Chapter 32: Q. 40 (page 925)

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

Short Answer

Expert verified

(a) The final expression for angular frequency is $ω = \frac{1}{\sqrt{3} R C}$

(b) The V_C at the given frequency is $V_{C} = \frac{\sqrt{3} ε_{0}}{2}$

Step by step solution

Part (a) Step 1: Given Information

We are given that for an RC circuit, the voltage across resistor is $V_{R} = \frac{1}{2} ε_{0}$ .

We have to find out the expression for angular frequency.

Part (a) Step 2: Simplification

We know,

$V_{R} = \frac{ε_{0} R}{\sqrt{R^{2} + \frac{1}{ω^{2} c^{2}}}}$ (where $V_{R} = \frac{1}{2} ε_{0}$ )

Here $V_{R}$ is voltage across resistor, R is the resistor, $ω$ is the angular frequency and $c$ is the capacitor.

On simplifying we get, $\sqrt{1 + \frac{1}{ω^{2} C^{2} R^{2}}} = 2$

$\frac{1}{ω^{2} C^{2} R^{2}} = 3$

which gives the result , $ω = \frac{1}{\sqrt{3} R C}$

Part (b) Step 1: Given Information

We are given that for an RC circuit, the voltage across the resistor is $V_{R} = \frac{1}{2} ε_{0}$ and we have already find out the angular frequency $ω = \frac{1}{\sqrt{3} R C}$ .

We have to find out the $V_{C}$ .

Part (b) Step 2: Simplification

We know, the formula for voltage across the capacitor is

$V_{C} = \frac{\frac{ε_{0}}{ω_{C}}}{\sqrt{R^{2} + \frac{1}{ω^{2} C^{2}}}}$

Substituting all the values in it, we get

$V_{C} = \frac{\sqrt{3} ε_{0}}{2}$ .

The voltage across capacitor $V_{C} = \frac{\sqrt{3} ε_{o}}{2}$

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a. For an RC circuit, find an expression for the angular frequency at which $V_{R} = \frac{1}{2} ε_{0}$
b. What is V_C 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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a. For an RC circuit, find an expression for the angular frequency at which VR=12ε0b. What is VC 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

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 $V_{R} = \frac{1}{2} ε_{0}$
b. What is V_C at this frequency?