To determine which quantity has the dimension of length, we need to analyze the dimensions of each option provided. We have:
−h as Planck's constant
-
m as the mass of the electron
−c as the velocity of light
First, we need to recall the dimensions of these constants:
- Planck's constant (h) has the dimension of action, which is energy multiplied by time:
[h]=[E][T]where
[E] is the dimension of energy and
[T] is the dimension of time.
- The dimension of energy
(E) is given by
[E]=[M][L]2∕[T]2, where
[M] is mass,
[L] is length, and
[T] is time.
- The dimension of mass
(m) is simply
[M].
- The dimension of the velocity of light
(c) is
[c]=[L]∕[T].
We can calculate the dimensions for each option:
Option A:
Substituting the known dimensions:
[]===[L]3Option B:
Substituting the known dimensions:
[]==| [M][L]2∕[T]2[T] |
| [M][L]2∕[T]2 |
=[T]=[1∕c]−1[L] Option D:
Substituting the known dimensions:
[]==| [M][L]2∕[T]2[T] |
| [M][L]∕[T] |
=[L]Thus, the quantity that has the dimension of length is Option D: