20.4.1 A.C. Through a Resistor

This section describes the behavior of an alternating current (A.C.) circuit that contains only a resistor connected to an A.C. voltage source.

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Key Concepts

• Circuit Setup: A resistor with resistance R is connected across an alternating voltage source.

*Figure 20.10 (a): A.C through a resistor.*

• Alternating Voltage: The instantaneous voltage provided by the source is given by:
  $$V = V_{0} \sin(\omega t)$$
  where $V_0$ is the peak (or maximum) value of the voltage and $\omega$ is the angular frequency.

• Alternating Current: According to Ohm's Law ($V=IR$), the instantaneous current I flowing through the resistor can be found:
  $$I = \frac{V}{R} = \frac{V_{0} \sin(\omega t)}{R}$$ By defining the peak current as , the equation for the current becomes:

• Phase Relationship:

  • Comparing the equations for voltage ($V = V_{0} \sin(\omega t)$) and current ($I = I_{0} \sin(\omega t)$), we can see that the sine function argument () is the same for both.

• Waveform Representation:
  The in-phase relationship between voltage and current is visualized below. Both waves are perfectly aligned.

*Figure 20.10 (b): Voltage and current are in phase.*

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Summary

• When an A.C. voltage is applied across a resistor, the resulting current is also alternating and sinusoidal.
• Ohm's law ($V=IR$) applies to the instantaneous and peak values of voltage and current in the resistor.
• The most important characteristic of a purely resistive A.C. circuit is that the voltage across the resistor and the current through it are in phase.

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References

(Derived from FBISE textbook)