Question 21.1: What Is the rms Current? Goal Perform basic AC circuit calcu...
What Is the rms Current?
Goal Perform basic AC circuit calculations for a purely resistive circuit.
Problem An AC voltage source has an output of Δv = (2.00 × 10² V)sin 2πft. This source is connected to a 1.00 × 10²-Ω resistor as in Active Figure 21.1. Find the rms voltage and rms current in the resistor.
Strategy Compare the expression for the voltage output just given with the general form, \Delta v=\Delta V_{\max } \sin 2 \pi f t , finding the maximum voltage. Substitute this result into the expression for the rms voltage.
Obtain the maximum voltage by comparison of the given expression for the output with the general expression:
\begin{aligned}\Delta v=&\left(2.00 \times 10^{2} V \right) \sin 2 \pi f t \quad \Delta v=\Delta V_{\max } \sin 2 \pi f t \\& \rightarrow \Delta V_{\max }=2.00 \times 10^{2} V\end{aligned}
Next, substitute into Equation 21.3 to find the rms voltage of the source:
\Delta V_{ rms }=\frac{\Delta V_{\max }}{\sqrt{2}}=0.707 \Delta V_{\max }
\Delta V_{ rms }=\frac{\Delta V_{\max }}{\sqrt{2}}=\frac{2.00 \times 10^{2} V }{\sqrt{2}}=141 V
Substitute this result into Ohm’s law to find the rms current:
I_{ rms }=\frac{\Delta V_{ rms }}{R}=\frac{141 V }{1.00 \times 10^{2} \Omega}=1.41 A
Remarks Notice how the concept of rms values allows the handling of an AC circuit quantitatively in much the same way as a DC circuit.