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Planck's constant has the dimension of:
1. Force
2. Energy
3. Linear momentum
4. Angular momentum

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Correct Answer - Option 4 : Angular momentum

Concept:

  • Planck's Constant (h): Light is said to be made up of small particles with zero rest mass known as photon. The energy of such photon is related with its frequency by Planck's constant.

E = h ν 

E is energy, ν is the frequency of the photon.

  • Dimensional Formula: It depicts the way to relate any unit to fundamental Units.
  • If we have to find The dimensional formula of any quantity, At first, we find its units in terms of these fundamental units, and then replace units with these dimensions.
  • Angular Momentum: The angular momentum of a body of mass m moving with speed v with radius of circular path r is 

L = m v r

  • The dimensional formula for angular momentum will be

(Dimension of mass)(Dimension of speed)(Dimension of radius) =  M L T -1 L = M L 2 T -1

Explanation:

Planck's constant can be given as 

\(h = \frac{E}{\nu}\)

The Dimensional Formula of Energy is M L 2 T - 2

The dimension of Frequency is T - 1 

The dimension of Planck's constant hence

\(H = \frac{ML^2 T^{-2}}{T^{-1}}\)

⇒ H = M L 2 T -1

This the dimension of angular momentum. 

So, angular momentum is the correct option.

  • Dimensional formula for Energy can be found using any expression for energy, for example, the expression for kinetic energy \(K = \frac{1}{2} mv^{2}\) 
  • Similarly, the dimensional formula of frequency can be obtained by relating it to the time period \(f = \frac{1}{T}\)
  • Similarly, the dimensional formula for any units can be obtained.

Fundamental Units Dimensions
Mass M
Length L
Time T
Temperature K
Current A

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