It is very important to know the electromagnetic wave theory, So that in this theory many important concept is hidden, as i have already stated above that changing electric field create magnetic field, Now you will see here its proof how changing electric field create magnetic field, so for this you have to understand concept of Displacement current and to understand displacement current you need to understand Ampere’s law according to this law.
Ampere’s law ∫ B.dl = 𝞵₀iₑₙ now you will understand this law meaning in depth, what Ampere’s law states by this mathematical expression so see the below picture all cases step by step.
Take first case a current i is following in the wire as shown, Now you take a loop as shown and apply Ampere’s law, as you know Ampere’s law is valid for closed loop.
∫ B.dl = 𝞵₀iₑₙ
For left hand side I have taken a loop but for right hand side expression you have to take a surface so i have taken a flat surface. iₑₙ represent the current cutting and passing through that flat surface, so here i is current cutting and passing through that flat surface.
Hence you can write Ampere’s ∫ B.dl = 𝞵₀iₑₙ = 𝞵₀i this is for first case.
Now take second case, As shown in figure same process but this time with different surface that is hemispherical now again apply Ampere’s law for this case you will get.
∫ B.dl = 𝞵₀iₑₙ = 𝞵₀i same value as previous case value, so you can take any surface for Ampere’s law, here loop is same hence result is same.
Now take third case as shown in figure now apply the Ampere’s law for this case.
∫B.dl = 𝞵₀iₑₙ now for the left hand side take a loop but for right hand side 𝞵₀iₑₙ you have to take a surface which is again flat, see clearly in this case iₑₙ is zero, how ? see the same amount of current is cutting the surface and same current return back as shown in figure, hence iₑₙ is zero.
So Ampere’s law ∫B.dl = 𝞵₀iₑₙ = 0.
Now take forth case as shown in figure with same loop but hemispherical surface and apply Ampere’s law, now in this case current is not cutting hemispherical surface, hence iₑₙ is also zero in this case.
So you can write ∫B.dl = 𝞵₀iₑₙ = 0 , now see the case third and forth value is same that is zero but the reason in both case is different, you can see in third case same current cutting the surface and same current return back, but in forth case current has not cut any surface, without cutting the surface current has returned back.
But anyway value of equation are coming same, whatever you are choosing the surface.
Now take fifth case as shown in figure, where two plate of capacitor are shown, basically this is a RC circuit, Now again take a loop for the wire, take a flat surface as shown and apply Ampere’s law.
→ → → →
∫B.dl = 𝞵₀iₑₙ see here i current is cutting the flat surface so ∫B.dl = 𝞵₀iₑₙ = 𝞵₀i which is correct .
Now take for sixth case as shown in figure take a loop and consider a hemispherical surface here it is important see iₑₙ is not cutting the hemispherical surface, Now apply Ampere’s law
So ∫B.dl = 𝞵₀iₑₙ = 0 because iₑₙ is zero .
Now you compare fifth and sixth case, in fifth case value ∫B.dl = 𝞵₀iₑₙ = 𝞵₀i where as in
sixth case ∫B.dl = 𝞵₀iₑₙ = 0 now this is contradiction different surface have different value where as you are taking same loop, but in rest cases first, 2nd, 3rd and 4th value was coming same for different surfaces hence it is clear that for fifth and sixth case Ampere’s law is WRONG. it must give the same value but not like this.
Actually when Ampere’s law was given at those time scientist were checked case first and third, no body was thought about case fifth and sixth but later a great scientist Maxwell thought about this case.
Hence Maxwell realized that Ampere’s law is not correct fully, so different value are coming for fifth and sixth case, hence Maxwell correct this equation and given the correct equation as given below.
Ampere’s Maxwell Law
∫B.dl = 𝞵₀iₑₙ + 𝞵₀𝝐₀d𝛟ₑ/dt Maxwell added new term 𝞵₀𝝐₀d𝛟ₑ/dt in the previous Ampere’s law hence this equation was called Ampere’s Maxwell law.
Now see the term d𝛟ₑ/dt this is electric flux, note flux may be electric as well as magnetic but here suffix e denote this flux is electric flux.
Now you can apply Maxwell law for all the above cases.
For first and second case there is no any electric field hence electric flux will be zero so the term 𝞵₀𝝐₀d𝛟ₑ/dt will be zero hence equation will be same as Ampere’s law and is correct as shown.
For third and forth case iₑₙ is zero as well as electric field is also zero so electric flux also zero hence.
∫B.dl = 𝞵₀iₑₙ = 0
Now for fifth case see there is a capacitor hence capacitor will charge and there will be electric field due to charge accumulation in between capacitors as shown in figure.
So what will be value of that electric field, Perhaps you know E = Q/𝝐₀A .
Now you see there is electric field lines but this field lines are not passing through the Ampere’s loop, electric field are between the capacitor hence flux through this surface is zero so 𝞵₀𝝐₀d𝛟ₑ/dt will be zero.
Hence ∫B.dl = 𝞵₀iₑₙ = 𝞵₀i.
