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Last lecture:
Light is an electromagnetic wave consisting of orthogonal E,B fieldsBoth E,B fields are orthogonal to the direction of propagationE,B have the same frequency and are in phaseThe relative magnitude is B=E/c
Lecture 9: Poynting vector
Describe what is meant by the Poynting vector
Write an expression in terms of electric and magnetic fields
Use it to calculate the instantaneous and average power associated with and EM wave propagating in vacuum
Brief notes on EM waves:
Similarly
Brief notes on EM waves:
Short cuts:For complex sinusoidal plane waves with wave vector k andangular frequency ω :
Let’s apply it:
Wave equation for LIH media with no free charge / current :
Take M3
Take the curl
Using M4
Similarly for B
with
Energy stored in electric and magnetic fields
Energy density in free spaceY1 EM
Generalized to energy density in materials:
LIH:
(energy stored in capacitor and inductor)
Energy density:
energyvolume
J m-3
Energy flow:e.g energy flow from the Sun to Earth through space provide approx 1 kW m-2
Energy flow has magnitude and direction - a vector quantity
Maxwell’s eqs alone do not provide a way to express energy flow.In 1884 Poynting applied concepts of energy conservation to EM fields and obtained expression for energy flow
At the time of this discovery he had become our 1st Professor of Physics here at Birmingham
Energy conservation: Poynting theorem
3 ingredients:The change in the stored energy in the region (1)The work done on the charges in the region (2)The energy flow out of the region (3)
(1) + (2) + (3) = 0
We can already write down (1) and (2)Hence we can calculate (3)
(1) Rate of change of stored energy in volume V:
Equal, LIH Equal, LIH
(2) Force acting on a charge q moving with speed ν
Rate of work done:
More generally for charge distribution density ρ
J = current density
(3) Energy flow is a vector, the Poynting vector N
Total energy flown out of the region considered
W m-2
Using divergence theorem
Requiring energy conservation: (1) + (2) + (3) = 0
True for any volume: Poynting theorem
Let’s now find a expression for N
Use M4
Take scalar product with E
Similarly, take scalar product of H with M3:
(i)
(ii)
(i)-(ii) :
Vector identity:
Compare to Poynting theorem:
Hence energy flow in EM system is:
Poynting vector
Represents the flow of energy per unit area per unit timei.e. the power per unit area at any point
Consider the case of monochromatic plane sinusoidal waves
Recall:
Time average energy density:
Time averaged energy flow:
Energy flow
Wave velocity
Energy density
Example
Calculate the electric field strength of a radio signal froma satellite if the magnitude of the time-averaged Poyntingvector is 1 pW m-2
Example
Calculate the electric field strength of a radio signal froma satellite if the magnitude of the time-averaged Poyntingvector is 1 pW m-2
Summary
Recommended readings:Grant+Phillips: 1.5.6 2.4, 2.4.1 6.3.1, 6.3.2, 6.3.4 11.4, 11.7
Next Lecture:
Polarization
Useful vector identities:
At the interface between 2 materials (1) and (2), with no free surface charge or current :
You will be expected to be ableto derive and remember theseboundary conditions,and to be able to apply themto simple conditions
