One of the most beautiful features of Maxwell’s equations is their symmetry. In this post, we will prove that Maxwell’s equations are invariant under a parity transformation (spatial inversion).
A parity transformation (spatial inversion) is defined by:
\[ \hat{\mathbf{P}} \;\longmapsto\; \mathbf{r}' = -\mathbf{r}, \qquad t' = t \]Transformation rules:
Gradient operator: \[ \nabla' = \frac{\partial}{\partial \mathbf r'} = -\nabla \]
In Maxwell’s theory:
By treating \( \mathbf{E} \) as a polar vector, \( \mathbf{B} \) as an axial vector, and assigning the correct transformation rules to sources, we find that all four Maxwell equations remain unchanged under parity transformation.
This symmetry is one of the many reasons Maxwell’s theory is so elegant: it respects both the structure of space and the distinction between vectors and pseudovectors.
A neat mathematical trick told through a timeless folk tale
Do you know the Camel Principle?
It is a nifty trick we often use in derivations in mathematics and physics. The trick is to add and subtract a term in an expression, usually resulting in a simplification that leads to the solution of the problem at hand.
The Camel Principle got its name from a tale in ancient Arabian folklore. Long ago, there lived a rich merchant in Arabia whose property consisted of an oasis and 17 camels. He had three sons. Unfortunately, he died while his sons were still young, leaving behind a will to be executed. The will was a bit strange.
It stated that the oasis would remain in the common possession of the sons, while the eldest son was to receive half of the 17 camels, the second son one third, and the youngest one ninth. When the sons tried to divide the camels, a great dilemma arose. The camels could not be perfectly divided into a half, a third, and a ninth. The elders they consulted could not resolve the problem either. A greedy relative even suggested killing all the camels for a feast and selling the meat, then dividing the money among the sons, but the brothers loathed the idea.
Finally, a wise man from India came along — and he turned out to be a mathematician as well. When the sons asked him how to solve their dilemma, he thought for a moment and replied, “Seventeen is a prime number, so it can’t be done in the normal way. Let me add and subtract.” Then he brought one of his own camels and added it to the herd. “Dear young chaps,” he said, “behold — we now have 18 camels. The first son gets half of these, which is 9 camels.” He led 9 camels to stand beside the eldest son. Then he called the second son and said, “Now you get one third of the camels.” He gave him 6 camels. Finally, he granted the youngest son 2 camels, which is one ninth of 18. After that, the wise man took back his own camel — the one remaining after the sons had received 9 + 6 + 2 = 17 camels in total.
If you think the wise man cheated by adding his own camel, remember that he only increased the count temporarily—something the boys themselves would readily agree to!
Mathematical takeaway: Sometimes a problem becomes simple if we temporarily add something convenient, solve in that setting, and then subtract it away — changing the form, not the value.
| Property | Alpha Rays | Beta Rays | Gamma Rays |
|---|---|---|---|
| Nature | Helium nuclei (2 protons, 2 neutrons) | High-energy electrons or positrons | High-energy electromagnetic waves (photons) |
| Charge | +2e (positive) | -e (electrons) or +e (positrons) | Neutral (0) |
| Mass | ~4 u (6.644 × 10⁻²⁷ kg) | ~1/1836 u (9.109 × 10⁻³¹ kg) | Massless |
| Penetration Power | Low (stopped by paper or a few cm of air) | Moderate (stopped by a few mm of aluminum) | High (requires several cm of lead or meters of concrete) |
| Ionization Ability | High (strong interaction with matter) | Moderate (less than alpha) | Low (minimal interaction) |
| Speed | ~5-10% of speed of light (~1.5-3 × 10⁷ m/s) | Up to 99% of speed of light (~3 × 10⁸ m/s) | Speed of light (3 × 10⁸ m/s) |
The D-line doublet (5890 Å and 5896 Å) belongs to the principal series, arising from transitions between:
P-state (L = 1): Spin-orbit coupling splits it into two fine-structure levels:
S-state (L = 0): Only one term:
The D-line doublet consists of two allowed transitions:
The energy splitting between ²P3/2 and ²P1/2 is due to spin-orbit coupling. This fine-structure splitting explains the two closely spaced D-line wavelengths.
Introduction to atom model – vector atom model – electron spin –spatial quantisation– quantum numbers associated with vector atom model – L-S and J-J coupling – Pauli's exclusion principle – magnetic dipole moment due to orbital motion and spin motion of the electron – Bohr magnetron – Stern-Gerlach experiment – selection rules – intensity rule.
Origin of atomic spectra – excitation and ionization potentials – Davis and Goucher's method – spectral terms and notations – fine structure of sodium D-lines – Zeeman effect –Larmor's theorem – quantum mechanical explanation of normal Zeeman effect – anomalous Zeeman effect (qualitative explanation) –Paschen-Back effect – Stark effect
Discovery of radioactivity – natural radio activity – properties of alpha rays, beta rays and gamma rays – Geiger-Nuttal law – alpha particle spectra –Gammow's theory of alpha decay (qualitative study) –beta ray spectra – neutrino theory of beta decay – nuclear isomerism – internal conversion – nonconservation of parity in weak interactions.
Conservation laws of nuclear reaction – Q-value equation for a nuclear reaction – threshold energy – scattering cross section – artificial radio activity – application of radio isotopes – classification of neutrons – models of nuclear structure – liquid drop model – shell model.
Classification of elementary particles – fundamental interactions – elementary particle quantum numbers –isospin and strangeness quantum number – Conservation laws and symmetry – quarks – quark model (elementary ideas only) – discovery of cosmic rays – primary and secondary cosmic rays – latitude effect– altitude effect.
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Text Books
1. R. Murugesan, Modern Physics, S. Chand and Co. (All units)
(Units IandII-Problems)
2. Brijlaland N. Subrahmanyam, Atomic and Nuclear Physics, S.
Chand and Co. (All units)
3.J. B. Rajam, Modern Physics, S. Chand and Co.
4.SehgalandChopra, Modern Physics, Sultan Chand, New Delhi
5.Arthur Beiser– Concept of Modern Physics, McGraw Hill
Publication, 6th Edition.
Reference Books
1. Perspective of Modern Physics, Arthur Beiser, McGraw Hill.
2. Modern Physics, S. Ramamoorthy, National Publishing and Co.
3. Laser and Non-Linear Optics by B.B.Laud, Wiley Easter
Ltd.,New York,1985.
4.Tayal, D.C.2000 – Nuclear Physics, Edition, Himalaya Publishing
House, Mumbai.
5.Irving Kaplan (1962) Nuclear Physics, Second Edition, Oxford
and IBH Publish and Co, New Delhi.
6.J.B. Rajam– Atomic Physics, S. Chand Publication, 7th Edition.
7.Roy and Nigam, – Nuclear Physics (1967) First edition, Wiley Eastern
Limited, New Delhi.
1.
Hyper Physics Site
2.
Making Physics Fun
3.
Types of Decay Khan Academy
4.
Nuclei Khan Academy
