2023 A/L Physics Past Paper · Question 06 (English Medium)

Answer: (4) sixteen times.

What the question asks

We want: how many times as big the energy radiated per unit time per unit area becomes, when the absolute temperature of a blackbody is made $2$ times as big.

The paper says "increased by $2$ times". It means the temperature becomes $2$ times as big. The options are read the same way.

What each letter means:

$E$ ($\text{W m}^{-2}$): energy radiated per unit time per unit area.

$T$ ($\text{K}$): absolute temperature of the blackbody.

$\sigma$ ($\text{W m}^{-2}\,\text{K}^{-4}$): Stefan's constant.

Relevant theory

The total power radiated from a unit area of a hot black body is directly proportional to

the fourth power of the absolute temperature of the body.

Required equations

$E = \sigma T^4$

Reasoning and calculation

What stays the same: the constant $\sigma$ and the blackbody.

What changes: the absolute temperature $T$, and with it the energy $E$.

For example, watch an electric stove ring as it heats up.

At first it glows a dull red. Then it glows bright orange, and every patch of it sends out much more heat and light each second.

It is not just a little more. The law says the energy goes with $T$ to the fourth power, so a small rise in $T$ makes a big rise in $E$.

Write the law for the first temperature and for the second, then divide.

$$E_1 = \sigma T_1^4 \qquad E_2 = \sigma T_2^4$$

$$\dfrac{E_2}{E_1} = \left(\dfrac{T_2}{T_1}\right)^4$$

The constant $\sigma$ cancels.

The new temperature is $2$ times the old one.

$$\dfrac{T_2}{T_1} = 2$$

Put it in:

$$\dfrac{E_2}{E_1} = 2^4 = 16$$

Reading the options

The energy becomes $16$ times as big. That is option (4).

Options (1), (2), (3) and (5) are $2^1$, $2^2$, $2^3$ and $2^5$. Each uses the wrong power.

Checks

Dimension check:

$\dfrac{E_2}{E_1}$ is $\text{W m}^{-2}$ divided by $\text{W m}^{-2}$. The units cancel, so it is a pure number, as a count of times should be.

$n * n * n * n$

Does it make sense?

The temperature went up, so the energy must go up.

The factor must be more than $2$, because the energy goes with the fourth power of $T$. $16$ is more than $2$.

Conclusion

$\dfrac{E_2}{E_1} = 16$

$\dfrac{E_2}{E_1} = \left(\dfrac{T_2}{T_1}\right)^4 = 2^4 = 16$.

The answer is (4).

Stefan's law has $T^4$. If $T$ becomes $n$ times as big, $E$ becomes $n^4$ times as big.

Write the ratio $\left(\dfrac{T_2}{T_1}\right)^4$ and put in the number.

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