Question

Consider two spherical perfect blackbodies with radii \(R_1\) and \(R_2\) at temperatures \(T_1 = 1000\, \text{K}\) and \(T_2 = 2000\, \text{K}\), respectively. They both emit radiation of power 1 kW. The ratio of their radii, \(R_1/R_2\), is given by ...............

Hint:

The total power radiated by a blackbody varies as \(A T^4\). For equal power output, the radius scales inversely with the square of temperature.

Answer 1

Correct Answer: 4

Physics Principle

The power ($P$) radiated by a perfect blackbody is given by the Stefan-Boltzmann Law:

$$P = \sigma A T^4$$

where:

$P$ is the total radiant power (in Watts).

$\sigma$ is the Stefan-Boltzmann constant ($\sigma \approx 5.67 \times 10^{-8}\ W\cdot m^{-2}\cdot K^{-4}$).

$A$ is the total radiating surface area (in $m^2$).

$T$ is the absolute temperature (in $K$).

For a sphere, the surface area $A$ is given by $A = 4\pi R^2$, where $R$ is the radius.

Substituting the area for a sphere into the Stefan-Boltzmann Law:

$$P = \sigma (4\pi R^2) T^4$$

$$P = 4\pi \sigma R^2 T^4$$

Given Data

The problem involves two blackbodies, body 1 and body 2, with the following properties:

Property	Body 1	Body 2
Radius	$R_1$	$R_2$
Temperature ($T$)	$T_1 = 1000\ K$	$T_2 = 2000\ K$
Power ($P$)	$P_1 = 1\ kW$	$P_2 = 1\ kW$

Since the power emitted is the same, $P_1 = P_2$.

Calculation

Set the power equations for both blackbodies equal to each other:

$$P_1 = P_2$$

$$4\pi \sigma R_1^2 T_1^4 = 4\pi \sigma R_2^2 T_2^4$$

The constants $4\pi$ and $\sigma$ cancel out:

$$R_1^2 T_1^4 = R_2^2 T_2^4$$

Rearrange the equation to find the ratio $R_1/R_2$:

$$\frac{R_1^2}{R_2^2} = \frac{T_2^4}{T_1^4}$$

Take the square root of both sides:

$$\frac{R_1}{R_2} = \sqrt{\frac{T_2^4}{T_1^4}}$$

$$\frac{R_1}{R_2} = \frac{T_2^2}{T_1^2}$$

Now, substitute the given temperature values:

$$\frac{R_1}{R_2} = \left(\frac{T_2}{T_1}\right)^2$$

$$\frac{R_1}{R_2} = \left(\frac{2000\ K}{1000\ K}\right)^2$$

$$\frac{R_1}{R_2} = (2)^2$$

$$\frac{R_1}{R_2} = 4$$

Final Answer

The ratio of the radii, $R_1/R_2$, is 4.