The dimensional formula of Planck's Constant is

Understanding the Dimensional Formula of Planck's Constant

The question asks for the dimensional formula of Planck's constant. Planck's constant, denoted by \(h\), is a fundamental constant in quantum mechanics. It relates the energy of a photon to its frequency. 

The relationship between the energy (\(E\)) of a photon and its frequency (\(\nu\)) is given by the equation:

\(E = h\nu\)

To find the dimensional formula of \(h\), we can rearrange this equation:

\(h = \frac{E}{\nu}\)

Now, we need to determine the dimensional formulas for energy (\(E\)) and frequency (\(\nu\)).

Dimensional Formula for Energy (E)

Energy has the same dimensions as work. Work is defined as force multiplied by distance (\(W = F \times d\)).

First, let's find the dimension of force (\(F\)). According to Newton's second law, Force = mass \(\times\) acceleration (\(F = ma\)).

• Dimension of mass [M] = [M]
• Dimension of acceleration [a] = \(\frac{\text{Dimension of velocity}}{\text{Dimension of time}}\). Velocity is distance/time, so dimension of velocity = \(\frac{\text{[L]}}{\text{[T]}} = \text{[LT\(^{-1}\)]}\). Therefore, dimension of acceleration = \(\frac{\text{[LT\(^{-1}\)]}}{\text{[T]}} = \text{[LT\(^{-2}\)]}\).

So, the dimension of force [F] = [M] \(\times\) [LT\(^{-2}\)] = [MLT\(^{-2}\)].

Now, the dimension of energy [E] = Dimension of Force \(\times\) Dimension of distance = [MLT\(^{-2}\)] \(\times\) [L] = [ML\({}^2\)T\(^{-2}\)].

Dimensional Formula for Frequency (\(\nu\))

Frequency is defined as the number of cycles per unit time. It is the reciprocal of the time period (\(T\)).

\(\nu = \frac{1}{T}\)

The dimension of time period [T] = [T].

So, the dimension of frequency [\(\nu\)] = \(\frac{1}{\text{[T]}} = \text{[T\(^{-1}\)]}\).

Deriving the Dimensional Formula for Planck's Constant (h)

Using the relation \(h = \frac{E}{\nu}\), we can now substitute the dimensional formulas for E and \(\nu\):

Dimension of \(h\) = \(\frac{\text{[E]}}{\text{[}\nu\text{]}} = \frac{\text{[ML\(^2\)T\(^{-2}\)]}}{\text{[T\(^{-1}\)]}}\)

Dimension of \(h\) = [ML\({}^2\)T\(^{-2}\) \(\times\) T\({}^{1}\)]

Dimension of \(h\) = [ML\({}^2\)T\(^{-2+1}\)]

Dimension of \(h\) = [ML\({}^2\)T\(^{-1}\)]

This dimensional formula can also be written as [MT\(^{-1}\)L\({}^2\)] by rearranging the terms, which represents Mass, Time to the power -1, and Length to the power 2.

Comparing with Options

Let's compare our derived dimensional formula [ML\({}^2\)T\(^{-1}\)] or [MT\(^{-1}\)L\({}^2\)] with the given options:

1. [MLT]
2. [MT\(^{-1}\)L\(^2\)]
3. [MT\(^2\)L\(^2\)]
4. [MT\(^{-2}\)L\(^2\)]

Our derived formula [ML\({}^2\)T\(^{-1}\)] matches the second option when the terms are reordered.

Revision Table: Key Concepts

Additional Information: Planck's Constant and Angular Momentum

Another way to think about the dimensions of Planck's constant is related to angular momentum. In quantum mechanics, angular momentum is often quantized in units of \(\hbar = h/(2\pi)\). Since \(2\pi\) is dimensionless, \(h\) has the same dimensions as angular momentum.

Angular momentum (\(L\)) for a particle with momentum (\(p\)) at a position (\(r\)) is \(L = r \times p\). Momentum \(p = mv\).