Now for sixth case see there is electric flux electric filed is passing through hemispherical surface to the capacitor so how much electric flux ? this will be 𝛟 =EA now from above
EA = Q/𝝐₀ so 𝛟 = Q/𝝐₀ now differentiate this with respect to time.
d𝛟/dt = 1/𝝐₀*dQ/dt = i/𝝐₀ , now put in equation 𝞵₀𝝐₀d𝛟ₑ/dt value of d𝛟/dt you will get
∫B.dl = 𝞵₀i
Now you see the magic in case fifth and sixth value is same that is ∫B.dl = 𝞵₀i .
now you can see when you were using Ampere’s law, the value was different for fifth and sixth cases, but using Maxwell law equation are giving same value.
So Ampere’s Maxwell law is correct because this law is correct for all the above cases .
∫B.dl = 𝞵₀iₑₙ + 𝞵₀𝝐₀d𝛟ₑ/dt = 𝞵₀(iₑₙ + 𝝐₀d𝛟ₑ/dt) , Now see the dimension of 𝝐₀d𝛟ₑ/dt is current dimension. well iₑₙ is called conduction current and 𝝐₀d𝛟ₑ/dt is called displacement current this name was given by Maxwell.
Now see the sixth case of circuit carefully, where capacitor plate having charge (+Q) and (-Q), As the current will flow through circuit the value of charge at capacitor (+Q) will increase, and due to this charge there will be a electric field between capacitor as shown and i have stated this earlier.
Now the term 𝝐₀d𝛟ₑ/dt is associated with electric field, because d𝛟ₑ/dt is electric flux, see the term 𝝐₀d𝛟ₑ/dt is displacement current name given by Maxwell but actually this is not current, This is simply associated with electric field as shown in sixth case, whereas Maxwell given this name as current, because its dimension is same as current.
Now you can write the above expression 𝞵₀(iₑₙ + 𝝐₀d𝛟ₑ/dt) = 𝞵₀(ic +id) where ic is conduction current i which is flowing in the circuit as shown.id is displacement current id current is in between the capacitor as told by Maxwell.
Now important concept conduction current is due to flow of charge, but when Maxwell is saying displacement current is in between the capacitors then there is flow of charge between the capacitor plate, Is it correct ? BIG no.
It is clear that there is no any charge flow between the two capacitor plate. So you must understand that displacement current does not mean flow of charge. Then what is meaning of displacement current here ?.Well displacement current meaning here is changing in electric field. But how ? see as the current will flow through the circuit, then charge (+Q) will accumulate at capacitor plate so charge (+Q) will increase and hence electric field will increase.
So it is very simple displacement current is due to changing electric field, you can see the mathematical expression for displacement current id = 𝝐₀d𝛟ₑ/dt , where d𝛟ₑ/dt is electric flux .
BIG Concept for displacement current
Due to changing electric field produce displacement current, Now you think electric field is a vector quantity, Hence electric field can change by two ways first either by changing its magnitude or second way by changing its direction only keeping magnitude constant, So here this is case of due to changing both magnitude as well as direction of electric field hence electric field are changing here due to both magnitude as well as direction, so finally changing electric flux.
For second case keep charge magnitude constant and accelerate that charge back and forth in this case electric field direction will change, so this is also changing electric field hence displacement current will also be there without flow of charge.
Now third case take a stationary charge will it change electric field ? No because neither electric field direction changing nor its magnitude is changing.
Hence a stationary charge will not change electric field, so no displacement current will be due to stationary charge.
Mathematically you can see if electric field is constant then electric flux will be also constant and hence d𝛟ₑ/dt will be zero, Hence displacement current will be zero.
Conduction and displacement current create magnetic field
Both current ic and id create magnetic field see below picture.
|Conduction and displacement current
From the first figure conduction current ic carries charge so at positive plate Q will accumulate and value of Q will go on increasing, Hence electric field will also increase as shown in figure. Hence changing electric field so create displacement current as shown.
Now you see the second above picture, As you know conduction current carrying in a wire create magnetic field as shown in red loop.
Similarly due to displacement current between the capacitor plat will also create magnetic field lines as shown in second picture. So as due to conduction current magnetic field lines are created in the same way due to displacement current magnetic field lines are created.
Note Electric field and magnetic field are perpendicular to each other, Now you can think due to displacement current (id) magnetic field are creating (B), and displacement current are created due to changing electric field (E).
So it is confirmed that when changing electric field create magnetic field and vice versa is also true changing magnetic field also create electric field how beautiful gift of nature electric and magnetic field.
How does electromagnetic wave is created ?
Source of electromagnetic wave is accelerating charge, Due to this accelerating charge electric field is changes and due to electric field changes magnetic field is created, Hence due to electric and magnetic field changes electromagnetic wave is created.
Hence to make accelerating charge you need a LC oscillation circuit where L is inductor and C is a capacitor in this case once inductor will charge and capacitor will discharge, in next time capacitor will charge and inductor will discharge, Hence charge will keep oscillation left and right.
But nature has many ways to create electromagnetic wave and it has been observed, but how you will understand this is electromagnetic wave.
