Question

When ${q_1}$ joules of radiation is incident on a body it reflects and transmits a total of ${q_2}$ joules. Find the emissivity of the body.

Answer 1

Hint: The formula for the emissivity of a body is applied which is given by the ratio of the amount of energy absorbed by the energy that is transmitted from the body.To find the amount of energy absorbed we need to subtract the energy that is transmitted from the surface from the total energy of the radiation that is incident on the surface.

Complete answer:
First, we must extract the data mentioned in the above question. The amount of radiation that is incident on a body is said to be ${q_1}$ joules. The amount of radiation that is reflected and transmitted from the body is given to be ${q_2}$ joules. The emissivity of the body is to be found out. The emissivity of a body is the measure of its ability to absorb or emit infrared radiation when compared with a black body under similar conditions. Thus the emissivity is defined as the ratio of the amount of radiation absorbed by the body by the incident radiation. The formula of emissivity is as follows:
$emissivity = \dfrac{{{q_{absorbed}}}}{{{q_{incident}}}}$ ---($1$)

Now, we calculate the amount of energy that is absorbed. Since the body transmits a total of ${q_2}$joules the absorption of the energy will be equivalent to the difference between the total amount of energy reflected and the amount which was incident on the body. Hence by this concept the following equation is observed:
The amount of absorbed radiation is: ${q_{absorbed}} = {q_{incident}} - {q_{transmitted}}$
Here, the radiation that is emitted is given by ${q_2}$ joules and the radiation that was incident is given by ${q_1}$. Hence:
${q_{transmitted}} = {q_2}$
$\Rightarrow {q_{incident}} = {q_1}$
$ \Rightarrow {q_{absorbed}} = {q_1} - {q_2}$ ----($2$)
The emissivity is equivalent to the coefficient of absorption that is given by the equation ($1$).
Substituting equation ($2$) in ($1$), we get:
$\therefore \text{emissivity} = \dfrac{{{q_1} - {q_2}}}{{{q_1}}}$

Thus the emissivity of the body in terms of ${q_1}$ and ${q_2}$ (in joules) is $\dfrac{{{q_1} - {q_2}}}{{{q_1}}}$.

Additional information: The amount of radiation reflected from the body signifies the temperature of the body which makes emissivity a useful concept. Usually, the emissivity of a material is calculated and compared with respect to a black body since it is one that absorbs all the radiation that is incident on it. This comparison is done in-order to determine the ability of the material to absorb or reflect infra-red radiation. The emissivity depends on the nature of the material as well wherein a polished material will have a lower amount of emissivity compared to a rough material.

Note:Emissivity is a dimensionless quantity and it lies in the range of zero to one. When the material or body is a perfect reflector then the amount of energy absorbed is zero and when the body that is considered is a black body then the amount of absorption is $100\% $ so emissivity is one.