Question

Obtain the expression for the applied emf and the effective resistance of the circuit when alternating emf is applied to a CR circuit.

Answer 1

1. Consider an alternating e.m.f. applied to a series combination of the pure capacitor of capacitance C and resistor of resistance R.
   
2. Let e_R and e_C be the r.m.s. the voltage across the resistor and capacitor respectively. As R and C are in series, the current at any instant is given as i = i₀ sin ωt. 
3. Vector diagram,
   
4. From the above figure,
   e² = `"e"_"R"^2 + "e"_"C"^2`
   ∴ e = `sqrt("e"_"R"^2 + "e"_"C"^2) = sqrt(("iR")^2 + ("iX"_"C")^2) = "i"sqrt("R"^2 + "X"_"C"^2)`
   ∴ e = iZ
   where Z = `sqrt("R"^2 + "X"_"C"^2)` is the impedance of the RC circuit.
5. a. Expression for applied emf in CR circuit,
   e = e₀sinωt = `sqrt("e"_"R"^2 + "e"_"C"^2)` = iZ
   b. Expression for effective resistance in CR circuit, Z = `sqrt("R"^2 + "X"_"C"^2)`