To find the dimensions of Planck's constant [tex]\( h \)[/tex] from the given equation [tex]\(\lambda = \frac{h}{P}\)[/tex], we will use dimensional analysis. Let's break this down step-by-step:
### Step 1: Understand the Given Equation
The given equation is:
[tex]\[
\lambda = \frac{h}{P}
\][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength.
- [tex]\( h \)[/tex] is Planck's constant.
- [tex]\( P \)[/tex] is the momentum.
### Step 2: Determine Dimensions of Known Quantities
We start by determining the dimensions of [tex]\(\lambda\)[/tex] and [tex]\( P \)[/tex].
#### Dimensions of Wavelength ([tex]\(\lambda\)[/tex]):
The wavelength is a measure of length. Therefore, its dimension is:
[tex]\[
[\lambda] = [L]
\][/tex]
#### Dimensions of Momentum ([tex]\(P\)[/tex]):
Momentum [tex]\( P \)[/tex] is defined as the product of mass and velocity. Hence, its dimensions are:
[tex]\[
[P] = [M][L][T]^{-1}
\][/tex]
where:
- [tex]\( [M] \)[/tex] represents mass.
- [tex]\( [L] \)[/tex] represents length.
- [tex]\( [T] \)[/tex] represents time.
### Step 3: Use Dimensional Analysis
Using the equation [tex]\(\lambda = \frac{h}{P}\)[/tex], we isolate [tex]\( h \)[/tex]:
[tex]\[
h = \lambda \cdot P
\][/tex]
### Step 4: Find the Dimensions of Planck's Constant ([tex]\( h \)[/tex])
Substitute the dimensions of [tex]\(\lambda\)[/tex] and [tex]\( P \)[/tex] into the equation:
[tex]\[
[h] = [\lambda] \cdot [P]
\][/tex]
Using the dimensions we identified earlier:
[tex]\[
[h] = [L] \cdot [M][L][T]^{-1}
\][/tex]
### Step 5: Simplify the Dimensions
Simplify the expression by combining like terms:
[tex]\[
[h] = [M][L] \cdot [L][T]^{-1}
\][/tex]
This further reduces to:
[tex]\[
[h] = [M][L]^2[T]^{-1}
\][/tex]
### Conclusion
The dimensions of Planck's constant [tex]\( h \)[/tex] are:
[tex]\[
[h] = [M][L]^2[T]^{-1]
\][/tex]
