} = 0.01 m \end{array}$

02

Step 2. Explanation

The inductance of the coil is,

$\begin{array}{l} L = \frac{渭_{0} N^{2} A}{l} \\ L = \frac{渭_{0} N^{2} 蟺 r^{2}}{l} \\ L = \frac{(4 蟺脳 10^{鈭� 7} T 鈰� m / A) {(300)}^{2} 蟺 {(0.01 m)}^{2}}{9 cm (\frac{1 m}{100 cm})} \\ = 3.94 脳 10^{鈭� 4} H \end{array}$

According to ohm's law, the initial current is,

$\begin{array}{l} I_{0} = \frac{V_{0}}{R} \\ I_{0} = \frac{6 V}{30 惟} = 0.2 A \end{array}$

The expression for the current in the coil is,

$\begin{array}{l} I = I_{0} e^{\frac{鈭� t}{蟿}} \\ I = I_{0} e^{\frac{鈭� t}{(\frac{L}{R})}} = I_{0} e^{\frac{鈭� R t}{L}} \\ I = I_{0} e^{\frac{鈭� R t}{L}} \\ I = (0.2 A) e^{鈭� (30 惟) (10 渭 s) (\frac{10^{鈭� 4} s}{1 / s})} \\ = 0.093 A \end{array}$

Hence, the required current is $0.093 A$

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Most popular questions from this chapter

The L-shaped conductor in $F I G U R E P 30.54$ moves at $10 m / s$ across and touches a stationary $L$ -shaped conductor in a $0.10 T$ magnetic field. The two vertices overlap, so that the enclosed area is zero, at $t = 0 s$ . The conductor has a resistance of $0.010$ ohms per meter.

a. What is the direction of the induced current?

b. Find expressions for the induced emf and the induced current as functions of time.

c. Evaluate $鈭�$ and $l$ at $t = 0.10 s$ .

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 $I_{0}$ ?

The switch in FIGURE P30.77 has been open for a long time. It is closed at $t = 0 s$ .

a. After the switch has been closed for a long time, what is the current in the circuit? Call this current $I_{0}$ .

b. Find an expression for the current $I$ as a function of time. Write your expression in terms of $I_{0,}, R, and L$ .

c. Sketch a current-versus-time graph from $t = 0 s$ until the current is no longer changing.

A 2.0 mH inductor is connected in parallel with a variable capacitor. The capacitor can be varied from 100 pF to 200 pF. What is the range of oscillation frequencies for this circuit?

The resistance of the loop in FIGURE EX30.15 is $0.20 惟$ Is the magnetic field strength increasing or decreasing? At what rate T/s ?

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