Q.The output of a step-down transformer is measured to be $24\ \text{V}$ when connected to a $12\ \text{W}$ light bulb. The value of the peak current is

Power Dissipation in Resistors — Why the Formula Holds

Let's build this from first principles. The goal is to understand why a resistor dissipates power as heat, and how the formula $P = I^2 R$ (and its equivalents) arise naturally.

1. What is "Power" in an Electrical Circuit?

Power is the rate of energy transfer — how much energy is converted from one form to another per unit time.

• In a resistor, electrical energy is converted into heat (thermal energy).
• The fundamental definition of electrical power is:

where:

• $P$ = power (watts, W)
• $V$ = voltage across the component (volts, V)
• $I$ = current through the component (amperes, A)

Why this definition?

Voltage is energy per unit charge ($V = \frac{W}{q}$), and current is charge per unit time ($I = \frac{q}{t}$). Multiplying them gives energy per unit time — exactly power.

2. How Does a Resistor Behave? — Ohm's Law

A resistor obeys Ohm's Law:

where $R$ is resistance (ohms, $\Omega$). This is an empirical law — it describes how real resistors behave: the voltage across them is proportional to the current through them.

3. Deriving the Power Dissipation Formulas

We start with $P = V I$ and substitute Ohm's Law in two ways.

Case A: Express power in terms of $I$ and $R$

Replace $V$ with $I R$:

Interpretation:

• For a fixed resistance, power grows with the square of current.
• Doubling current quadruples the heat generated — this is why high currents cause wires to overheat.

Case B: Express power in terms of $V$ and $R$

Replace $I$ with $\frac{V}{R}$:

Interpretation:

• For a fixed voltage, power is inversely proportional to resistance.
• A low-resistance resistor (like a short circuit) dissipates huge power at a given voltage — that's why short circuits are dangerous.

4. The Physical "Why" — Energy Conversion at the Atomic Level

Why does this energy turn into heat?

• Electrons moving through a resistor collide with the atoms of the material.
• Each collision transfers kinetic energy from the electron to the atom, making the atom vibrate more — i.e., heating up the resistor.
• The rate at which this energy is lost by the electrons (and gained by the lattice) is exactly $P = I^2 R$.

Key insight:

The $I^2$ term appears because:

• More current = more electrons per second. …