AC Voltage Applied to a Resistor

Figure 7.1 shows a simple series circuit consisting of an AC voltage source (drawn as a circle containing a sine-wave symbol, labelled $\varepsilon$) on the left, connected by plain conducting wires to a resistor (drawn as a zig-zag element, labelled $R$) on the right. The top and bottom wires complete the single-loop rectangular path. There are no current arrows.

The physical idea is that a sinusoidally varying voltage drives a sinusoidally varying current through a pure resistor, and the two quantities remain in phase — they reach zero, maximum, and minimum at the same instants.

The textbook uses this figure to derive the key relations:

• The applied AC voltage is

where $v_m$ is the amplitude (peak voltage) and $\omega$ is the angular frequency.

• Applying Kirchhoff’s loop rule gives

so the current is

with current amplitude

This is Ohm’s law for AC — the same form as for DC.

• The instantaneous power dissipated in the resistor is

• Averaging over one cycle gives the average power

where $I = \frac{i_m}{\sqrt{2}}$ is the rms current (root mean square).

• Similarly, the rms voltage is …

What the plot shows

The figure is a standard time‑domain plot of two sinusoidal waves on the same set of axes. The horizontal axis is labelled $\omega t$ (angular frequency times time), with tick marks at $0$, $\pi$, and $2\pi$ — representing one complete cycle of the alternating quantity. The vertical axis is the amplitude, with the peak of the larger wave marked $v_m$ and the peak of the smaller wave marked $i_m$.

Two curves are drawn:

• The larger sinusoid represents the instantaneous voltage $v = v_m \sin \omega t$.
• The smaller sinusoid represents the instantaneous current $i = i_m \sin \omega t$.

Both curves start at zero at $\omega t = 0$, rise together to their positive peaks near $\omega t = \pi/2$, cross zero together at $\omega t = \pi$, dip together to their negative minima, and return to zero at $\omega t = 2\pi$. The key visual fact is that the two waves are perfectly aligned — they reach zero, maximum, and minimum at exactly the same instants. This is what the textbook means by “the voltage and current are in phase.”

Physical idea taught by the figure

For a pure resistor, there is no energy storage (no inductance or capacitance). The opposition to current is purely resistive, given by Ohm’s law. Therefore, the current responds instantaneously to the applied voltage — there is no lag or lead. The figure makes this concrete: at every moment, the ratio $v/i$ is constant and equal to the resistance $R$. The fact that both waves cross zero together shows that when the voltage is zero, the current is also zero; when the voltage is maximum, the current is also maximum. This “in‑phase” behaviour is the defining characteristic of a purely resistive AC circuit.

Key formulas developed with this figure

From the circuit analysis using Kirchhoff’s loop rule, the applied voltage is

and the current through the resistor is

where the current amplitude is

This is Ohm’s law for AC — the same form as for DC, but now relating the amplitudes.

The instantaneous power dissipated is

and its average over one cycle is

To express this in the familiar DC form $P = I^2 R$, the root mean square (rms) current is defined: …

What the Figure Shows

The figure plots the instantaneous alternating current $i(\omega t)$ as a function of the phase angle $\omega t$ (in radians) on the horizontal axis. The axis is marked at $0$, $\pi$, and $2\pi$, covering one complete cycle of the sinusoidal waveform. The vertical axis represents the current, with the peak values $+i_m$ and $-i_m$ clearly labelled at the top and bottom of the oscillation.

A single sine curve oscillates symmetrically between these two extremes. Superimposed on this curve is a horizontal dashed line drawn at a constant level between $0$ and the positive peak $i_m$. This dashed line represents the root mean square (rms) current, denoted by $I$. Its position is at $I = i_m / \sqrt{2} \approx 0.707\, i_m$, as stated in the caption.

Physical Idea Taught

The figure visually demonstrates the concept of rms current — a way to express an alternating current in terms of an equivalent direct current that would produce the same average heating effect in a resistor. Since the instantaneous power dissipated in a resistor is $i^2 R$, and $i^2$ is always positive, the average power is not zero even though the average current over a cycle is zero. The rms value is the effective value of the ac current, and the dashed line shows how it compares to the peak swing of the waveform.

Key Formula Developed

The textbook derives the relation between the peak current $i_m$ and the rms current $I$:

where:

• $i_m$ is the amplitude (peak value) of the sinusoidal current,
• $I$ (or $I_{\text{rms}}$) is the root mean square current — the constant dc current that would dissipate the same average power in a resistor. …