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- Electromagnetic radiation travels in waves, but it interacts with matter in discrete energy transfers via
**photons**. The energy of a photon is proportional to its frequency:E = h ν,

where h is**Planck's constant**= 6.626 * 10^{-34}J s. - This quantized transfer of energy explains the spectrum of radiation emitted by a
**black body**: an object which absorbs*all*radiation incident on it. The radiation increases the thermal motion of the atoms and molecules in the body, which then re-radiates energy in a**thermal spectrum**. The resulting**Planck (black body) distribution**of intensity per unit wavelength, as a function of wavelength and temperature (K), isI(λ,T) = 2 π h c

where k^{2}/ (λ^{5}(e^{h c / (λ kB T)}- 1))_{B}is**Boltzmann's constant**= 1.381 * 10^{-23}J / K:(The solid

For a given temperature, the maximum is located at (2.9 * 10*normalized*curves correspond to the surface temperatures of Sirius A (9200 K), Sol (5800 K) and Betelgeuse (3800 K). The dashed curves are proportional intensities for Sol and Betelgeuse.)^{-3}m K) / T (**Wien's law**). - In the
**photoelectric effect**, light incident on a material can free electrons from the material via the momentum transfer from photons:p = h / λ.

If φ energy is required to eject an electron, a potential ΔV_{s}can be applied which will stop the process:e ΔV

where ΔV_{s}= h ν - φ_{s}is the**stopping potential**and φ is the**work function**. The frequency corresponding to zero stopping potential is ν_{cutoff}: the minimum frequency required to eject an electron. Note that this relationship is*independent of intensity*. - The
**Heisenberg uncertainty principle**states thatΔx Δp

and_{x}≥ h / (4 π)ΔE Δt ≥ h / (4 π)

Think in terms of the wavelength required to resolve a particle, and the resulting recoil. -
We now know four fundamental constants of nature:
- the speed of light, c = 299792458 m/s, which is the greatest possible speed;
- the electrical permittivity of the vacuum, ε
_{0}= 8.85418782 * 10^{-12}C^{2}/(N m^{2}), which determines the strength of the electrical field; - Newton's gravitational constant, G = 6.67428 * 10
^{-11}N m^{2}/ kg^{2}, which determines the strength of the gravitational field; and - Planck's constant, h = 6.62606896 * 10
^{-34}J s; the smallest possible angular momentum is h / 2.

These constants can be combined to set natural scales for our fundamental units:

- the
**Planck length**, L_{P}= √ (h G / c^{3}) = 4.05134 * 10^{-35}m; - the
**Planck time**, t_{P}= √ (h G / c^{5}) = 1.35138 * 10^{-43}s; - the
**Planck mass**, m_{P}= √ (h c / G) = 5.45552 * 10^{-8}kg; and - the
**Planck charge**, q_{P}= √ (ε_{0}h c) = 1.32621 * 10^{-18}C.

(Some sources define these quantities in terms of h / (2 π) instead of h, or 4 π ε

L_{0}instead of ε_{0}, but the orders of magnitude are essentially unchanged.)_{P}and t_{P}are often interpreted as the smallest measurable intervals of distance and time; m_{P}is the mass of 5.9989 * 10^{22}electrons, and q_{P}is the charge on 8.27756 electrons (as defined here).It is also interesting to note that these constants can be combined in easy ways so as to "convert" one fundamental unit into any other; in General Relativity (a geometric theory), it is usually convenient to express everything in terms of length, so that

- mass * G / c
^{2}→ length; - time * c → length;
- angular momentum / unit mass / c → length; and
- charge * √ (G / ε
_{0}) / c^{2}→ length.

This suggests that if indeed there is a theory which encompasses all physical phenomena (at least at the smallest scales), that all of the fundamental physics quantities (length, time, mass and charge) will be interrelated.

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©2011, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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