15:I[8216,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"BackButton"] 16:I[1553,["972","static/chunks/972-b28da0de92681928.js","47","static/chunks/47-a7f06e9eda1ac756.js","115","static/chunks/app/%5Bexam%5D/questions/%5Byear%5D/%5Bsubject%5D/%5Bchapter%5D/%5BquestionId%5D/page-7b6f0a12b9cc0206.js"],"QuestionViewer"] 17:T789,

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:

$$\frac{\mathrm{hc}}{\mathrm{m}}$$

Substituting the known dimensions:

$$[\frac{\mathrm{hc}}{\mathrm{m}}] = \frac{[E][T][L]/[T]}{[M]} = \frac{[M][L]^2/[T]^2[L]/[T]}{[M]} = [L]^3$$

Option B:

$$\frac{\mathrm{h}}{\mathrm{mc}^2}$$

Substituting the known dimensions:

$$[\frac{\mathrm{h}}{\mathrm{mc}^2}] = \frac{[E][T]}{[M][L]^2/[T]^2} = \frac{[M][L]^2/[T]^2[T]}{[M][L]^2 / [T]^2} = [T] = [1 / c]^{-1}[L]$$

Option C:

$$\frac{\mathrm{h}^2}{\mathrm{mc}^2}$$

Substituting the known dimensions:

$$[\frac{\mathrm{h}^2}{\mathrm{mc}^2}] = \frac{([E][T])^2}{[M][L]^2/[T]^2} = \frac{[M]^2[L]^4/[T]^4

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