CBSE Class 12-science Answered

Electromagnetic waves are transversal in nature? Please provide me the mathamatical derivation

with most simple method

Asked by Anshul | 08 Sep, 2017, 06:25: PM

Expert Answer

1) Electromagnetic waves are transverse in nature. This statement can be proved by using Maxwell's equations.

2) According to Maxwell's equations, a region which is devoid of electric charge has

begin mathsize 12px style nabla times straight E equals 0 space and space nabla times straight B equals 0 end style

3) The above equations can be written as

begin mathsize 12px style fraction numerator partial differential straight E subscript straight x over denominator partial differential straight x end fraction plus fraction numerator begin display style partial differential straight E subscript straight y end style over denominator begin display style partial differential straight y end style end fraction plus fraction numerator begin display style partial differential straight E subscript straight z end style over denominator begin display style partial differential straight z end style end fraction equals 0 fraction numerator begin display style partial differential straight B subscript straight x end style over denominator begin display style partial differential straight x end style end fraction plus fraction numerator begin display style partial differential straight B subscript straight y end style over denominator begin display style partial differential straight y end style end fraction plus fraction numerator begin display style partial differential straight B subscript straight z end style over denominator begin display style partial differential straight z end style end fraction equals 0 end style

4) Let us assume that the electromagnetic wave propagates in the x-direction. Therefore, the components of the field are of the form

begin mathsize 12px style cos open parentheses kx minus ωt close parentheses end style

5) The variations along y and z directions are already zero. So, we have

begin mathsize 12px style fraction numerator partial differential straight E subscript straight x over denominator partial differential straight x end fraction equals 0 fraction numerator begin display style partial differential straight B subscript straight x end style over denominator begin display style partial differential straight x end style end fraction equals 0 end style

6) The above equations suggest that the electric and magnetic field components along x-direction are constant with respect to x. This statement directly means that only the y and z components of the field vary with x, thereby proving that the em waves are transverse in nature.

Answered by Romal Bhansali | 11 Sep, 2017, 11:39: AM