AC Voltage Applied to a Resistor
Resistance is the opposition to the flow of the current offered by any substance. The substance that limits the flow of electric current in the circuit is the resistor. If the source is producing a current that varies with time periodically, then this current is called an alternating current. Here, we are considering that the source is producing sinusoidal varying potential differences or voltage across its terminals. We can represent it as
V = Vm Sin ωt
where Vm represents the amplitude of the oscillating potential difference and represents the angular frequency. In the below article, we will describe what will happen when we apply the AC voltage source to a resistor.
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The above image represents the circuit diagram of a resistor connected to a source that is producing AC voltage.
AC Voltage Applied to a Resistor Equation
Please note that the resistance value of the resistor in both AC and DC circuits is likewise without considering the frequency of the AC supply voltage. Also, the alteration in direction of current in AC supply does not impact resistors' behaviour.
To calculate the current through the resistor due to the present voltage source, we will apply Kirchhoff's loop rule which is
∑ V(t)=0
By using this equation, we can write
V = Vm Sin ωt = iR
Or,
i = Vm RSin ωt
where i represents the current, and R represents the resistance of the given resistor.
According to ohm's law,
V = iR
Or
\[i = \frac{V}{R}\]
Ohm law works equally where the source is producing AC Voltage or DC voltage. Hence, we can also write the above equation as
\[i = \frac{V_{m}}{R}\]
By using this equation, we can also write \[i = \frac{V_{m}sin \omega t}{R} \] as;
i = im nSi ωt
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The above image shows the graphical representation of the variation in current and voltage with respect to time.
From the above image, it is clear the current and voltage reach maximum and minimum values at the same time. Both the current passing through the resistor and voltage across it are the sinusoidal quantities and are in phase with each other.
Average Value of the Power
AC Voltage Applied to a Resistor derivation shows us the potential difference, and current is in phase with each other. Also, we know that p = i2R. Hence, we can write:
p = im2Rωt
We can represent the average value of power generated in the complete cycle as;
p = i2R = im2Rωt
The above quantities im and R are constant.
Using trigonometry, \[\omega t = \frac{1}{2} (1 - cos 2 \omega t)\]
From the above graph, we can see that cos2ωt = 0. Now, we put this value of cos2ωt in the above equation. Now, we get:
\[\omega t = \frac{1}{2} (1 - 0)\]
Or,
\[\omega t = \frac{1}{2}\]
Now, we will put this value of ωt in the equation p = im2Rωt. Thus we get:
\[\overline{p} = \frac{1}{2} i_{m}^{2} R \omega t\]
It is also important to note we can express AC power into DC power by denoting the current in terms of root-mean = square current or effective current.
\[I_{rms} = \sqrt{i^{-2}} = \sqrt{\frac{1}{2} i_{m}^{2}}\]
By evaluating, we can also write the above equation as;
\[I_{rms} = \frac{i_{m}}{\sqrt{2}} = 0.707 i_{m}\]
Note on Power Consumption in AC Voltage Applied to a Resistor
The sum of instantaneous current over one complete cycle is zero, and hence, the average current is also zero. However, it does not mean that there is no dissipation of electrical energy. We know that the joule heating is i2R, and thus it is always positive because it depends on i2. The thermal energy developed during the time t to t+dt in the resistor is:
\[i^{2}Rdt = i_{m}^{2} (\omega t) Rdt\]
The thermal energy developed in one time period is:
U = \[\int i^{2} Rdt = \int i_{m}^{2} (\omega t) Rdt\]
Over the limit 0 to T
= R\[\int i_{m}^{2} (\omega t) Rdt\]
= \[\int i_{rms}^{2} (\omega t) Rdt\]
It means that the root-mean-square value of alternating current is the steady current that would generate the same amount of heat in the resistor in a given amount.
Solved Examples
Example 1: What is the approx peak value of an alternating current producing four times the heat produced per second by a steady current of 2 A in a resistor?
Ans. The equation that represents the relation between RMS current and peak current is:
\[I_{0} = I_{rms} \sqrt{2}\]
The expression for the heat produced by the resistor is
W = \[I^{2} Rt\]
Hence, the heat generated by the resistor when the steady current of 2A flows through it is
\[W_{2A} = (2A)^{2} RT\]
\[W_{2A} = 4R\]
The heat produced by the alternating current is four times the heat generated by the constant current. Hence,
\[W_{alternating} = 4\times W_{2A}\]
\[W_{alternating} = (4\times 4) R = 4^{2}R\]
Therefore, the RMS value of the alternating current is 4A.
Hence,
\[I_{0} = (4A)\times \sqrt{2}\]
\[I_{0} = 5.6 A\]
FAQs on AC Voltage Resistor
1. What happens when an alternating voltage is applied across a resistor in a circuit?
When a sinusoidal AC voltage is applied across a resistor, both the current and voltage follow sinusoidal patterns and remain in phase with each other. The current at any instant can be expressed as i = (Vm/R) sinωt, where Vm is the peak voltage and R is the resistance. The resistor’s opposition to current remains unchanged compared to a DC circuit, as it only depends on resistance, not on the frequency of the AC source.
2. How do resistance and frequency of AC supply relate when calculating current through a resistor?
The current amplitude depends on the applied voltage and resistance, but the frequency of the AC supply does not affect the value of resistance. The resistor behaves the same way in both AC and DC circuits, showing that its value is independent of frequency for ideal resistors.
3. Why is the average value of alternating current over one complete cycle zero?
Over a complete AC cycle, the current changes direction every half cycle, producing equal positive and negative values that cancel each other out. Therefore, the sum of instantaneous values results in a zero average current for the cycle.
4. What is the difference between RMS voltage and average voltage in an AC circuit?
The RMS (Root Mean Square) voltage is the value of AC voltage that produces the same heating effect as a corresponding DC voltage and is calculated as 0.707 times the peak voltage for pure sine waves. The average voltage, taken over a complete cycle, is zero for a pure AC sine wave, as positive and negative halves cancel.
5. How is electrical power consumed by a resistor in an AC circuit determined?
The electrical power consumed over a cycle is calculated using the RMS current: P = Irms2R. Although average current is zero, the resistor dissipates energy as heat throughout the cycle, since power depends on the square of instantaneous current, which is always positive.
6. What does it mean when we say current and voltage are 'in phase' in a resistor connected to AC supply?
'In phase' means that the peaks and zero crossings of the current and voltage waveforms occur at the same times. Both reach their maximum and minimum values together, indicating there is no time lag between the voltage across and the current through the resistor.
7. If an AC current produces four times the heat per second as a steady current of 2A in a resistor, what is the peak value of the AC current?
Given the steady current I = 2A, the corresponding heat by AC is four times greater, so Irms = 4A. The peak value I0 is then I0 = Irms√2 = 4 × 1.414 = 5.6A.
8. Why does power dissipation occur in a resistor with alternating current even though average current is zero?
Power dissipation occurs because it is proportional to the square of the current (i2R). Both positive and negative currents contribute equally to heating, so energy is always converted to heat, regardless of current direction, making average power nonzero.
9. Can Ohm's Law be applied in an AC circuit with a pure resistor? Why or why not?
Yes, Ohm's Law applies in an AC circuit containing only a resistor, because the relationship V = iR holds at every instant. Both voltage and current change simultaneously and proportionally with time, without any phase shift.
10. What misconceptions might students have about the effect of AC frequency on resistor behavior?
A common misconception is that the frequency of AC changes the resistance of a resistor. In reality, ideal resistance is independent of frequency; only inductors and capacitors show frequency-dependent behavior. Real-world resistors are nearly unaffected by practical AC frequencies used in school physics.
