Q27 / 110 · NTA Abhyas (NEET) 2020 · Physics · Units 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:

  • Correct answer
    A

      G 1 2   h 1 2   c −5 2

  • B

      G −1 2   h 1 2   c 1 2

  • C

      G 1 2   h 1 2   c −3 2

  • D

      G 1 2   h 1 2   c 1 2

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+3Y+2Z T −X−2Y−Z ]

Equating powers of M, L, and T, on both sides, we get –

  −Y+Z=0     ...(1)

  X+3Y+2Z=0      ...(2)

  −X−2Y−Z=1       ...(3)

From equation (i)

Adding (2) and (3)

  Y+Z=1

or            2Y=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]=[ G 1 2   h 1 2   c −5 2 ]

Hence, the correct answer is option: 'A'.

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