Planck's Constant Dimensions Explained

The question asks us to identify which pair of physical quantities has the same dimensions as Planck's constant ($h$). We are given a key formula relating energy ($E$), Planck's constant ($h$), and frequency ($\nu$): $E=h\nu$. This formula will be crucial for our dimensional analysis.

Deriving Dimensions of Planck's Constant

First, let's find the dimensional formula for Planck's constant ($h$) using the provided equation $E = h\nu$. We can rearrange this to solve for $h$:

$h = \frac{E}{\nu}$

To proceed, we need the dimensions of Energy ($E$) and frequency ($\nu$).

• Energy ($E$): Energy has the same dimensions as work done. Work is defined as Force $\times$ Distance. The dimensions of Force are $MLT^{-2}$ (Mass $\times$ Acceleration), and the dimensions of Distance are $L$. Therefore, the dimensions of Energy are: $[E] = [Force] \times [Distance] = (MLT^{-2}) \times L = ML^2T^{-2}$
• Frequency ($\nu$): Frequency represents cycles per unit time. Its dimensions are: $[\nu] = T^{-1}$

Now, we substitute these dimensions back into the equation for $h$:

$[h] = \frac{[E]}{[\nu]} = \frac{ML^2T^{-2}}{T^{-1}}

Simplifying this expression gives us the dimensions of Planck's constant:

$[h] = ML^{2}T^{-2 - (-1)} = ML^{2}T^{-1}

Our goal is to find which of the given options results in these dimensions: $ML^2T^{-1}$.

Analyzing Options for Physical Quantity Dimensions

Let's calculate the dimensions for each combination given in the options:

Identifying Correct Dimension Match

After evaluating each option:

• Option 1 (Energy $\times$ frequency) yields dimensions of $ML^2T^{-3}$.
• Option 2 (Momentum $\times$ velocity) yields dimensions of $ML^2T^{-2}$.
• Option 3 (Force $\times$ time) yields dimensions of $MLT^{-1}$.
• Option 4 (Energy $\times$ time) yields dimensions of $ML^2T^{-1}$.

Comparing these results with the dimensions of Planck's constant ($[h] = ML^2T^{-1}$), we find that only Option 4 matches.

Therefore, the combination of Energy $\times$ time has the same dimensions as Planck's constant ($h$). These dimensions are also known as action or angular momentum.