If `P` represents radiation pressure , `C` represents the speed of light , and `Q` represents radiation energy striking a unit area per second , then non - zero integers `x, y, z` such that `P^(x) Q^(y) C^(z)` is dimensionless , find the values of `x, y , and z`.

Options

A. `x = 1, y = 1, z = 1`
B. `x =- 1, y =1, z = 1`
C. `x = 1,y =- 1, z = 1`
D. `x = 1, y = 1, z =- 1`

Answer

C

Explanation

Radiation pressure, `p = [ML^(-1)T^(-2)]`
Velocity of light, `c = [LT^(-1)]`
Energy striking unit area per second
`S = ([ML^(2)T^(-2)])/([L^(2)T]) = [MT^(-3)]`
Now, `p^(x)S^(y)c^(z)` is dimensionless.
`:. [M^(0)L^(0)T^(0)] = p^(x) S^(y) c^(z)`
or `[M^(0)L^(0)T^(0)] = [M^(1)L^(-1)T^(-2)]^(x) [M^(1) T^(-3)]^(y) [L^(1)T^(-1)]^(z)`
or `[M^(0)L^(0)T^(0)] = [M]^(x+y) [Li]^(-x+z) [T]^(-2x-3y -z)`
From principle of homogeneity of dimensions
`x = y = 0` ....(i)
`- x + z = 0` ...(ii)
`2x - 3y - z = 0`...(iii)
Solving Eqs. (i),(ii) and (iii), we get
`x = 1, y =- 1, z = 1`