Monday, 2 December 2019

electromagnetism - Prove EM Waves Are Transverse In Nature


Why we say that EM waves are transverse in nature? I have seen some proofs regarding my question but they all calculate flux through imaginary cube. Here is My REAL problem that I can't here imagine infinitesimal area for calculating flux because em line of force will intersect (perpendicular or not) surface at only one point so $E.ds$ will be zero so even flux through one surface of cube will always be zero. I am Bit Confused. I DON'T KNOW VECTOR CALCULUS BUT KNOW CALCULUS.




Answer




Why we say that em waves are transverse in nature?



In a region empty of electric charge, we have, from Maxwell's equations:


$$\nabla \cdot \vec E = \nabla \cdot \vec B = 0$$


Since you don't yet know vector calculus, let's rewrite these divergence equations as so:


$$\frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} = 0 $$


$$\frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} + \frac{\partial B_z}{\partial z} = 0 $$


Now, assume an electromagnetic wave is propagating in the $z$ direction so that the space and time variation of the field components are of the form



$$\cos(kz - \omega t)$$


Since the spatial variation is zero in the $x$ and $y$ directions, our equations become


$$\frac{\partial E_z}{\partial z} = 0$$


$$\frac{\partial B_z}{\partial z} = 0$$


Which means that electric and magnetic field components in the $z$ direction, the direction of propagation, must be constant with respect to $z$.


In other words, only the electric and magnetic field components transverse to the direction of propagation vary with respect to $z$. i.e., the electromagnetic wave is transverse.




Addendum to address a comment:



Why spatial Variations are zero in both x and y directions.




We stipulated that the field components are of the form $\cos(kz - \omega t)$ which means the wave is propagating in the $z$ direction.


Clearly, the partial derivative of $\cos(kz - \omega t)$ with respect to $x$ and $y$ is zero.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...