Planck's Constant Formula: Explained Simply
Hello there! I understand you're curious about the dimensional formula of Planck's constant. Don't worry, I'm here to provide you with a clear, detailed, and correct explanation. Let's dive in!
Correct Answer
The dimensional formula of Planck's constant is [M¹L²T⁻¹].
Detailed Explanation
Let's break down why the dimensional formula is [M¹L²T⁻¹] and what it means. We'll explore the concept of Planck's constant and its significance in physics.
What is Planck's Constant?
Planck's constant, denoted by h, is a fundamental physical constant that governs the behavior of energy at the atomic and subatomic levels. It plays a crucial role in quantum mechanics, which describes the nature of matter and energy at the smallest scales.
It was introduced by Max Planck in 1900 while studying the problem of black-body radiation. Planck's constant helps us understand that energy is not emitted or absorbed continuously, but in discrete packets called quanta.
Importance of Planck's Constant
- Quantum Mechanics: It's a cornerstone of quantum mechanics. It links a particle's energy to its frequency, which is fundamental in understanding the behavior of particles and waves. Think of it like the key to unlocking the quantum world. Without it, we can't explain how atoms work, or how light interacts with matter. For example, the energy of a photon (a particle of light) is directly proportional to its frequency, and the proportionality constant is Planck's constant.
- Wave-Particle Duality: It helps describe the wave-particle duality of matter, meaning particles can behave as waves, and waves can behave as particles. The constant connects a particle's momentum and wavelength.
- Uncertainty Principle: It's central to the Heisenberg Uncertainty Principle, which states that we cannot know both the position and momentum of a particle with perfect accuracy. The precision of one measurement affects the precision of the other, and Planck's constant sets the limit.
The Formula
The energy (E) of a photon is related to its frequency (ν) by the following equation:
E = hν
Where:
- E is the energy of the photon (measured in Joules, J)
- h is Planck's constant (measured in Joule-seconds, J⋅s)
- ν (nu) is the frequency of the photon (measured in Hertz, Hz or s⁻¹)
Deriving the Dimensional Formula
To find the dimensional formula of Planck's constant, we'll rearrange the energy equation and then use the dimensions of the other quantities.
- Rearrange the Formula: Solve for h: h = E / ν
- Dimensional Analysis: Now, let's look at the dimensions of each term:
- Energy (E): Energy is defined as the ability to do work. The dimensional formula for energy is [M¹L²T⁻²]. This comes from the formula for kinetic energy (KE = 1/2 * mv²), where m is mass, and v is velocity.
- Frequency (ν): Frequency is the number of cycles per second. The dimensional formula for frequency is [T⁻¹].
- Substitute Dimensions: Substitute the dimensions into the rearranged equation: h = E / ν becomes h = [M¹L²T⁻²] / [T⁻¹]
- Simplify: Divide the dimensions: h = [M¹L²T⁻²] * [T¹] = [M¹L²T⁻¹]
Therefore, the dimensional formula of Planck's constant is [M¹L²T⁻¹].
Breakdown of the Dimensions
- M: Represents mass (measured in kilograms, kg).
- L: Represents length (measured in meters, m).
- T: Represents time (measured in seconds, s).
The exponents (1, 2, and -1) indicate how these fundamental dimensions are related. The formula [M¹L²T⁻¹] tells us how Planck's constant depends on mass, length, and time.
Real-World Example: The Photoelectric Effect
A classic example where Planck's constant is crucial is the photoelectric effect. When light shines on a metal surface, electrons are emitted. The energy of these emitted electrons depends on the frequency of the light and Planck's constant, not the intensity of the light (which was a problem for classical physics). This experiment demonstrated the quantum nature of light, where light behaves as a stream of photons, each with energy dependent on Planck's constant.
Another Example: Atomic Spectra
Planck's constant is also fundamental to understanding atomic spectra. Atoms absorb and emit light at specific frequencies, creating unique spectral lines. These frequencies are directly related to the energy levels of the electrons within the atom, and the differences between these energy levels are quantized by Planck's constant.
Comparing Planck's Constant to Other Constants
Planck's constant is one of several important physical constants. Others include:
- Speed of Light (c): Represents how fast light travels in a vacuum. (≈ 299,792,458 m/s). The dimensional formula is [L¹T⁻¹].
- Gravitational Constant (G): Describes the strength of gravitational attraction. (≈ 6.674 × 10⁻¹¹ N⋅m²/kg²). The dimensional formula is [M⁻¹L³T⁻²].
These constants, along with Planck's constant, are essential for building a complete picture of the universe.
Key Takeaways
- Planck's constant, denoted as h, is a fundamental constant in physics.
- Its dimensional formula is [M¹L²T⁻¹].
- It is crucial in understanding quantum mechanics and governs energy at the atomic and subatomic levels.
- It links energy to frequency in the equation E = hν.
- It explains phenomena like the photoelectric effect and atomic spectra.
I hope this explanation has been helpful. If you have any more questions, feel free to ask!
