Dimensions of Constants for a Dimensionless Quantity – Rankers Physics
Topic: Unit And Dimensions
Subtopic: Dimensions

Dimensions of Constants for a Dimensionless Quantity

(P) represents radiation pressure, (c) represents speed of light and (S) represents radiation energy striking per unit area per sec. The non-zero integers (x, y, z) such that \(P^x S^y c^z\) is dimensionless are:

[1992]

(x = 1, y = 1, z = 1)
(x = -1, y = 1, z = 1)
(x = 1, y = -1, z = 1)
(x = 1, y = -1, z = -1)

Solution:

Dimensions: \(P = [ML^{-1}T^{-2}]\), \(c = [LT^{-1}]\), \(S = [MT^{-3}]\). For \(P^x S^y c^z\) to be dimensionless, powers of M, L, T must be zero. \(M: x+y=0\). \(L: -x+z=0\). \(T: -2x-3y-z=0\). Solving gives \(y=-x\) and \(z=x\). Taking \(x=1\) yields \(y=-1\), \(z=1\).

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