Q119 / 155 · WBJEE 2024 · Physics
Which of the following quantity has the dimension of length?
(h is Planck's constant, m is the mass of electron and c is the velocity of light)
- A$\frac{\mathrm{hc}}{\mathrm{m}}$
- B$\frac{\mathrm{h}}{\mathrm{mc}^2}$
- C$\frac{\mathrm{h}^2}{\mathrm{mc}^2}$
- Correct answerD$\frac{\mathrm{h}}{\mathrm{mc}}$
Solution
$$
\begin{aligned}
& \text { Hint : }[\mathrm{h}]=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right] \\
& {[\mathrm{c}]=\left[\mathrm{LT}^{-1}\right]} \\
& {[\mathrm{m}]=\left[\mathrm{M}^1 \mathrm{~L}^0 \mathrm{~T}^0\right]} \\
& \text { Dimension of length } \\
& {[\mathrm{L}]=\mathrm{h}^{\mathrm{x}} \mathrm{c}^{\mathrm{y}} \mathrm{m}^z} \\
& {\left[\mathrm{~L}^1\right]=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]^{\mathrm{x}}\left[\mathrm{LT}^{-1}\right]^{\mathrm{y}}\left[\mathrm{M}^1 \mathrm{~L}^0 \mathrm{~T}^0\right]^{\mathrm{z}}}
\end{aligned}
$$
Dimensional analysis on mass
$$
0=x+z \Rightarrow x=-z \Rightarrow z=-1
$$
Dimensional analysis on length
$$
1=2 x+y \Rightarrow 2 x-x=1 \Rightarrow x=1
$$
Dimensional analysis on time
$$
0=-x-y \Rightarrow-x=+y \Rightarrow y=-1
$$
Therefore final formula is $\frac{h}{\mathrm{mc}}$
Hence, the correct answer is option: 'D'.