1. The Maxwell equations

2. James Clerk Maxwell (13 June 1831 – 5 November 1879)
Scottish theoretical physicist and mathematician.
The most significant achievements:
the development of the classical electromagnetic theory, synthesizing all previous unrelated observations, experiments and equations of electricity,magnetism and even optics into a consistent theory.

3. Scientific disciplines
Scientific career :
Professor of Aberdeen University
Professor of Kings College in London
Professor of Cambridge University
Founder of Cavendish Laboratory
Scientific disciplines
Molecular physics
Optics
Electrodynamics
Important dates
1864 – Maxwell’s equations
1865 – concept of electromagnetic
nature of light

4. James Clerk Maxwell
(13 June 1831 – 5 November 1879)

5. The Maxwell equations
Maxwell’s equations are a generalization of all electrostatic and magnetic phenomena known before.
Maxwell’s equations express the inseparable relation between electric and magnetic fields.
From Maxwell’s equations come directly:
law of energy conservation
law of charge conservation
electromagnetic waves existence and propagation (proved by Hertz in 1886)

6. Electromagnetic waves
Electric
field
Magnetic field
E-M wave
Propagation
direction
Electromagnetic waves 6

7. Oscilating electrons
WdWI 2015 PŁ 7

8. Maxwell’s equations are the basis of electromagnetism
These two equations and additional
six equations form a set of 8 equations
fully describing electromagnetic field.
Three Maxwell’s postulates concerning divergence of the field:
So called „point equations” inform us about a medium in which the field
exists:

9. I Maxwell’s equation Differential form of Maxwell’s equation
Rotational magnetic field (vortex field) is generated by currents
and/or by changing electric field.
is the conduction current density
The notion
is called „density of displacement current”

10. JT expresses a density
of total current
Density of
the conduction current
Density
of the displacement current

11. Integral form of I Maxwell’s equation
From differential form of Maxwell’s equation
after integration over the surface S with the bound c we receive a new form:
The order of operations can be changed,
because the integration over the surface
and differentiation with respect of time
are independent.
Stokes
theorem

12. I Maxwell’s equation in integral form:
The line integral:
- represents the magnetic voltage between points A and B
represents a total current: the sum
of conduction current
and displacement current
Electric field flux

13. Let’s consider the field in capacitor when the electric field changes in time:

14. Displacement current exists in the capacitor
The conduction current flowing in wires is motion of the charges.
Changing electric field exists between the capacitor plates. The electric flux goes through the surface S:
Displacement current exists in the capacitor

15. A displacement current is the extension of current flow in dielectric, where there is no way for conduction current.
The rule: the current flows by closed path is realised like this:
in wires – a conduction current
in dielectric – a displacement current
This current can reach great values if the field is changing with great frequency.

16. II Maxwell’s equation Differential form of II Maxwell’s equation
Rotational electric field (vortex field) is generated
by changing magnetic field.

17. Integral form of II Maxwell’s equation
From differential form of II Maxwell’s equation
after integration over the surface S with the bound c we receive the integral form:
Stokes
theorem
Magnetic flux

18. The line integral:
- represents the electric voltage between points A and B
This equation represents the Faraday’s law of induction
Emf – induced electromotive force e

19. The integral form of Maxwell’s equations

20. I Maxwell’s equation - Ampère’s law
A magnetic voltage along the closed curve c is equal to the total current – the sum of the conduction current and displacement current passing through the surface which bound is the curve c.

21. II Maxwell’s equation - Faraday’s induction law
Emf – induced electromotive force e
A voltage along the closed curve c is equal to electromotive force induced by time changes of magnetic flux passing through the surface bounded by this curve.

22. Wave in an ideal dielectric
We will derive the wave equations for electric and magnetic field from Maxwell’s equations.
It will be a proof that the electromagnetic field is a wave.
Assumptions:
Considered area is unlimited
Considered medium is linear, homogenous and isotropic
γ=0, it means a medium is lossless
3’. There are no currents and no charges in the considered area

23. We have a set of four differential equations with two unknown vector functions.
In a general case we have to find 6 unknown scalar functions – components of searched vectors.

24. Assumptions: μ=const., ε=const., γ=0, ρ=0
Let’s calculate the rotation of both sides
of both equations

25. From vector identity: =0 =0

26. We have received two wave equations.
Two vector functions E(P,t) i H(P,t) which satisfy this equations form electromagnetic wave.
Attention:
This means there might exist such solutions of wave equations which don’t satisfy Maxwell’s equations.
Solutions E(P,t) i H(P,t) must satisfy condition coming from Maxwell’s equations.

27. Three Maxwell’s postulates concerning divergence of the field:
and