Electromagnetic

Magnetic Flux

Let a uniform magnetic field passing through a surface S, as shown in Figure. The area vector be A nA; where A is the area of the surface and n its unit normal. The magnetic flux through the surface is given by

  B.A  ABcos

where  is the angle between Band n. For non – uniform fields



 s

A d B. 



The SI unit of magnetic flux is the weber (Wb); 1Wb = 1T.m2.

Faraday’s Law of Induction

In 1831, Michael Faraday discovered that, time varying magnetic field, could generate an electric field. The phenomenon is known as electromagnetic induction. Faraday’s experiment demonstrates that an electric current is induced in the loop by changing the magnetic field. The coil behaves as if it were connected to an emf source. Experimentally it is found that the induced emf depends on the rate of change of magnetic flux through the coil. A.C. currents or A.C Field are examples of a time varying fields.

It states that time varying magnetic field induces an electromotive force (emf), which may establish a current in closed circuit. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc. Produced emf is given by

dt d

emf  

The -ve sign indicates that em.f, opposes the changes in flux, which produces it.

Also, emf 



E.dl

This integral is along closed loop. Therefore this relation is valid only for time varying fields, since for static field it is zero for a closed loop





  BdS

dt d dt d l d

E.   . 





  dS

dt B d S

d

E 

 



.

. (By Stoke’s Theorem) 

dt B d E

 

   

Faraday's law is a fundamental relationship, which comes from Maxwell's equations. It explains the ways in which a voltage (or emf) may be generated by a changing magnetic environment.

Displacement Current

Maxwell postulated that a changing electric field is equivalent to a current, which flows as long as the electric field is changing. This equivalent current produces the same magnetic effect as an ordinary current in a conductor. This equivalent current is known as displacement current.

Thus there is a need to a modification in Ampere’s law for time varying fields. To establish the concept of displacement current, Let a parallel plate capacitor is been charged through a series circuit as shown in the Figure. As soon as the key is pressed the capacitor start charging and the current flows through the connecting wires. This current reduces as the amount of charge increases on the plates. When the capacitor is fully charged to the e.m.f. of the cell, the current stops. During the charging process, no actual flow of charges between the plates during charging. If we place a compass needle in the space between the plates, the needle deflects. This indicates that there is a magnetic field between the capacitor plates even though there is no motion of charge. This indicates that there must be some other source of magnetic field in the gap. This additional source is due to changing electric field between the plates.

In figure plane surface S1 and hemispherical S2 are around the capacitor plate. Let both the surface are bounded by the same closed path l. Applying Ampere’s circuital law for

both surfaces.

For surface S1,





1

0 .

S

I l d

B  

For surface S2,





2

. S

o l d B 

Because during the charging process, current I is flowing through plane surface S1 and I is zero at all points on S2. Thus above two statements contradict each other and both can’t be correct. Maxwell removed this inconsistency by adding a new term,

displacement current, in main electric current. It is generated due to changing electric field in a source of magnetic field in the gap between the capacitor plates (during charging). The magnitude of displacement current is given by,₀d_E /dt, where E is the flux of the electric field through an area bounded by the closed curve. Therefore for changing electric field along with electric current, the modified Ampere’s law is given by,



 

1

(

. 0

S

I l

d

B   ₀d_E /dt)

Thus time varying electric fields, produce a magnetic field lines that curl around the electric field lines. If we have electric fields, which are varying sinusoidally with time, they will produce magnetic fields also varying sinusoidally with time, etc. That is, a varying field can produce other varying fields ad infinitum. This will give rise to electromagnetic radiation. In fact, this was the big breakthrough of Maxwell's work: he predicted the existence of electromagnetic waves. Now this is kind of like J. J. Thomson's discovery of the electron. Optics is all about electromagnetic waves. Visible light, infrared, ultraviolet, radio, and other such waves are electromagnetic waves.

To find the displacement current between the plates of capacitor, let q be the charge collected on the capacitor plates at any instant t. then the electric field developed between the plates will be

A q E

0



  I

A dt

dq A dt

dE

0 0

1 1



 

 

dt dE A I ₀

This conduction current (I = ID) is known as displacement current. It is an apparent

current, which represents the rate at which charge flow from one electrode to another in the external circuit. The term dE/dt is the rate at which electric field is changing between the plates of the capacitor.

