Chapter 32: Q. 42 (page 925)

For an RC filter circuit, show that $V_{R} = V_{C} = \frac{ε_{0}}{\sqrt{2}}$ at $ω = ω_{c}$ .

Short Answer

Expert verified

The expression is true $V_{R} = V_{C} = \frac{ε_{0}}{\sqrt{2}}$ .

Step by step solution

Given Information

We are given that $ω = ω_{c}$ and

we have to find out that $V_{R} = V_{C} = \frac{ε_{o}}{\sqrt{2}}$ .

Simplification

As we know that for an RC filter circuit,

$ω_{c} = \frac{1}{R C}$

Next we will find V_R :

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

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

Finally, we will find V_C:

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

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

We get, $V_{C} = \frac{ε_{0}}{\sqrt{2}}$

As $V_{R} = V_{C} = \frac{ε_{o}}{\sqrt{2}}$

Hence, it is proved that $V_{R} = V_{C} = \frac{ε_{0}}{\sqrt{2}}$ .

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For an RC filter circuit, show that $V_{R} = V_{C} = \frac{ε_{0}}{\sqrt{2}}$ at $ω = ω_{c}$ .

Short Answer

Step by step solution

Given Information

Simplification

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Most popular questions from this chapter

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For an RC filter circuit, show that $V_{R} = V_{C} = \frac{ε_{0}}{\sqrt{2}}$ at $ω = ω_{c}$ .