In a series RC circuit, express the steady-state current as a function of V, R, C, and frequency.

In a series RC circuit, express the steady-state current as a function of V, R, C, and frequency.

• A. I = V / sqrt(R^2 + (1/(ω C))^2)
• B. I = V / sqrt(R^2 + (ω C)^2)
• C. I = V / (R + 1/(ω C))
• D. I = V / R

Answer: (A)

In a series RC circuit with an AC source, the current is determined by the total impedance, which combines resistance and the capacitor’s reactance. The capacitive reactance is X_C = 1/(ωC), so the impedance magnitude is Z = sqrt(R^2 + X_C^2) = sqrt(R^2 + (1/(ωC))^2). The steady-state current magnitude is the source voltage divided by this impedance: I = V / sqrt(R^2 + (1/(ωC))^2). As frequency rises, X_C shrinks and the current approaches V/R; as frequency falls, X_C grows and the current decreases. The other forms misrepresent the impedance (they either use ωC instead of 1/(ωC) or sum magnitudes instead of combining in quadrature) and the simple V/R form applies only to a pure resistor.

In a series RC circuit with an AC source, the current is determined by the total impedance, which combines resistance and the capacitor’s reactance. The capacitive reactance is X_C = 1/(ωC), so the impedance magnitude is Z = sqrt(R^2 + X_C^2) = sqrt(R^2 + (1/(ωC))^2). The steady-state current magnitude is the source voltage divided by this impedance: I = V / sqrt(R^2 + (1/(ωC))^2). As frequency rises, X_C shrinks and the current approaches V/R; as frequency falls, X_C grows and the current decreases. The other forms misrepresent the impedance (they either use ωC instead of 1/(ωC) or sum magnitudes instead of combining in quadrature) and the simple V/R form applies only to a pure resistor.