• Dimension of position [r] = [L]
• Dimension of mass [m] = [M]
• Dimension of velocity [v] = [LT\(^{-1}\)]
• Dimension of momentum [p] = [M] \(\times\) [LT\(^{-1}\)] = [MLT\(^{-1}\)]

Dimension of Angular Momentum [L] = Dimension of position \(\times\) Dimension of momentum = [L] \(\times\) [MLT\(^{-1}\)] = [ML\({}^2\)T\(^{-1}\)].

This confirms that the dimension of Planck's constant [h] is indeed [ML\({}^2\)T\(^{-1}\)] or [MT\(^{-1}\)L\({}^2\)].

Understanding Planck's Constant Dimensions

The question asks for the dimensional formula of Planck's Constant. Finding the dimension of planck's constant involves using a known physical formula that includes this fundamental constant. 

Calculating the Dimensional Formula of Planck's Constant

We can determine the dimensional formula of Planck's constant ($h$) by using a well-known relationship from quantum physics, such as the formula for the energy of a photon:

\(E = h\nu\)

Here, \(E\) represents energy, \(h\) is Planck's constant, and \(\nu\) is the frequency of the photon. To find the dimensional formula of planck's constant h, we need to know the dimensions of energy and frequency.

Dimensions of Energy (E)

Energy has the same dimensions as work. Work is defined as Force multiplied by Distance.

• Dimensions of Force: Force is mass times acceleration. Acceleration is velocity change over time, and velocity is displacement over time. So, dimensions of acceleration are \([LT^{-2}]\). Dimensions of Force are \([M][LT^{-2}] = [MLT^{-2}]\).
• Dimensions of Work (Energy): Work = Force \(\times\) Distance. So, dimensions of Energy are \([MLT^{-2}][L] = [ML^2T^{-2}]\).

The dimensions of Energy are \([ML^2T^{-2}]\). This will be crucial in determining the planck's constant dimensional formula.

Dimensions of Frequency (\(\nu\))

Frequency is the number of cycles per unit time, which is the reciprocal of the time period.

• Dimensions of Time Period: \([T]\).
• Dimensions of Frequency: Frequency = 1 / Time Period. So, dimensions of Frequency are \([T^{-1}]\).

The dimensions of Frequency are \([T^{-1}]\). Now we have the components needed to find the dimension of planck's constant.

Deriving Dimensions of Planck's Constant (h)

From the formula \(E = h\nu\), we can rearrange it to solve for \(h\):

\(h = \frac{E}{\nu}\)

Now we substitute the dimensions we found for \(E\) and \(\nu\) into this equation:

Dimensions of \(h = \frac{\text{Dimensions of } E}{\text{Dimensions of } \nu}\)

Dimensions of \(h = \frac{[ML^2T^{-2}]}{[T^{-1}]}\)

To simplify, we move the \([T^{-1}]\) from the denominator to the numerator, which changes the sign of the exponent:

Dimensions of \(h = [ML^2T^{-2}] \times [T^{1}]\)

Dimensions of \(h = [ML^2T^{-2+1}]\)

Dimensions of \(h = [ML^2T^{-1}]\)

So, the calculated dimensional formula of planck constant is \([ML^2T^{-1}]\). This gives us the core planck constant dimension.

Matching the Dimensions of Planck's Constant with Options

Our calculated dimension for Planck's constant is \([ML^2T^{-1}]\). We need to check the given options to see which one matches this result. Remember that the order of the fundamental dimensions (M, L, T) does not change the overall dimension.

Comparing our derived dimension \([ML^2T^{-1}]\) with the options, we see that Option 2 is \([MT^{-1}L^{2}]\). Although the order of \(T^{-1}\) and \(L^2\) is swapped compared to our standard format, the powers of M, L, and T are the same ([M¹L²T⁻¹]). Therefore, this is the correct planck’s constant dimension.

The dimensional formula of planck's constant is indeed \([ML^2T^{-1}]\), which corresponds to option 2 \([MT^{-1}L^2]\).

Understanding the dimensional formula of planck's constant h is important for various calculations and dimensional analysis in physics. The planck's constant dimensions are fundamental to quantum mechanics.

To summarize, by using the energy-frequency relation, we found the dimensional formula of planck constant to be \([ML^2T^{-1}]\), confirming the dimensions of planck's constant match option 2.