Well in every electromagnetic wave weather it is created by LC oscillation or by nature but one property will be same that is velocity of electromagnetic wave is always a constant and it is equal to speed of light C = 3×10⁸ m/s.
So we assume that light is an electromagnetic wave in which electric and magnetic field are present.
Now you must understand charge properties in different states.
1 A static charge create a electric field around it, This electric field apply force on a static electric charge.
2 A moving charge with constant velocity (electric current) create magnetic field around it, This magnetic field exert force on a moving charge. but this magnetic field can’t exert force on a static charge. This is as simple as that a moving charge will experience force in magnetic field but static charge will not experience any force in magnetic field.
3 An accelerating charge create electric field as well as magnetic field and push ahead both electric and magnetic field. So this accelerating charge create electromagnetic wave.
Now think what will do this electromagnetic wave ?
Wherever this electromagnetic wave will fall, it will accelerate charge, Hence this electromagnetic wave comes in antenna and accelerate charge of antenna.
Now this acceleration create force and this force create work and energy, But think from where this energy is coming in electromagnetic wave.
Well think at the time of LC oscillation circuit we use battery source energy to create electromagnetic wave from this battery source energy is taken and carries this energy in the form of electromagnetic wave and finally deliver this energy to antenna.
Hence in this way electromagnetic wave carries energy and transfer to your mobile antenna.
Properties of electromagnetic wave
Is there momentum in electromagnetic wave ? As you know that momentum mathematical formula is p = mv that indicates if any particle having velocity as well as mass has momentum, Now you think light has no mass, so it should not has momentum, but it has. because momentum concept is changed after Albert Einstein.
light is packet of photon and photon has mass but can’t measured, so for that momentum concept was changed rather than mass concept.Hence momentum defined as again .
Momentum is a property of a moving particle, which will not change when moving particle will go from one point to other point, so momentum is conserved.
So momentum is a physical property, which is conserved between two location. Hence anything having velocity and energy has momentum.
So electromagnetic wave has velocity and energy so it has momentum.
Hence momentum p = Energy/Velocity , So energy with photon = E = h𝝼 ,
now velocity of electromagnetic wave c = speed of light so momentum p = h𝝼/c
you also know c = 𝝺𝝼 so put in momentum p = h/𝝺 where 𝝺 is wavelength of electromagnetic wave and h is plank’s constant.
Electromagnetic wave create pressure, because due to change in momentum create force and force per unit area is pressure but it value is so small that you don’t feel.
Electromagnetic wave equation
See below wave equation varying with time magnitude of electric field as well as magnetic field change,according to displacement and time.
You should remember that electric field and magnetic field are always perpendicular to each other and wave propagation equation also perpendicular to both electric and magnetic field
where 𝛚 is angular frequency, k = 2𝛑/𝛌 is called wave constant or wave number .
E₍ₓ,ₜ₎ = Eₘcos(kx-𝛚t +𝛟) , B₍z,ₜ₎ = Bₘcos(kx-𝛚t +𝛟) , This is the equation of electric and magnetic field.
questions are asked on the basis of this equation, compare and find the required parameter 𝛌 , 𝛚 , 𝛋 and 𝛎 , 𝛟 So remember the equation to find the asked value above relations.
Electromagnetic wave source
The source of electromagnetic wave is electric charge, You know there are two basic characteristics of material one is mass and other is electric charge. Mass is that characteristics, which creates gravitational field. you can refer gravitational field concepts.
Charge is that characteristics, which creates electric field around it. there can be three state of a charge.
1 A stationary charge ( Velocity is zero )
2 A moving charge ( Velocity is constant )
3 An accelerating charge ( Velocity is variable )
According to this velocity there is effect around it. A stationary charge create stationary electric field. And a moving charge create electric field as well as magnetic field.
Sample of moving charge is current in a conductor see below picture.
|Electromagnetic wave source
Conductor carrying current is not charge body because every time number of proton and number of electron in conductor is same, so it is not a charge body.So it can’t create electric field.
see at one end electron inter in conductor and same time other end electron out from conductor hence net charge on conductor is always zero, electron in motion but number of electron and proton are always same on conductor body.
So current in a conductor only create magnetic field. Hence current in a conductor does not create electromagnetic wave.
Accelerating charge as shown in figure q create electric field, due to motion of charge create magnetic field and due to increase in charge velocity both electric and magnetic field are pushed forward.
So electric and magnetic field start moving forward. Hence accelerating charge create electromagnetic wave.
Electromagnetic wave propagation
Propagation means spreading in all the directions, motion word is not used because normally motion word is used for one particular direction, But for a point source electromagnetic wave goes in all the directions, So you use the word propagation.
Propagate means spread out, Now how this electromagnetic wave spread out ? well you have learned, when an electric charge accelerate or oscillate a magnetic field is created and oscillating charge also create a varying electric field.
You know if there is a positive and negative charge then electric field direction is from positive charge to negative charge like this .
(+q)→(-q) see below picture.
I will continue this post with more important concept. I hope you have enjoyed learning electromagnetic wave. I want your feedback, comments, like and share, learn more and grow thanks for sharing.
Dated 16th Dec 2018