Q.Both alternating current and direct current are measured in amperes. But how is the ampere defined for an alternating current?

The key idea is that the ampere for AC is defined by the heating effect (power dissipation) it produces in a resistor, not by the direction of charge flow.

Reasoning:

1. Direct current (DC) is steady; 1 A DC through a resistor $R$ dissipates power $P = I^2 R$ continuously.
2. Alternating current varies sinusoidally: $i(t) = I_0 \sin(\omega t)$. Its instantaneous power is $p(t) = i^2 R = I_0^2 R \sin^2(\omega t)$.
3. The average power over a full cycle is $\langle P \rangle = \frac{1}{2} I_0^2 R$. …

The key idea is that an ampere is a measure of current, but alternating current changes direction and magnitude continuously. So we can't just say "1 A AC" means the same instantaneous flow as 1 A DC — that would be meaningless because the AC value is always changing. Instead, we define the AC ampere by what it does: how much heat it produces in a resistor.

This is the RMS (root-mean-square) definition. It's the single most important concept in AC measurements.

1. The problem with measuring AC directly

A direct current of 1 A means a steady flow of $1\ \text{C/s}$ past a point. But an alternating current like $i(t) = I_0 \sin(\omega t)$ varies between $+I_0$ and $-I_0$. Its average value over a full cycle is zero — so the average current tells us nothing useful about its ability to do work.

2. The physical basis: power dissipation in a resistor

When current flows through a resistor $R$, the instantaneous power dissipated is $p(t) = i^2(t) R$. This is always positive (since $i^2$ is always positive), even when the current reverses direction. The heating effect depends on the square of the current.

For a DC current $I_{\text{DC}}$, the power is constant: $P_{\text{DC}} = I_{\text{DC}}^2 R$.

For an AC current $i(t)$, the power varies with time. But the average power over a complete cycle is what matters for heating:

3. Defining the RMS current

We want an AC current to be called "1 A" if it produces the same average heating as a 1 A DC current. So we set:

Cancelling $R$:

The right-hand side is the mean of the square of the current. Taking the square root gives the root-mean-square (RMS) value:

This $I_{\text{rms}}$ is the AC ampere — the value that, when used in $P = I^2 R$, gives the correct average power.

4. The standard example: sinusoidal AC

For $i(t) = I_0 \sin(\omega t)$ with period $T = 2\pi/\omega$:

Using $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$: …

Method: Defining a Time-Varying (AC) Quantity by Its Equivalent Steady-State Effect

This is the general reasoning pattern for any question asking how a fluctuating/alternating quantity can be assigned a single meaningful numerical value — here, how an alternating current is "measured in amperes" at all.

Steps

Step 1: Identify why a simple time-average fails

The plain arithmetic average of a symmetric AC waveform over a full cycle is zero (positive and negative halves cancel), so it cannot define a meaningful "1 ampere of AC" — that would wrongly suggest no current flows at all.

Step 2: Choose a physical EFFECT that stays meaningful for alternating quantities

Instead of averaging the current directly, average something that does NOT change sign and cancel — here, the heating (power dissipation) it produces in a resistor, since $p(t) = i^2(t)R$ is always non-negative regardless of the current's direction.

Step 3: Define the AC quantity as "the value of steady DC that produces the same effect"

Set the average AC heating equal to the constant DC heating an equivalent steady current $I_{eq}$ would produce: …