Q.A metal cylinder 0.628 m long and 0.04 m in diameter has one end in boiling water at 100°C and the other end in melting ice. The coefficient of thermal conductivity of the metal is 378 Wm⁻¹K⁻¹. Latent heat of ice is 3.36 × 10⁵ Jkg⁻¹. Find the mass of ice that melts in one hour.

Thermal Conduction — From Intuition to the Law

Imagine holding a metal rod. You put one end into a fire. After a few seconds, the end you're holding gets hot — even though it's nowhere near the flame. Something travelled through the rod. That something is heat, and the process is thermal conduction.

At the microscopic level, the atoms in the hot end vibrate violently. They bump into their neighbours, passing on some of that energy. Those neighbours bump into the next ones, and so on. In metals, free electrons also carry energy rapidly, which is why metals feel cold to the touch (they steal heat from your hand fast) and conduct heat so well. In non-metals, only the atomic vibrations (phonons) do the job, so conduction is slower.

The Precise Statement

Now for the formal description. When heat flows steadily through a slab of material — say a wall, a window, or a metal bar — experiments show that the rate of heat transfer depends on three things:

1. The temperature difference across the material — bigger difference, faster flow.
2. The cross-sectional area — a thicker rod or a larger wall lets more heat through.
3. The thickness — a longer path slows the flow down.

Combine these observations into a single proportionality:

where $Q/t$ is the heat transferred per unit time (the heat current), $A$ is the area, $T_1 - T_2$ is the temperature difference, and $d$ is the thickness.

The ratio $(T_1 - T_2)/d$ is the temperature gradient — how sharply the temperature changes with distance. The steeper the gradient, the faster heat flows.

Fourier's Law of Heat Conduction

The proportionality becomes an equation when we introduce a material-dependent constant, $k$, called the thermal conductivity:

Or, in differential form for a continuous medium:

The minus sign is a convention: heat flows from hot to cold, so the temperature gradient $dT/dx$ is negative in the direction of flow. The minus sign makes the heat current positive.

What "Steady State" Means

The law above assumes steady-state conduction — the temperatures at each point in the material do not change with time. Heat flows in at one end and out at the other at the same constant rate. The rod doesn't keep getting hotter; it reaches a stable temperature profile. This is the simplest case and the one you'll meet first in exams.

A Quick Example …