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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

01

Given Information

We are given that $ω = ω_{c}$ and

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

02

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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Most popular questions from this chapter

A generator consists of a $12 c m$ by $16 c m$ rectangular loop with $500$ turns of wire spinning at $60 H z$ z in a $25 m T$ uniform magnetic field. The generator output is connected to a series RC circuit consisting of a $120 o h m$ resistor and a $35 m F$ capacitor. What is the average power delivered to the circuit?

The tuning circuit in an FM radio receiver is a series RLC circuit with a $0.200 m H$ inductor.

a. The receiver is tuned to a station at $104.3 m H Z$ . What is the value of the capacitor in the tuning circuit?

b. FM radio stations are assigned frequencies every $0.2 m H Z$ , but two nearby stations cannot use adjacent frequencies. What is the maximum resistance the tuning circuit can have if the peak current at a frequency of $103.9 m H Z$ , the closest frequency that can be used by a nearby station, is to be no more than $0.10 %$ of the peak current at $104.3 m H Z$ ? The radio is still tuned to $104.3 M H z$ , and you can assume the two stations have equal strength.

Use a phasor diagram to analyze the RL circuit of FIGURE P $32.49$ . In particular,

a. Find expressions for $I, V_{R} a n d V_{L}$ .

b. What is VR in the limits $ω \to 0$ and $ω \to \infty$ ?

c. If the output is taken from the resistor, is this a low-pass or a high-pass filter? Explain.

d. Find an expression for the crossover frequency $ω_{C}$ .

The small transformers that power many consumer products produce a $12.0 V_{r m s}$ , $60 H z$ emf. Design a circuit using resistors and capacitors that uses the transformer voltage as an input and produces a $6.0 V_{r m s}$ output that leads the input voltage by $45 °$ .

A series $R L C$ circuit consists of a $75 Ω$ resistor, a $0.12 H$ inductor, and a $30 m F$ capacitor. It is attached to a $120 V / 60 H z$ power line. What are (a) the peak current $I$ , (b) the phase angle $ϕ$ , and (c) the average power loss?

See all solutions

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