AC Voltage Applied to a Resistor

 In this section, we will deal with an electric circuit, where an AC voltage is applied across a resistor. Before applying AC voltage across the resistor, let us know what an alternating current (AC) is. Electric current, which reverses its direction periodically and changes its magnitude continuously with time, is known as an alternating current.

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AC Voltage Applied to a Resistor Derivation

Let us consider the circuit as shown below. We have a resistor and an AC voltage V, represented by the symbol ~, that produces a potential difference across its terminals that varies sinusoidally. Here, the potential difference or the AC voltage can be given as

v=vmsinωt

Here, vis the amplitude of the oscillating potential difference and the angular frequency is given by ω.

The current through the resistor due to the present voltage source can be calculated using Kirchhoff’s loop rule, as under,

V(t)=0

Here, using this equation, we can write,

vmsinωt=iR

Or,

i=vmRsinωt

Here, R is the resistance of the given resistor, and hence we can write,

i=imsinωt

From the Ohm’s law, we can write,

im=vmR

As it works equally well for AC voltages and DC voltages. We saw that the voltage across the resistor and the current passing through it are both sinusoidal quantities and are represented by the graph shown in the figure above. Both the quantities are in phase with each other.

The instantaneous power in the circuit is given by,

P=i2R=im2Rsin2ωt

The average value of power over the complete cycle is given as,

P¯=i2R=im2Rsin2ωt

Here the quantities im and R are constants, and hence, the above equation is evaluated as,

P¯=im2Rsin2ωt

Using trigonometry, we have

sin2ωt=12(1cos2ωt)

, and we can write

sin2ωt=12(1cos2ωt)

And since the integration of cos 2ωt over the complete cycle is zero. So we can write

cos2ωt=0

, we can write,

sin2ωt=12

Thus we can also write,

P¯=12im2R

It is important to note that the AC power can also be expressed as DC power if we denote the current in terms of the effective current or the root-mean-square current, which is given as,

Irms=i2¯=12im2=im2=0.707im

    1,94,536


    Frequently Asked Questions – FAQs

    What is direct current?

    The unidirectional flow of electric charges is known as direct current.

    Does AC change its magnitude continuously with time?

    Yes, alternating current changes its magnitude continuously with time.

    Define frequency of the electrical signal.

    The number of cycles per second is known as frequency. Frequency is measured in hertz.

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