JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Units & Measurement

Expression for time in terms of G (universal gravitational constant), h (Planck constant) and c (speed of light) is proportional to:

Answer: (C)

Solution

Build a time out of

G

h

and

c

by dimensions alone. Write

t \propto G^{a} h^{b} c^{d}

and use

[G] = M^{- 1} L^{3} T^{- 2}

[h] = M L^{2} T^{- 1}

[c] = L T^{- 1}

, matching powers of

M

L

T

against

[T]

. Mass gives

- a + b = 0

, length gives

3 a + 2 b + d = 0

, and time gives

- 2 a - b - d = 1

. Solving,

a = b = \frac{1}{2}

and

d = - \frac{5}{2}

, so

t \propto \frac{G h}{c ^{5}}

-- this is the Planck time.

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Single correct option