Characteristics of Displacement current

1. Displacement current is a current in the sense that it produces a magnetic field. As it not linked with the motion of charge, it has no other properties of current. It has a finite value even in perfect vacuum, where there is no charge at all.

2. It serves the purpose to make the total current across the discontinuity in conduction current.

3. In good conductors, it is almost nil as compared to the conduction current at frequencies less than optical frequency ( 1015 Hz).

Continuity equations

In electromagnetic theory, the continuity equation can either be regarded as an empirical law expressing (local) charge conservation. It states that the divergence of the current density is equal to the negative rate of change of the charge density,

To derive the equation of continuity, using Maxwell’s equation, Ampere’s law

E J

B₀₀₀ 



  .B₀.J₀₀.E

But the divergence of a curl is zero  .J₀.E 0

Using Gauss’s law .E/₀  .J/t 0

t J

    . 

Which is continuity equation

Current density is the movement of charge density. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore the continuity equation amounts to a conservation of charge. In steady state . 0

   

t

J 



(otherwise charge would be piling up some where).

MAXWELL'S EQUATIONS

Maxwell related the interactions between charges, currents, electric field and magnetic field by four fundamental equations called "Maxwell's equations". These equations represent the experimentally achieved fundamental laws of electromagnetism.

Law Equation Physical Interpretation Gauss’s law for E

0 .



Q A d E

S





  Electric flux through a closed surface is proportion to the charge enclosed

Faraday’s Law





dt d s d

E.  B Changing magnetic flux produces _{an electric field} Gauss’s law for B

_

_{. 0}

S A d

B  The total magnetic flux through a

closed surface is zero Ampere–Mawell’s

law _dt

d I

s d

B E

S

   _{0 0 0}

.  



  Electric electric flux produces a magnetic current and changing field

In differential form these equations can also be rewritten as

0

/

    E

B0

B E

 

E J

B000  



These Maxwell’s equations are written for the fields in vacuum where no material is present. The quantities 0 &0are called the permeability and permittivity respectively.

Plane Electromagnetic Waves in Free space or Vacuum and its solution:

In regions of free space where there is no charge or current, Maxwell’s equations for E and B are

E 0 (i) B0 (ii)

B E



 (iii) B00E (iv)

Taking curl of equation (iii)

     

         

t B E

 

) (



B



t E

E   

       

 2

)

( Using vector product rule

Substituting equation E 0 from equation (i) and B 



₀



₀E from equation (iv)

2 2 0 0 2

t E E

  



 

 

Now applying curl operator to equation (iv)

   

 

  

     

t E B

 

0 0

)

(  



E



t B

B   

  

   

 2 _{0 0}

)

(   Using vector product rule

Substituting equation B0 from equation (ii) and E B from equation (iii)

These equations are plane electromagnetic wave equations governing electromagnetic fields E and B, called. They represent relation between the space and time variation in 3-D case. These equations are identical in form and in general form can be written

0 2 2 0 0

2 

  



t u

u  

Here u may be scalar or vector quantity. The well-known equation of a wave propagating with a velocity v is given by

0 1

2 2 2

2 

   

t u v u

Thus Maxwell’s equations imply that empty space supports the propagation of electromagnetic waves traveling at a speed.

0 0

1

  

v = 8

10

3 m sec-1

Hence electromagnetic wave propagates with the speed of light.

Transverse Nature of electromagnetic wave

To find the solution of electromagnetic wave let’s consider an electromagnetic wave is propagating in the +x-direction, with the electric field E pointing in the +y-direction and the magnetic field B in the +z -direction, as shown in Figure. The plane wave solution of above equations may be written in the well-known form

) . ( 0

) ,

(r t E eikr t

E  

 

  

( . )

0

) ,

(r t Bei kr t B  



  

where E₀and B₀ are complex amplitudes of the electric and magnetic fields. The wave propagation vector is given by

n

k 

  2 

Here n is a unit vector in the direction of propagation of electromagnetic wave.

