Explanation:
When two bodies are at different temperatures, they exchange heat through radiation. The rate of heat transfer by radiation depends on the emissivity of the bodies involved. Emissivity is a dimensionless property that quantifies the ability of a surface to emit thermal radiation.

In this case, we have a small body placed inside a very large enclosure. The emissivity of the enclosure is given as 0.5. We need to determine the equivalent emissivity for the radiant heat exchange between the small body and the enclosure.

Definition of Equivalent Emissivity:
The equivalent emissivity is a concept used to simplify the analysis of heat transfer between two bodies with different emissivities. It represents an effective emissivity value that takes into account the radiative properties of both bodies.

Method to Calculate Equivalent Emissivity:
To calculate the equivalent emissivity, we can use the following formula:

1/ε_eq = (1/ε_1) + (1/ε_2) - 1

Where:
ε_eq = Equivalent emissivity
ε_1 = Emissivity of body 1
ε_2 = Emissivity of body 2

In this case, body 1 is the small body and body 2 is the enclosure.

Substituting the values:
ε_1 = Emissivity of the small body = ?
ε_2 = Emissivity of the enclosure = 0.5

We can rearrange the equation to solve for ε_1:

1/ε_eq = (1/ε_1) + (1/0.5) - 1

1/ε_eq = (1/ε_1) + 2 - 1

1/ε_eq = (1/ε_1) + 1

Now, we can substitute the given values:

1/ε_eq = (1/ε_1) + 1

Since ε_eq = 0.4 (given in option B), we can substitute this value:

1/0.4 = (1/ε_1) + 1

2.5 = (1/ε_1) + 1

Subtracting 1 from both sides:

1.5 = 1/ε_1

Cross-multiplying:

ε_1 = 1/1.5

ε_1 = 2/3

Therefore, the emissivity of the small body is 2/3, which is approximately 0.667. This value does not match any of the given options, so the correct answer cannot be determined based on the information provided.