99 Percentile Qs Bank for JEE MainPhysicsUnits and Dimensions
The speed of light c , gravitational constant G and Planck’s constant h are taken as fundamental units in a system. The dimensions of time in this new system should be:
Options
- AG 1 2 h 1 2 c − 5 2
- BG − 1 2 h 1 2 c 1 2
- CG 1 2 h 1 2 c − 3 2
- DG 1 2 h 1 2 c 1 2
Correct answer
A. G 1 2 h 1 2 c − 5 2
Step-by-step solution
Let time, T ∝ c X G Y h Z ⇒ T = k c X G Y h Z Taking dimensions on both sides [ M 0 L 0 T 1 ] = [ L T − 1 ] X [ M − 1 L 3 T − 2 ] Y [ M L 2 T − 1 ] Z i.e., [ M 0 L 0 T 1 ] = [ M − Y + Z L X + 3 Y + 2 Z T − X − 2 Y − Z ] Equating powers of M , L , and T , on both sides, we get – − Y + Z = 0 ... ( 1 ) X + 3 Y + 2 Z = 0 ... ( 2 ) − X − 2 Y − Z = 1 ... ( 3 ) From equation (i) Adding (2) and (3) Y + Z = 1 or 2 Y = 1 ( ∵ Z = Y ) ∴ Z = Y = 1 2 Putting these values in (2), we get – X + 3 2 + 1 = 0 or X = − 5 2 Hence, [ T ]