2 2 0 0 2

t B B

  



 

 

Now





_{0x oy oz}



. i(kxx kyy kzz t)



e

k E j E i E k z j y i x

E _     

  

 

   

  

  

       

=E_0xei(kxxkyykzzt)(ik_x)E_oyei(kxxkyykzzt)(ik_y)E_ozei(kxxkyykzzt)(ik_z)

) (

0 ]

[ _{x x oy y oz z} ikxx kyy kzz t e

k E k E k E

i     



) . ( 0)

.

(kE ei kr t

i 

    )

. (k E i E   

 

Thus for a free space E 0 when k.E0

Similarly results for B0 we have k.B 0

These equations shows that electric and magnetic fields are perpendicular to the direction of propagation vector k, i.e. transverse in nature.

Similarly using Maxwell’s third and fourth equation we can arrive the following

B i E k i

E     



 ( )  kE B

E B 



0



0  

  kB₀₀E

These equations shows that electric and magnetic fields are perpendicular to the direction of propagation vector k.

From above relation k(nE)B  B  (k /



)(nE) (

k n k    )

Where n is a unit vector along the direction of propagation of electromagnetic wave?

) ( 0

E n k

H   





B H

 

0   )

( 1

0

E n c

H   





k c



Magnitude E

c H

0 1   

0 0 0

0 0

0 _



 

 

 c

H E Z H

E

=376.72 ohms

This clearly indicates that not only E and B are interrelated but their ration (E/B or E/H) also remains constant value Z0, called the intrinsic impedance or wave impedance of the

free space.

Nature of Electromagnetic Waves

2. The direction of flow of electromagnetic energy is along the direction of propagation of electromagnetic wave.

3. The electromagnetic field vectors E, H and the direction of propagation of the wave are mutually perpendicular to each other, i.e. electromagnetic waves are transverse in nature.

4. The electromagnetic energy flow per unit are per second is

2 2 oEo

c

S  

5. The energy density associated with an electromagnetic wave in free space travels with a speed equal to the velocity of light with which the field vectors propagate. 6. The electrostatic energy density is equal to the magnetic energy density.

Poynting theorem

Electric and magnetic fields store energy. This, energy can also be carried by the electromagnetic waves, which consist of both fields. The rate of flow of energy per unit area in a plane electromagnetic wave can be described by a vector S, called the Poynting vector, which is represented by

H E S  

The Unit of Sin MKS is watt/m2.

Let an electromagnetic field interacts with a particle of charge q traveling at a velocity v. The Lorentz force on this particle

) ( )

( mv

dt d E B v q

F_Lorentz      

To obtain the energy relation, multiply this equation by.

E v q mv dt

d 

 ) 2 1

( 2

Here v.(vB)is zero as the magnetic field does not contribute to the particle's energy. Multiply by the particle density n and introduce the current density J nqv, we obtain

E J dt

dT  

. 

where T is the kinetic energy of the ensemble of particles. Using fourth Maxwell's equations to express J in terms of the magnetic and electric fields.

E B

J( )/₀ ₀   

      



 2

0 0

2 1 /

) .(

. E

dt d B

E E

J     

Using vector identity (AB)B.(A)A.(B) above equation can be written

0 2

0

0 .( )/

2 1 /

) .(

.   E B E 

dt d B

E E

J       

      

 

The last term in the equation above is actually the time derivative of the magnetic field energy density. This can be shown by using Faraday's law to substitute

dt B d



for the curl of E. The first term on the R.H.S contains the Poynting vector S.

   

 

 



 2

0 2 0

1 2

1 .

. E B

dt d S E

J

 

 



The electromagnetic field energy density U is given by

   

 



 2

0 2 0

1 2

1

B E

U





We get the Poynting theorem for the case of an ensemble of free particles in an electromagnetic field in its most compact form.

S J E t

U _ ___ 

 

total energy rate mechanical work with the Poynting vector S EH

This theorem states that the work done on the charge by an electromagnetic force is equal to the decrease in energy stored in the field, less than the energy, which flowed out through the surface. It is also called the energy conservation law in electrodynamics. Above relation for pointing vector show the instantaneous rate of flow of energy per unit area. Average value of S can be obtained for more than one cycle of the wave. The average values of E and B for the cycle are E0/2 and B0/2 respectively. Therefore the average value of the Poynting vector

2 0

0 0 0

0

0 2 2

1 2

1

o oE c c E E B

E

S 



  