Electrostatic Potential and Capacitance

Question

A 100 volt a.c. source of frequency 500 hertz is connected to LCR circuit with L = 8.1 millihenry, C = 12.5 micro farad and R = 10 ohm, all connected in series. Find the potential difference across the resistance.

Solution

Given, an LCR series circuit.
Rms value of voltage, V = 100 V
Frequency = 500 Hz

The impedance of LCR circuit is given by

box enclose straight Z space equals space square root of open square brackets straight R squared plus open parentheses straight X subscript straight L minus straight X subscript straight C close parentheses squared close square brackets end root end enclose

where

box enclose straight X subscript straight L space equals space ωL end enclose space equals space 2 πfL space space space space

equals space 2 space cross times space 3.14 space cross times space 500 space cross times space left parenthesis space 8.1 space cross times space 10 to the power of negative 3 end exponent right parenthesis equals space 25.4 space ohm

and,

box enclose straight X subscript straight C space equals space 1 over ωC end enclose space equals space fraction numerator 1 over denominator 2 πfC end fraction

equals space fraction numerator 1 over denominator 2 cross times 3.14 cross times 400 cross times left parenthesis 12.5 space cross times space 10 to the power of negative 6 end exponent right parenthesis end fraction

= 25.4 ohm

therefore

Impedence,

straight Z space equals space square root of open square brackets left parenthesis 10 right parenthesis squared plus left parenthesis 25.4 minus 25.4 right parenthesis squared close square brackets end root space equals space 10 space ohm

Rms value of current,

straight I subscript rms space equals space straight E subscript rms over straight Z space equals space fraction numerator 100 space volt over denominator 10 space ohm end fraction space equals space 10 space amp.

Potential difference across resistance

straight V subscript straight R space equals space straight I subscript rms space cross times space straight R space space space space space space space equals space 10 space amp space cross times space 10 space ohm space space space space space space equals space 100 space volt.