WBJEE 2024 Physics & Chemistry Paper

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:
‌

hc

m

‌

hc

m

Substituting the known dimensions:
[‌

hc

m

]=‌

[E][T][L]∕[T]

[M]

=‌

[M][L]2∕[T]2[L]∕[T]

[M]

=[L]3
Option B:
‌

h

mc2

Substituting the known dimensions:
[‌

h

mc2

]=‌

[E[T]

[M][L]2∕[T]2

=‌

[M][L]2∕[T]2[T]

[M][L]2∕[T]2

=[T]=[1∕c]−1[L]

Option D:
‌

h

mc

Substituting the known dimensions:
[‌

h

mc

]=‌

[E][T]

[M][L]∕[T]

=‌

[M][L]2∕[T]2[T]

[M][L]∕[T]

=[L]
Thus, the quantity that has the dimension of length is Option D:
‌

h

mc