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Poynting Theorem

Energy is transported through space by means of electromagnetic waves, let the material inside the surface S be isotropic homogeneous and characterised by permeability µ and permittivity ε and conductivity σ

 

Maxwell’s IIIrd and IVth equations are as follows

 

Curl E = -dB/dt

Curl H = J + dB/dt

 

 

Taking scalar product of both sides of equation (1) and equation (2) respectively with Hand E

 

H cual E = – H dB/dt

 

E cual H =E(J+ dB/dt)

 

Now, subtracting equation (3) from equation (4), we get

 

(curl H –H curl E ) =E .J +E . dB/dt  + H dB/dt

 

J . dB/dt  + H dB/dt

 

Now, using vector identity

 

Div (E x H ) =H curl E –E  curl E

 

putting above value in L.H.S. equation (5) may be expressed as

 

-div (E x H) = J.E+ E . dB/dt  + H dB/dt

 

Now, if the medium is linear so that the relations

Now, integrating this equation over the volume V bounded by the surface S, we get

By applying Gauss transformation formula in L.H.S.

The above equation (10) represents the poynting theorem.

7.20.1 Physical Significance or Interpretation of terms of Poynting Theorem

 

(a) Interpretation of term ʃ V. Ed V: The current distribution represented by the vector J can be considered as made up of various changes qi moving with velocity

where E j denotes the electric field at the position of charge q j According to Lorentz force equation, the force experienced by point charge is given by

 

f =q  (E +v + B)

 

We know that workdone is equal to force x displacement

Dw =F .dl

= q (E +v x B )dl

= q .E dl +q (v x B) dl

 

= q E dl +dl (dl/dt x B) q

=q E dl + dl +(dl x dl/dt) .B q

Dw = q E dl + 0

Dw/dt =q E dl/dt

Dw /dt =q E .v

 

taking all the charges into consideration, we get

Dw /dt =Ʃqj(Ej vj)

From equations (1) and (2)

 

This shows that this term represents the rate at which the work is done by the field

on the charges.

 

(b) Interpretation of term

The above equation represents the time rate of increase in energy stored in the electric and magnetic field respectively in volume.

 

(c) Interpretation of the term

Since L.H.S. shows the rate of change of work done or power so this term must shows the power flow into volume V across the surface 5 or the power flow out of volume V across the surface S

 

where P =Ex H

 

where P = E x H is known as Poynting vector and hence the meaning of power density associated with electromagnetic field at a point

 

 

 

Dimension of P = [M -3]

unit of P = Watt/meter2

where P =Ex H

where P = E x H is known as Poynting vector and hence the meaning of power density associated with electromagnetic field at a point

 

Dimension of P = [M -3]

unit of P = Watt/meter2