In a series RLC circuit at frequency ω, which expression correctly gives the current I in terms of V and the impedance Z = R + j(ωL − 1/(ωC))?

In a series RLC circuit at frequency ω, which expression correctly gives the current I in terms of V and the impedance Z = R + j(ωL − 1/(ωC))?

• A. I = Z / V
• B. I = V / (R − j(ωL − 1/(ωC)))
• C. I = V / (R + j(ωL + 1/(ωC)))
• D. I = V / (R + j(ωL − 1/(ωC)) )

Answer: (D)

In AC circuits, current is found by dividing the applied voltage by the circuit’s impedance. For a series RLC, the total impedance is the sum of the resistor, the inductor’s impedance, and the capacitor’s impedance. The inductor contributes jωL, while the capacitor contributes −j/(ωC) (since 1/(jωC) = −j/(ωC)). So the impedance is R + j(ωL − 1/(ωC)). Therefore the current is I = V / Z = V / [R + j(ωL − 1/(ωC))]. This matches the given impedance expression. The other forms would either invert the ratio, use the complex conjugate of the impedance, or flip the sign of the capacitor’s reactance, all of which don’t reflect the actual impedance of the circuit.

In AC circuits, current is found by dividing the applied voltage by the circuit’s impedance. For a series RLC, the total impedance is the sum of the resistor, the inductor’s impedance, and the capacitor’s impedance. The inductor contributes jωL, while the capacitor contributes −j/(ωC) (since 1/(jωC) = −j/(ωC)). So the impedance is R + j(ωL − 1/(ωC)). Therefore the current is I = V / Z = V / [R + j(ωL − 1/(ωC))]. This matches the given impedance expression. The other forms would either invert the ratio, use the complex conjugate of the impedance, or flip the sign of the capacitor’s reactance, all of which don’t reflect the actual impedance of the circuit.