Maxwell Equations
The CBSE conducts the class 12 physics exams, and students are advised to know about the CBSE class 12 syllabus. One can score good marks in CBSE class 12 if they go through the textbook properly.
Moreover, they should also focus on test videos and other platforms and practice previous year's question papers as this would help them grow their concept and help them score good marks.
There are many online platforms that students can refer to to get good marks in physics exams. Once upon reading, this article will get to know about one of the most important physics topics: maxwell equation derivation.
There are 4 maxwell equations, but you will be discussed about the first two equations of Maxwell in detail. Please read the below sections carefully to have a piece of good knowledge about the two equations of Maxwell.
The equations have been explained briefly. Deriving maxwell's equation is one of the most important concepts in the class 12 physics syllabuses.
Equation of Maxwell
The first person to calculate the speed of propagation of electromagnetic waves was Maxwell. The speed of propagation of electromagnetic waves was the same as the light speed, and thus it can be concluded that EM waves and visible light are somewhat similar. Using Maxwell equations, one can derive maxwell equations most of the working relationships in the electricity and magnetism fields.
Maxwell's First Equation
The first maxwells equations' integral form is based on the Gauss law of electrostatic. The Gauss law of electrostatic states that in a closed surface integral of electric flux density is always equal to the charge enclosed over the surface.
Gauss Law Can Be Expressed Mathematically As
∯D⃗ .ds =Qenclosed (1)
The product of electric flux density vector and surface integral over a close surface is equal to the charge in enclosed
∯ D.ds =∭▽.D⃗ dv⃗ -(2)
There are multiple surfaces in any closed system, and it will have a single volume. Hence the surface integral mentioned above can be converted into a volume integral by considering the divergence of the same vector. This can be represented mathematically as below:
Hence combining equation one and equation 2, the following can be derived
∭▽.D⃗ dv⃗ =Qenclosed (3)
The charges, including a close surface, will get distributed over its volume. Hence one can define volume charge density in the following ways.
ρv=dQdv measured using C/m3
On rearranging the above equation, we get
dQ=ρvdv
After integrating the above equation, the following maxwell derivation is derived,
Q=∭ρvdv -(4)
The charges surrounding the close surface can be given by volume charge density over that particular volume
Upon substituting the above equation, we get;
∭▽.Ddv=∭ρvdv
Now canceling the integral volume present on both sides, we land upon Maxwell’s first equation, which is represented as below;
⇒▽.Ddv=ρv
Maxwell Second Equation
The second maxwells equation integral form is completely based on the Gauss law of magnetostatics. According to Gauss's law of magnetostatics, the close surface integral of magnetic flux density is equal to the total scalar magnetic flux surrounded within that surface.
Mathematically derivation of maxwell equations can be expressed as the following
∯B⃗ .ds=ϕenclosed (1)
What is Scalar electric flux?
Scalar electric fluxes are imaginary lines of force that radiate in the outward direction.
What is scalar magnetic flux?
Scalar magnetic flux is circulating magnetic flux that comes around a current-carrying conductor
Hence from the above equation, it can be concluded that magnetic flux cannot be enclosed in a close surface of any shape.
∯ B⃗ .ds=0 (2)
Upon applying the Gauss divergence theorem to equation 2, the surface integral can be converted into volume integral if divergence is considered.
⇒∯ B⃗ .ds=∭▽.B⃗ dv (3)
Upon substituting equation 3 into 2 we get the following
∭▽.B⃗ dv=0 -(4)
To satisfy the above equation, we get this equation, ∭dv=0 or ▽.B⃗ =0. Moreover, the volume of any object cannot be equal to 0. Thus we arrive at the second derivation of maxwell equations which is represented as below.
▽.B⃗ =0
