C
CraftExam
medium difficulty10 min

AC Voltage Across a Pure Resistor

When a sinusoidal AC voltage is applied across a pure resistor, the current is also sinusoidal and remains in phase with the applied voltage.

In one line: In R, response is regular: voltage and current rise and fall together.

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Understand it

  • For a resistor, Ohm's law applies at every instant.
  • If the applied voltage is V = V0 sin ω t, then the instantaneous current is I = V/R = (V0/R) sin ω t = I0 sin ω t.
  • Since voltage and current have the same sine factor, their maxima, minima, and zero values occur at the same instant.
  • The phase difference between voltage and current is zero.
  • RMS values are used because AC changes continuously; Vrms = V0/√2, Irms = I0/√2, and Vrms = Irms R.

Example

If a 220 V RMS AC supply is connected to a 110 ohm resistor, the RMS current is I = V/R = 220/110 = 2 A. Voltage and current reach their peak values together.

Formula and rule

Instantaneous voltage: V = V0 sin ω t.

Instantaneous current: I = I0 sin ω t, where I0 = V0/R.

RMS values: Vrms = V0/√2, Irms = I0/√2.

R is resistance in ohm, V in volt, I in ampere, and ω in rad s−1.

Use these relations only for a pure resistor with negligible inductance and capacitance.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

A 50 ohm resistor is connected to V = 100 sin(314t) V.

  1. 1Here V0 = 100 V and R = 50 ohm.
  2. 2I0 = V0/R = 100/50 = 2 A.
  3. 3Therefore I = 2 sin(314t) A.
  4. 4Vrms = 100/√2 = 70.7 V and Irms = 2/√2 = 1.41 A.
  5. 5The current is in phase with voltage.

Method you can reuse

  1. 1Identify V0 and R.
  2. 2Apply I0 = V0/R.
  3. 3Write current with the same phase as voltage.
  4. 4Convert peak values to RMS only when the question asks for RMS or average power.
  5. 5State phi = 0 for a pure resistor.

Visual tip

Useful diagrams include the simple AC source-resistor circuit, sinusoidal V-t and I-t graphs on the same axes, and a phasor diagram with V and I along the same line. Labels should include V, I, R, time axis, amplitude, and phase difference zero.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In a pure resistor connected to AC, what is the phase relation between voltage and current?

Think first, then show answer
In a pure resistor, voltage and current are in the same phase. Both become zero, maximum, and minimum at the same instants because I = V/R at every instant.

Common wrong answer

Writing I = I0 sin(ω t - π/2).

Why it loses marks: That lag occurs for a pure inductor, not for a pure resistor.

Don't mix this up

Students often use peak voltage as if it were RMS voltage, or say that current lags in a resistor. In a pure resistor, phase difference is zero.

Exam tip

Write both instantaneous equations and RMS relation when asked to explain AC through a resistor. Mention phase difference explicitly: phi = 0.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions usually ask for the expression of current, RMS value, phase difference, average power, or a graph/phasor showing V and I in a resistor.
  1. 1

    Enough for 1 mark

    When a sinusoidal AC voltage is applied across a pure resistor, the current is also sinusoidal and remains in phase with the applied voltage.

  2. 2

    Add for 2 marks

    Example: If a 220 V RMS AC supply is connected to a 110 ohm resistor, the RMS current is I = V/R = 220/110 = 2 A. Voltage and current reach their peak values together.

  3. 3

    Add for 3 marks

    For a resistor, Ohm's law applies at every instant.

Practice

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