Jackson’s Laplacian in spherical Coordinates

If you took a look at one of the previous posts on how to remember the Laplacian in different forms by using a metric,  you will notice that the form of  the Laplacian that we get is:

\nabla^2 \psi = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \psi}{\partial r} \right) + \frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta} \left( sin(\theta)  \frac{\partial \psi}{\partial \theta} \right)   + \frac{1}{r^2 \sin^2(\theta)} \frac{\partial^2 \psi}{\partial \phi^2}   

But in Jackson’s Classical Electrodynamics, III edition he notes the following:


This is an interesting form of the Laplacian that perhaps not everyone has encountered. This can obtained from the known form by making the substitution u = r \psi and simplifying. The steps to which have been outlined below:




Can you crush a soda can with lightning ?

Here you can see the lightning striking the empire state building three times.

As you might know skyscrapers like these have a lightning rod that will carry the lightning bolt’s electrical charge through
the path of least resistance along the cable into the ground, reducing
the risk of fire or heat damage from the strike.


But you might know from high school or college physics that when you have two current carrying wires parallel to each other, then they experience an attractive force towards each other.


                                               Source Video                    

Lightning carries 30000 amps

An average bolt of negative lightning carries an electric current of 30,000 amperes (30 kA).

So if you were to pass such high currents through a rod, then surely they must experience a substantial magnetic force towards each other and get crushed right ?


Absolutely, Physics always works ! Have a look at this metal tube:


This is what would happen when electric lightning is passed through a metal tube. Using magnetic forces to compress electrical filaments is known as ‘Pinching’.

Can Crusher

One cool application for this would be for forming metal cans into interesting shapes.


As soon as the spark gap fires the capacitor discharges an enormous amount
of current through the coil (tens of thousands of amperes).  This
discharge creates a magnetic field around the coil. 

As the flux lines
pass through the cross section of the can, current is induced and flows
around the can.

 This induced current creates its own magnetic field
which opposes the magnetic field from the coil.  Between the two magnetic
fields there is now a force pushing inward on the can and outward on the
coil.  Once the force is strong enough the can is crushed.


And with large enough voltage, one can blow the can in opposite directions too!

From generating X-rays to forming of metal into shapes, Pinches have a lot of interesting applications. The references section of the wikipedia is surely a fun ride; check it out and have a good one!


I have always been fascinated by Pokemon.

Tiding through the waves of time, now that I think about it : Pokemon did teach me a lot about physics, especially electricity. 

What is Electricity ?


Electricity stems from a potential difference between two areas,
which allows for electromotive force to ensue in mobile electrons.



In biological cells, a voltage imbalance or a cell potential difference exists between the inside and outside of a a cell.

The cell achieves this by removing 3 sodium ions for every 2 potassium ions allows into the system. The removing process consumes energy ( ATP ).


                               The sodium ions leaving the cell 


                            The Potassium ions entering the cell

Source Video

Pikachu and Bioelectrogenesis

Where does pikachu gets it’s electrical powers ? 


Its by a process known as bioelectrogenesis.

Bioelectrogenesis is the generation of electricity by living organisms

How it works is rather blunt. Remember I told you that the cells are maintained in a potential difference.

There are passageways /electrolytes that are present that allows a flow of ions through them.


                                            Ion Passageways

When required,
the brain of the pikachu sends a signal through the nervous system to these
electrolytes, opening ions channels and reversing charge polarity,
causing an abrupt difference in electric potential.

The final effect is the generation of electric current, capable of going up to 100,000 Volts during its thunderbolt move.

Result : Opponent stupefied.


Water is better conductor than air

Most of animals that bioelectrogenic in nature are aquatic creatures ( electric eels, rays, cattlefish, etc ) . This is because water is a much better electrical conductor than air, therefore electrical signals signals can be transmitted through water.

This betters the chance for the organism to protect itself against predators. Pikachu is not aquatic because probably the writes didn’t want it be so – Poetic License ;P


Some other pokemons that were also bioelectrogenic  were:
Eelektrik and the Eelektross


The voice of Pikachu – must watch

Electrogenic Humans

The one that ash has is a male pikachu. There is a female to the species as well. ( Look at the tail )


That’s pokemon physics for you folks.

Hope you enjoyed reading this post as much i did drafting it. Oh boy! There is physics just in about everything !

Electroluminescent wires.

Electroluminescent wire (often abbreviated as EL wire) is a thin copper wire coated in a phosphor which glows when an alternating current is applied to it.

A Star Wars Geek?

In the spirit of the new Star Wars movie, you can even make your own lightsaber with these EL wires. 🙂


Have a Great day!


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