A sphere, a cube and a thin circular plate, all of same material and same mass are initially heated to same hi

NCERT Exemplar · Q8

Q. A sphere, a cube and a thin circular plate, all of same material and same mass are initially heated to same high temperature.

(a) Plate will cool fastest and cube the slowest

(b) Sphere will cool fastest and cube the slowest

(d) Cube will cool fastest and plate the slowest.

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Concept understanding — Thermal Radiation Properties

Thermal Radiation Properties

Imagine holding your hand near a hot iron — you feel warmth without touching it. That warmth travels through empty space, not through air or metal. This is thermal radiation: energy emitted by any object purely because it has a temperature.

Unlike conduction (touching) or convection (fluid flow), radiation doesn't need a medium. It's electromagnetic waves — mostly infrared, but at high temperatures, visible light too. A glowing red-hot steel rod is radiating; so is your own body, though you can't see it.

The Core Idea: Every Object Radiates

Every object above absolute zero ( $0\ \text{K}$ ) emits radiation. The amount and type depend on two things:

Temperature — hotter objects radiate more energy, and at shorter wavelengths.
Surface properties — some surfaces are good emitters, others are poor.

This leads to three key properties that describe how a real surface behaves compared to an ideal "perfect" radiator.

1. Emissivity ( $\varepsilon$ )

Intuition: A black matte surface feels hotter in sunlight than a shiny white one. Why? The black surface emits radiation more efficiently.

Definition: Emissivity is the ratio of radiation emitted by a real surface to the radiation emitted by an ideal blackbody at the same temperature.

\varepsilon = \frac{E_{\text{real}}}{E_{\text{blackbody}}}

$\varepsilon = 1$ for a perfect blackbody (ideal emitter).
$0 < \varepsilon < 1$ for all real surfaces.
$\varepsilon$ depends on material, surface finish, and wavelength.

Note

A good emitter is also a good absorber. This is Kirchhoff's law: at thermal equilibrium, $\varepsilon = \alpha$ (absorptivity) for the same wavelength and direction.

2. Absorptivity ( $\alpha$ )

Intuition: A black car roof gets hotter in summer than a white one. It absorbs more sunlight.

Definition: Absorptivity is the fraction of incident radiation that a surface absorbs.

\alpha = \frac{\text{absorbed radiation}}{\text{incident radiation}}

$\alpha = 1$ for a perfect blackbody (absorbs everything).
$\alpha = 0$ for a perfect reflector.
For opaque surfaces: $\alpha + \rho = 1$ , where $\rho$ is reflectivity.

Watch out

Don't confuse absorptivity with emissivity. A shiny metal has low $\alpha$ (reflects most light) and low $\varepsilon$ (emits poorly). A black surface has high $\alpha$ and high $\varepsilon$ .

3. Reflectivity ( $\rho$ ) and Transmissivity ( $\tau$ )

Intuition: A mirror reflects; a glass window transmits; a brick wall does neither.

Definition: For any surface, incident radiation is either absorbed, reflected, or transmitted:

\alpha + \rho + \tau = 1

Reflectivity $\rho$ : fraction reflected.
Transmissivity $\tau$ : fraction transmitted through.
For opaque solids: $\tau = 0$ , so $\alpha + \rho = 1$ .

The Blackbody: The Ideal Reference

A blackbody is a theoretical surface that:

Absorbs all incident radiation ( $\alpha = 1$ ).
Emits the maximum possible radiation at any temperature.
Follows Planck's law, Stefan-Boltzmann law, and Wien's displacement law.

Real surfaces are compared to this ideal. The blackbody is a standard — like comparing a real engine to a Carnot engine.

E_b = \sigma T^4

Putting It Together: A Real Surface

A hot metal plate at $500\ \text{K}$ with $\varepsilon = 0.3$ emits:

$E = 0.3 \times 5.67 \times 10^{-8} \times (500)^4 \approx 1063\ \text{W/m}^2$ …

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The rate of cooling for an object of a given material and mass is directly proportional to its surface area. A thin circular plate has the largest surface area for a given volume, while a sphere has the smallest. Therefore, the plate cools fastest and the sphere cools slowest. The correct option is (C).

When an object cools, it loses heat to its surroundings. This heat loss occurs primarily through radiation and convection. The rate at which an object loses heat is directly proportional to its surface area exposed to the surroundings, assuming the material, temperature difference, and emissivity are the same.

Important

For objects made of the same material and having the same mass, their volumes are identical. The rate of heat loss (and thus cooling) is directly proportional to the surface area of the object. An object with a larger surface area for a given volume will cool faster, and an object with a smaller surface area for a given volume will cool slower.

Let's break down the comparison:

Identify the governing principle: The rate of heat loss, $dQ/dt$ , from an object is proportional to its surface area $A$ and the temperature difference between the object and its surroundings. For radiation, Stefan-Boltzmann Law states $dQ/dt = \epsilon \sigma A (T^4 - T_0^4)$ , where $\epsilon$ is emissivity, $\sigma$ is Stefan-Boltzmann constant, $T$ is object temperature, and $T_0$ is surrounding temperature. For convection, Newton's Law of Cooling states $dQ/dt = hA(T - T_0)$ , where $h$ is the convective heat transfer coefficient. In both cases, the rate of heat loss is directly proportional to the surface area $A$ . A higher rate of heat loss means faster cooling.
Establish common parameters: We are given that all three objects (sphere, cube, and thin circular plate) are made of the same material and have the same mass. Since mass $m = \rho V$ (where $\rho$ is density and $V$ is volume), and both $m$ and $\rho$ are the same for all objects, their volumes $V$ must also be identical.
Compare surface areas for the same volume: Now we need to compare the surface areas of a sphere, a cube, and a thin circular plate, all having the same volume $V$ .
- Sphere: A sphere is the geometric shape that encloses the maximum volume for a given surface area, or equivalently, has the minimum surface area for a given volume.
- Cube: A cube has a larger surface area than a sphere of the same volume. …

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Q.A sphere, a cube and a thin circular plate, all of same material and same mass are initially heated to same high temperature.

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