Question

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

Answer 1

1. Consider an alternating e.m.f. applied to a series combination of the pure inductor of inductance L and resistor of resistance R.
   
2. Let e_R and e_L be the r.m.s. the voltage across resistor and inductor respectively. As R and L 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"_"L"^2` 
   ∴ e = `sqrt("e"_"R"^2 + "e"_"L"^2) = sqrt(("iR")^2 + ("iX"_"L")^2) = "i"sqrt("R"^2 + "X"_"L"^2)`
   ∴ e = iZ
   where Z = `sqrt("R"^2 + "X"_"L"^2)` is the impedance of the RL circuit.
5. a. Expression for applied emf in RL circuit,
   e = e₀sinωt = `sqrt("e"_"R"^2 + "e"_"L"^2)` = iZ
   b. Expression for effective resistance in RL circuit, Z = `sqrt("R"^2 + "X"_"L"^2)`