# Physics > Alternating Current

• An alternating voltage  applied to a resistor R drives a current  in the resistor,

The current is in phase with the applied voltage.

• For an alternating current  passing through a resistor R, the average power loss P (averaged over a cycle) due to joule heating is  . To express it in the same form as the dc power (P = I 2R), a special value of current is used. It is called root mean square (rms) current and is denoted by I:

Similarly, the rms voltage is defined by

We have P = IV = I2R

• An ac voltage  applied to a pure inductor L, drives a current in the inducto , where
• is called inductive reactance. The current in the inductor lags the voltage by  The average power supplied to an inductor over one complete cycle is zero.
• An ac voltage  applied to a capacitor drives a current in the capacitor:  Here,

is called capacitive reactance.

The current through the capacitor is  ahead of the applied voltage. As in the case of inductor, the average power supplied to a capacity or over one complete cycle is zero.

• For a series RLC circuit driven by voltage , the current isgiven by

is called the impedance of the circuit.

The average power loss over a complete cycle is given by

The term  is called the power factor.

• In a purely inductive or capacitive circuit,  and no power is dissipated even though a current is flowing in the circuit. In such cases, current is referred to as a wattless current.
• The phase relationship between current and voltage in an ac circuit can be shown conveniently by representing voltage and current by rotating vectors called phasors. A phasor is a vector which rotates about the origin with angular speed ω. The magnitude of a phasor represents the amplitude or peak value of the quantity (voltage or current) represented by the phasor.

The analysis of an ac circuit is facilitated by the use of a phasor diagram.

• An interesting characteristic of a series RLC circuit is the phenomenon of resonance. The circuit exhibits resonance, i.e., the amplitude of the current is maximum at the resonant frequency,

The quality factor Q defined by

is an indicator of the sharpness of the resonance, the higher value of Q indicating sharper peak in the current.

• A circuit containing an inductor L and a capacitor C (initially charged) with no ac source and no resistors exhibits free oscillations. The charge q of the capacitor satisfies the equation of simple harmonic motion:

and therefore, the frequency ω of free oscillation is

The energy in the system oscillates between the capacitor and the inductor but their sum or the total energy is constant in time.

• A transformer consists of an iron core on which are bound a primary coil of  turns and a secondary coil of turns. If the primary coil is connected to an ac source, the primary and secondary voltages are related by  and the currents are related by

If the secondary coil has a greater number of turns than the primary, the voltage is stepped-up  This type of arrangement is called a step up transformer. If the secondary coil has turns less than the primary, we have a step-down transformer.

### Examples

#### Question

A sinusoidal voltage of peak value 283 V and frequency 50 Hz is applied to a series LCR circuit in which R = 3 Ω, L = 25.48 mH, and C = 796 μF. Find (a) the impedance of the circuit; (b) the phase difference between the voltage across the source and the current; (c) the power dissipated in the circuit; and (d) the power factor.

#### Solution

(a) To find the impedance of the circuit, we first calculate XL and XC.

= 2

=

=

=

Since  is negative, the current in the circuit lags the voltage across the source.

(c) The power dissipated in the circuit is

Therefore,

(d) Power factor =