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Chapter 32: Q. 41 (page 925)

a. For an RC circuit, find an expression for the angular frequency at which $V_{C} = \frac{1}{2} 蔚_{0}$ .
b. What is V_R at this frequency ?

Short Answer

Expert verified

(a) The expression for angular frequency is $蝇 = \frac{\sqrt{3}}{R C}$

(b) The expression for voltage across the resistor is $V_{R} = \frac{\sqrt{3}}{2} 蔚_{0}$

Step by step solution

01

Part (a) Step 1 : Given information

We are given that for an RC circuit, the voltage across a capacitor $V_{C} = \frac{1}{2} 蔚_{0}$ and

we have to find out the angular frequency $蝇$ .

02

Part (a) Step 2 : Simplification

As the expression for impedance of the circuit is,

$R = \frac{\sqrt{3}}{2} Z$

On rearranging we get,

$Z = \frac{\sqrt{3}}{2} R$

Now, the expression for angular frequency is,

$Z^{2} = R^{2} + \frac{1}{{(蝇 C)}^{2}}$

Put the value of Z,

${(\frac{2 R}{\sqrt{3}})}^{2} = R^{2} + \frac{1}{蝇^{2} C^{2}}$

Simplify it, $\frac{1}{蝇^{2} C^{2}} = (\frac{4}{3} - 1) R^{2}$

$蝇^{2} = \frac{3}{R^{2} C^{2}}$

On solving we get,

$蝇 = \frac{\sqrt{3}}{R C}$

03

Part (b) Step 1: Given Information

We are given that the voltage across a capacitor $V_{c} = \frac{1}{2} 蔚_{0}$ and

we have to find out the $V_{R}$ i.e., potential across resistor.

04

Part (b) Step 2: Simplification

As we know,

The expression for voltage across resistor is,

$蔚_{0}^{2} = V_{R}^{2} + V_{C}^{2}$

Put the value of $V_{C}$

$蔚_{0}^{2} = V_{R}^{2} + {(\frac{蔚_{0}}{2})}^{2}$

On simplifying, $V_{R}^{2} = \frac{3}{4} 蔚_{0}^{2}$

We get $V_{R} = \frac{\sqrt{3}}{2} 蔚_{0}$

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Most popular questions from this chapter

FIGURE EX32.24 shows voltage and current graphs for an inductor.

a. What is the emf frequency $f$ ?

b. What is the value of the inductance $L$ ?

The resonance frequency of a series $R L C$ circuit is $1000$ Hz. What is the resonance frequency if:

a. The resistance $R$ is doubled?

b. The inductance $L$ is doubled?

c. The capacitance $C$ is doubled?

Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf, three separate wires carry currents for the emfs $E_{1} = E_{0} \cos v t, E_{2} = E_{0_{}} \cos 1 v t + 120 掳, a n d E_{3} = E_{0} \cos 1 v t - 120 掳$ over three parallel wires, each of which supplies one-third of the power. This is why the long-distance transmission lines you see in the countryside have three wires. Suppose the transmission lines into a city supply a total of $450 M W$ of electric power, a realistic value.

a. What would be the current in each wire if the transmission voltage were $E_{0} = 120 V_{r m s}$ ?

b. In fact, transformers are used to step the transmission-line voltage up to $500 k V_{r m s}$ . What is the current in each wire?

c. Big transformers are expensive. Why does the electric company use step-up transformers?

A motor attached to a $120 V / 60 H Z$ power line draws an $8 A$ current. Its average energy dissipation is $800 W$ .

a. What is the power factor?

b. What is the rms resistor voltage?

c. What is the motor鈥檚 resistance?

d. How much series capacitance needs to be added to increase the power factor to $1$ ?

a. Evaluate $V_{R}$ in figure at emf frequencies $100, 300, 1000, 3000, 10000 H z$

b. Graph $V_{R}$ versus frequency. Draw a smooth curve through your five points.

See all solutions

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