Question

If P represents radiation pressure, c represents the speed of light and Q represents radiation energy striking a unit area per second, then the values of non-zero integers x, y and z, such that  P^xQ^yC^z is dimensionless, are: 

1. x = 1, y = 1, z = -1
2. x = 1, y = -1, z = 1
3. x = -1, y = 1, z = 1
4. x = 1, v = 1, z = 1

Answer 1

Correct Answer - Option 2 : x = 1, y = -1, z = 1

CONCEPT:

Principle of homogeneity of dimensions:

• According to this principle, a physical equation will be dimensionally correct if the dimensions of all

          the terms occurring on both sides of the equation are the same.

• This principle is based on the fact that only the physical quantities of the same kind can be

           added, subtracted, or compared.

• Thus, velocity can be added to velocity but not to force.

EXPLANATION:

Given that, P = pressure, Q = Energy striking per unit area per second, C = speed of light

Let k = P^xQ^yC^z....(1)

Dimensions of k = [M⁰L⁰T⁰]

Therefore Dimensions of \(Pressure=\frac{Force}{Area}=\frac{MLT^{-2}}{L^2}\) = ML^-1T^-2

\(Q=\frac{Energy}{Area \times time}=\frac{MLT^{-2}}{L^2}=\frac{ML^{2}T^{-2}}{L^{2}T}=MT^{-3}\)

\(C=LT^{-1}\)

Substituting these values in equation (1)

\(M^{0}L^{0}T^{0}=[ML^{-1}T^{-2}]^{x}[MT^{-3}]^{y}[LT^{-1}]^{z}\)

Applying the principle of homogeneity of dimensions, we get

x + y = 0...(i)

-x + z = 0...(ii)

-2x - 3y - z = 0...(iii)

solving (i),(ii),(iii), we get

x = 1,y = -1,z = 1

The correct x = 1,y = -1,z = 1