Chapter 32: Q. 41 (page 925)

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

Short Answer

Expert verified

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

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

Step by step solution

Part (a) Step 1 : Given information

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

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

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,

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

Put the value of Z,

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

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

$ω^{2} = \frac{3}{R^{2} C^{2}}$

On solving we get,

$ω = \frac{\sqrt{3}}{R C}$

Part (b) Step 1: Given Information

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

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

Part (b) Step 2: Simplification

As we know,

The expression for voltage across resistor is,

$ε_{0}^{2} = V_{R}^{2} + V_{C}^{2}$

Put the value of $V_{C}$

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

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

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

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

Short Answer

Step by step solution

Part (a) Step 1 : Given information

Part (a) Step 2 : Simplification

Part (b) Step 1: Given Information

Part (b) Step 2: Simplification

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Short Answer

Step by step solution

Part (a) Step 1 : Given information

Part (a) Step 2 : Simplification

Part (b) Step 1: Given Information

Part (b) Step 2: Simplification

One App. One Place for Learning.

Most popular questions from this chapter

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