Determine the dimensions of Planck's constant h using the equation E = hc/ extlambda.

Question

ScanSolve AI Tutor · Accepted Answer

To find the dimensions of Planck's constant $h$ , we use the given energy equation:

E = \frac{hc}{\lambda}

where $E$ is energy, $h$ is Planck's constant, $c$ is the velocity of light, and $\lambda$ is wavelength.

Rearrange the equation to solve for $h$ .
$h = \frac{E \lambda}{c}$
Determine the dimensions of each known quantity.
- Energy ( $E$ ): Energy has the dimensions of work, which is force times distance. Force is mass times acceleration ( $F=ma$ ). $[E] = [M][L T^{-2}][L] = [M L^2 T^{-2}]$
- Wavelength ( $\lambda$ ): Wavelength is a measure of length. $[\lambda] = [L]$
- Velocity of light ( $c$ ): Velocity is distance over time. $[c] = [L T^{-1}]$
Substitute these dimensions into the equation for $h$ .
$[h] = \frac{[M L^2 T^{-2}] [L]}{[L T^{-1}]}$
Simplify the expression.
$[h] = \frac{M L^{2+1} T^{-2}}{L T^{-1}}$ $[h] = \frac{M L^3 T^{-2}}{L T^{-1}}$ $[h] = M L^{3-1} T^{-2 - (-1)}$ $[h] = M L^2 T^{-2+1}$ $[h] = M L^2 T^{-1}$
Comparing this with the given options:
A. $MLT^{-1}$
B. $ML^1T^{-1}$
C. $ML^2T^{-1}$
D. $ML^2T^{-2}$

The calculated dimension matches option C.

The final answer is $\boxed{\text{C. } ML^2T^{-1}}$ .

Step-by-step solution