Planck Constant's Dimensional Formula Explained

by Wholesomestory Johnson 48 views

Hello there! I understand you're curious about the dimensional formula for the Planck constant. Don't worry; I'll provide a clear, detailed, and correct answer to help you understand this fundamental concept in physics.

Correct Answer

The dimensional formula for Planck's constant is [M¹L²T⁻¹]. This represents the dimensions of mass (M), length (L), and time (T).

Detailed Explanation

Let's delve into why this is the correct answer and break down the concept step by step.

What is Planck's Constant?

Planck's constant, denoted by h, is a fundamental physical constant that plays a crucial role in quantum mechanics. It relates the energy of a photon to its frequency. In simpler terms, it's a tiny number that helps us understand the quantized nature of energy, meaning energy isn't continuous but comes in discrete packets.

The Significance of Dimensions

Before we dive into the specifics, let's quickly review what dimensional formulas are. Dimensional formulas express a physical quantity in terms of the fundamental dimensions: mass (M), length (L), and time (T). They help us understand the nature of a quantity and ensure that equations are dimensionally consistent. Think of it like this: you can't add apples and oranges; similarly, you can't add quantities with different dimensions unless you convert them to a common dimension.

How to Determine the Dimensional Formula

To find the dimensional formula for Planck's constant, we need to start with an equation that involves h. The most common and fundamental equation is:

  • E = hf

Where:

  • E is the energy of a photon
  • h is Planck's constant
  • f is the frequency of the photon

Let's rearrange the equation to solve for h:

  • h = E / f

Now, we need to determine the dimensions of energy (E) and frequency (f).

Dimensions of Energy (E)

Energy can be expressed in various forms, such as kinetic energy or potential energy. A convenient formula for kinetic energy is:

  • E = (1/2)mv²

Where:

  • m is the mass
  • v is the velocity

The dimensional formulas for mass (m) and velocity (v) are:

  • [M] for mass
  • [LT⁻¹] for velocity (length per time)

Therefore, the dimensional formula for energy (E) is:

  • E = (1/2)mv² → [M][LT⁻¹]² → [M][L²T⁻²] = [M¹L²T⁻²]

Dimensions of Frequency (f)

Frequency (f) is the number of cycles per unit of time. Its dimensional formula is:

  • [T⁻¹] (inverse of time)

Calculating the Dimensional Formula for Planck's Constant

Now we can substitute the dimensional formulas of energy (E) and frequency (f) into the equation h = E / f:

  • h = E / f
  • [h] = [M¹L²T⁻²] / [T⁻¹]
  • [h] = [M¹L²T⁻²T¹]
  • [h] = [M¹L²T⁻¹]

Therefore, the dimensional formula for Planck's constant is [M¹L²T⁻¹].

Another Approach: Using Angular Momentum

Another way to determine the dimensional formula for h is through the concept of angular momentum (L). In quantum mechanics, the angular momentum of a particle is quantized, meaning it can only take on specific discrete values. The formula for angular momentum is:

  • L = nh/2π

Where:

  • L is the angular momentum
  • n is an integer (quantum number)
  • h is Planck's constant

Since 2π and n are dimensionless constants, the dimensions of h must be the same as the dimensions of L.

Angular momentum (L) is given by:

  • L = r x p

Where:

  • r is the radius (distance)
  • p is the linear momentum

Linear momentum (p) is given by:

  • p = mv

Where:

  • m is the mass
  • v is the velocity

Now let's determine the dimensions for each quantity:

  • r (radius): [L]
  • p (linear momentum): [M][LT⁻¹] = [MLT⁻¹]

So, the dimensions of angular momentum (L) are:

  • [L] = [r x p] = [L][MLT⁻¹] = [M¹L²T⁻¹]

As you can see, using this approach also results in the dimensional formula for h being [M¹L²T⁻¹].

Units of Planck's Constant

The SI unit of Planck's constant is the joule-second (J·s). This unit is consistent with the dimensional formula we derived:

  • Energy (E) is measured in joules (J), which is equivalent to kg·m²/s² (kilogram meter squared per second squared).
  • Frequency (f) is measured in Hertz (Hz), which is equivalent to s⁻¹ (per second).

Therefore:

  • h = E / f → (kg·m²/s²) / (1/s) = kg·m²/s (which aligns with [M¹L²T⁻¹])

Real-World Examples and Applications

Planck's constant is a cornerstone of quantum mechanics and is used in various fields, including:

  1. Quantum Physics: It is central to understanding wave-particle duality, where particles can exhibit both wave-like and particle-like behavior. The energy of a photon is directly proportional to its frequency, with Planck's constant as the proportionality factor.
  2. Spectroscopy: Planck's constant helps scientists analyze the spectra of light emitted or absorbed by atoms and molecules, allowing them to identify substances and study their properties.
  3. Solid-State Physics: Planck's constant is used to understand the behavior of electrons in solids, crucial for developing semiconductors, transistors, and other electronic devices.
  4. Cosmology: It plays a role in understanding the very early universe and the formation of the first particles.

Common Mistakes to Avoid

  • Confusing Units: Always keep track of the units. The joule-second (J·s) is a specific unit for Planck's constant. Forgetting the units can lead to incorrect calculations.
  • Incorrect Formulas: Ensure you are using the correct formulas for energy, frequency, and other related quantities.
  • Dimensional Inconsistency: Always check for dimensional consistency in your calculations. Make sure the dimensions on both sides of an equation match.
  • Misunderstanding the Concept of Quantization: Remember that energy, at the quantum level, is not continuous but comes in discrete packets (quanta). Planck's constant helps define the size of these packets.

Key Takeaways

  • The dimensional formula for Planck's constant (h) is [M¹L²T⁻¹].
  • This formula can be derived using the relationship E = hf (Energy = Planck's constant x frequency) or through angular momentum.
  • The SI unit for h is the joule-second (J·s).
  • Planck's constant is fundamental to quantum mechanics, helping us understand the quantized nature of energy and wave-particle duality.
  • It's used in many fields, including quantum physics, spectroscopy, solid-state physics, and cosmology.

I hope this explanation clarifies the dimensional formula for Planck's constant. If you have any more questions, feel free to ask!