What is the magnitude of the impedance Z for a series RC circuit?

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Multiple Choice

What is the magnitude of the impedance Z for a series RC circuit?

Explanation:
In AC circuits, impedance is a complex quantity. A resistor contributes a real part R, while a capacitor contributes an imaginary part -1/(ωC) (since Z_C = 1/(jωC) = -j/(ωC)). In series, these add as complex numbers: Z = R - j(1/(ωC)). The magnitude of Z is the length of this complex number, found by the Pythagorean relation |Z| = sqrt(R^2 + [1/(ωC)]^2). This is why the correct form is sqrt(R^2 + (1/(ωC))^2). The other forms ignore the phase and treat impedance as a simple scalar sum, which isn’t how the magnitude is determined.

In AC circuits, impedance is a complex quantity. A resistor contributes a real part R, while a capacitor contributes an imaginary part -1/(ωC) (since Z_C = 1/(jωC) = -j/(ωC)). In series, these add as complex numbers: Z = R - j(1/(ωC)). The magnitude of Z is the length of this complex number, found by the Pythagorean relation |Z| = sqrt(R^2 + [1/(ωC)]^2). This is why the correct form is sqrt(R^2 + (1/(ωC))^2). The other forms ignore the phase and treat impedance as a simple scalar sum, which isn’t how the magnitude is determined.

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