Correct Answer - Option 4 : None of the above
The correct answer is option 4) i.e. None of the above
CONCEPT:
- The dimensional formula is used to express any physical quantity in terms of fundamental quantities.
-
Fundamental quantity: A quantity that can be measured is called a physical quantity.
-
Fundamental quantities are those which cannot be expressed or measured in terms of other physical quantities.
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The SI system has seven fundamental quantities i.e. time, length, mass, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
The seven fundamental quantities along with their dimensions are as follows:
| Fundamental quantity |
Dimension |
| Time |
T |
| Mass |
M |
|
Electric current
|
A |
| Thermodynamic temperature |
K |
| Amount of substance |
mol |
| Luminous intensity |
cd |
| Length |
L
|
EXPLANATION:
| Gravitational constant (G) |
- From the universal law of gravitation, \(F = \frac{Gm_1m_2}{R^2}\)
- \(\Rightarrow G =\frac{Fr^2}{m_1m_2} \)
- Dimensional formula of force, F = mass × acceleration ⇒ [M1 L1T-2]
- The Dimensional formula for mass is [M1]
- \(\Rightarrow G = \frac{[M^1 L^1T^{-2}] [L^2]}{[M^2]} = [M^{-1}L^3T^{-2}]\)
- G has dimensions.
|
| Planck's constant (h) |
- From the equation for the energy of a photon, \(E =\frac{hc}{λ}\)
-
\(\Rightarrow h =\frac{Eλ}{c}\)
- Dimensional formula of energy (E) = work done = force × displacement
= [M1L1T-2] × [L1] = [M1 L2T-2]
- Dimensional formula of wavelength (λ) = [L1]
- Dimensional formula of speed of light (c) = [L1T-1]
- \(\Rightarrow h =\frac{[M^1 L^2T^{-2}][L^1]}{[L^1T^{-1}]} = [M^1L^2T^{-1}]\)
- h has dimensions.
|
| Power of lens (P) |
- Power of a lens, \(P = \frac{1}{focal\:length}\)
- The focal length is a distance and has dimensions [L1].
- \(\Rightarrow P = \frac{1}{[L^1]} = [L^{-1}]\)
- P has dimensions.
|
- All the given quantities have dimensions.
- Hence, none of the above is dimensionless.
