Question

The temperature of a black body is increased by 50%, then the percentage increase in the rate of radiation by the body is approximated as

• A. 50%
• B. 100%
• C. 400%
• D. 150%

Hint:

For scaling problems with a power of 4, you can quickly estimate the answer using simple fractions: a 50% increase scales the variable by a factor of $\frac{3}{2}$. Raising this to the fourth power gives $\frac{81}{16} \approx 5$. A final value that is 5 times the original implies a net increase of exactly 4 times the baseline, which immediately converts to 400

Answer 1

The Correct Option is C

Step 1: Recall Stefan's law.
The energy radiated per second per unit area by a black body depends on the fourth power of its absolute temperature: $$E \propto T^4.$$
Step 2: Express the new temperature.
A $50\%$ rise means $$T_2 = T_1 + 0.5\,T_1 = 1.5\,T_1.$$
Step 3: Form the ratio of radiation rates.
Using the fourth-power law, $$\frac{E_2}{E_1} = \left(\frac{T_2}{T_1}\right)^4 = (1.5)^4.$$
Step 4: Evaluate $(1.5)^4$.
$1.5^2 = 2.25$, and $2.25^2 = 5.0625$, so $E_2 \approx 5.06\,E_1.$
Step 5: Find the fractional increase.
The extra radiation is $$\frac{E_2 - E_1}{E_1} = 5.0625 - 1 = 4.0625.$$
Step 6: Convert to a percentage.
Multiplying by $100$ gives about $406\%$, which rounds to the nearest option of $400\%$.
\[ \boxed{\approx 400\%} \]