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Chapter 30: Q95P (page 902)

In Fig. 30-77, $R_{1} = 8.0 惟, R_{2} = 10 惟, L_{1} = 0.30 H, L_{2} = 0.20 H$ and the ideal battery has $蔚 = 6.0 V$ . (a) Just after switch S is closed, at what rate is the current in inductor 1 changing? (b) When the circuit is in the steady state, what is the current in inductor 1?

Short Answer

Expert verified

a) The rate of change of current in inductor 1 is,

$\frac{d i_{b a t}}{d t} = 20 A / s$

b) The current in the inductor 1 is $i = 0.75 A$

Step by step solution

01

Given

$R_{1} = 8.0 惟 R_{2} = 10 惟 L_{2} = 0.20 H 蔚 = 6.0 V$

02

Understanding the concept

By using the equation 30-35, we can find the rate of current. After that by using the KVL

We can find the current in the inductor 1.

Formula:

$i_{b a t} = \frac{蔚}{R} (1 - e^{\frac{R t}{L}}) \frac{d i_{b a t}}{d t} = \frac{|蔚|}{L_{1}}$

03

(a) Calculate the rate of change of current in inductor 1.

We have, equation for self-induced emf as,

$L_{1} \frac{d i_{b a t}}{d t} = |蔚|$

So that the rate of current is,

$\frac{d i_{b a t}}{d t} = \frac{|蔚|}{L_{1}}$

By substituting the value

$\frac{d i_{b a t}}{d t} = \frac{6.0}{0.30} \frac{d i_{b a t}}{d t} = 20 A / s$

04

(b) Calculate the current in the inductor 1 at steady state.

Now we have to find the current in inductor 1 so that by Appling the KVL for outer loop we can writer as

$蔚 - {iR}_{1} - |蔚_{L_{1}}| - |蔚_{L_{2}}| = 0$

But as $t 鈫� 鈭� e m f$ in the coil vanishes

$蔚 - i R_{1} = 0 蔚 - {iR}_{1} i = \frac{蔚}{R_{1}} i = \frac{6.0}{8.0} i = 0.75 A$

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Most popular questions from this chapter

Two solenoids are part of the spark coil of an automobile. When the current in one solenoid falls from $6.0 A$ to zero in 2.5ms, an emf of 30 kVis induced in the other solenoid. What is the mutual inductance M of the solenoids?

A coil with 150turns has a magnetic flux of $50.0 n T . m^{2}$ through each turn when the current is 2.00mA . (a) What is the inductance of the coil? What are the (b) inductance and (c) flux through each turn when the current is increased to i = 4.00mA ? (d) What is the maximum emf across the coil when the current through it is given by i= (3.00mA)cos(377 t) , with t in seconds?

At a given instant the current and self-induced emf in an inductor are directed as indicated in Figure. (a) Is the current increasing or decreasing? (b) The induced emf is 17 V, and the rate of change of the current is 25 KA/s; find the inductance.

If the circular conductor in Fig. 30-21 undergoes thermal expansion while it is in a uniform magnetic field, a current is induced clockwise around it. Is the magnetic field directed into or out of the page?

Figure (a) shows, in cross section, two wires that are straight, parallel, and very long. The ratio $i_{1} / i_{2}$ of the current carried by wire 1 to that carried by wire 2 is $\frac{1}{3}$ . Wire 1 is fixed in place. Wire 2 can be moved along the positive side of the x-axis so as to change the magnetic energy density u_B set up by the two currents at the origin. Figure (b) gives u_B as a function of the position x of wire 2. The curve has an asymptote of $u_{B} = 1.96 nJ / m^{3} as x 鈫� 鈭�$ , and the horizontal axis scale is set by $x_{s} = 60.0 cm$ . What is the value of (a) i₁ and (b) i₂?

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