The famous Stefan's law of radiation states that the rate of emission of thermal radiation per unit by a black body is proportional to area and fourth power of its absolute temperature that is `Q=sigmaAT^(4)` where A = area, T = temperature and `sigma` is a universal constant. In the 'energy-length-time temperature' (E-L-T-K) system the dimension of `sigma` is

Options

A. `E^(2)T^(2)L^(-2)K^(-2)`
B. `E^(-1)T^(-2)L^(-2)K^(-1)`
C. `ET^(-1)L^(-3)K^(-4)`
D. `ET^(-1)L^(-2)K^(-4)`

Answer

To find the dimension of the universal constant \(\sigma\) in Stefan's law of radiation, we start with the equation:

\[ Q = \sigma A T^4 \]

where:
- \(Q\) is the rate of emission of thermal radiation,
- \(A\) is the area,
- \(T\) is the absolute temperature.

### Step 1: Rearranging the equation for \(\sigma\)

We can rearrange the equation to express \(\sigma\):

\[ \sigma = \frac{Q}{A T^4} \]

### Step 2: Finding the dimensions of each term

1. **Dimension of \(Q\)**:
- \(Q\) represents energy per unit time (rate of emission).
- The dimension of energy (\(E\)) is \([E]\) and the dimension of time (\(T\)) is \([T]\).
- Therefore, the dimension of \(Q\) is:

\[ [Q] = \frac{[E]}{[T]} = [E][T]^{-1} \]

2. **Dimension of \(A\)**:
- Area (\(A\)) is given by length squared.
- The dimension of length (\(L\)) is \([L]\).
- Therefore, the dimension of area is:

\[ [A] = [L]^2 \]

3. **Dimension of \(T^4\)**:
- Temperature (\(T\)) is represented by \(K\) (Kelvin).
- Therefore, the dimension of temperature is:

\[ [T] = [K] \]

- Thus, the dimension of \(T^4\) is:

\[ [T^4] = [K]^4 \]

### Step 3: Substituting the dimensions into the equation for \(\sigma\)

Now we can substitute the dimensions back into the equation for \(\sigma\):

\[ [\sigma] = \frac{[Q]}{[A][T^4]} = \frac{[E][T]^{-1}}{[L]^2[K]^4} \]

### Step 4: Simplifying the expression

This simplifies to:

\[ [\sigma] = [E][T]^{-1}[L]^{-2}[K]^{-4} \]

### Conclusion

Thus, the dimension of \(\sigma\) in the energy-length-time-temperature (E-L-T-K) system is:

\[ [\sigma] = [E][T]^{-1}[L]^{-2}[K]^{-4} \]