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 and simplifying. The steps to which have been outlined below:
In this post, let’s derive a general solution for the Laplacian in Spherical Coordinates. In future posts, we shall look at the application of this equation in the context of Fluids and Quantum Mechanics.
The Laplacian in Spherical coordinates in its ultimate glory is written as follows:
To solve it we use the method of separation of variables.
Plugging in the value of into the Laplacian, we get that :
Dividing throughout by and multiplying throughout by , further simplifies into:
It can be observed that the first expression in the differential equation is merely a function of and the remaining a function of and only. Therefore, we equate the first expression to be and the second to be . The reason for choosing the peculiar value of is explained in another post.
The first expression in (1) the Euler-Cauchy equation in .
The general solution of this has been in discussed in a previous post and it can be written as:
The second expression in (1) takes the form as follows:
The following observation can be made similar to the previous analysis
The first expression in the above equation (2) is the Associated Legendre Differential equation.
The general solution to this differential equation can be given as:
The solution to the second term in the equation (2) is a trivial one:
Therefore the general solution to the Laplacian in Spherical coordinates is given by:
Mars has always been an interesting planet to us earthlings. The possibility of life, rovers leaving no stone unturned(literally), it’s demanding reddish appearance and now those breathtaking sunsets.Mesmerizing isn’t it ? But,
Why are martian sunsets blue?
Here on earth, sunsets are bright with Yellow, Orange and Red colors dazzling in the sky. During sunsets, the light from the sun has to travel a longer distance in our atmosphere to reach the earth.
Consequently, all the blue and violet light is scattered( thrown in various directions) by the particles in our atmosphere leaving behind only shades of yellow, orange and red, which is what you see. This phenomenon is known as Rayleigh scattering.
On mars, the reverse effect occurs. The martian dust is smaller and more abundant than on earth and it incidentally happens to be just the right size that it absorbs the blue light whilst scattering the red ones across the sky. This makes martian sunsets blue :).
Stay tuned, there is more space stuff coming your way.