Characteristics of an alternating current
Section: Alternating Currents | Syllabus: Cambridge A Level Physics (9702)
Characteristics of an alternating current
Alternating e.m.f. and potential difference
Definitions:
- Alternating Current (a.c.): A current that periodically varies from positive to negative and changes its direction.
- Alternating Voltage: A potential difference that changes polarity with time.
Waveform Terminology:
- Peak Value (x_0): The maximum magnitude of the quantity (current or voltage) from zero.
- Peak-to-Peak Value: The difference between the maximum positive and maximum negative values (2x_0).
- Period (T): The time taken for one complete cycle of oscillations. Unit: second (s).
- Frequency (f): The number of complete cycles per unit time. Unit: Hertz (Hz).
Relationship between Period and Frequency:
T = (1/f)Angular Frequency (ω):
ω = 2π f = (2π/T)Unit of ω: radians per second (rad \ s^-1).
Sinusoidal Nature
An alternating current is often produced by a coil rotating at a constant speed in a uniform magnetic field. This uniform circular motion results in a sinusoidal waveform.
One complete rotation (2π radians) corresponds to one complete cycle of the alternating quantity.
Instantaneous values at time t:
ε = _0 sin ω tV = V_0 sin ω t
I = I_0 sin ω t Important
Since ω is in rad \ s^-1, your calculator MUST be in RADIANS mode when performing calculations involving sin ω t.
Worked Examples
Worked ExampleQuestion: The mains supply in a country has a peak voltage of 311 V and a frequency of 50 Hz. Write an equation for the potential difference V in terms of time t.
Solution:
- Identify constants: V_0 = 311 V, f = 50 Hz.
- Calculate angular frequency: ω = 2π f = 2π(50) = 100π ≈ 314 rad s^-1.
- Substitute into general equation: V = V_0 sin ω t.
Answer: V = 311 sin(100π t)
Common Exam ErrorDo not confuse frequency (f, in Hz) with angular frequency (ω, in rad s⁻¹). The equation is sin(2π f t), simply writing sin(ft) is incorrect.
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