Maxwell's equations

The science of physics is often divided into Classical Physics and Modern Physics.  Classical physics works well describing and predicting almost all everyday phenomena.  It breaks down when things get too ... .  (You can substitute here almost any very large deviation from out everyday experience.)

In classical physics the world is made of matter particles that behave according to Newton's laws of motion.  To use Newton's laws, we need to know which forces are acting on the particles.  The interactions that give rise to the forces between the particles are represented by fields.  Electric and magnetic fields represent the electromagnetic interactions.  If we know the fields, we know the electric and the magnetic forces acting on charged particles.

F = Felectric +  Fmagnetic  = qE + qv × B = q(E + v × B).

In classical physics our model for the electromagnetic fields is a set of four equations, called Maxwell's equations.  They let us predict E(r) and B(r) if we know the charge densities ρ(r) and the current densities j(r) in some volume V, and we have information about the fields at the boundaries of V.  Maxwell's equations have a broader range of applicability than Newton's laws.  They are relativistically correct and correctly describe the fields when relative speeds approach the speed of light.  But the classical model for the electromagnetic fields no longer describes observations correctly on the scale of atoms or elementary particles.

Maxwell's equations (in SI units) are
(1)  ∇∙E = ρ/ε0,
(2)  ∇×E = -∂B/∂t,
(3)  ∇·B = 0, 
(4)  ∇×B = μ0j + (1/c2)∂E/∂t,

or,  in integral form

(1)  ∮A E∙dA = Qinside/ε0,
(2)  ∮Γ E∙ds = -∂/∂t∫AB×dA,
(3)  ∮A B∙dA = 0,
(4)  ∮Γ B∙ds =  μ0Ithrough Γ  + (1/c2)∂/∂t∫AE×dA.

Here ρ = charge density, j = current density, ε0 = 8.854*10-12 C2/(Nm2), μ0 = 4π*10-7 N/A2.

Maxwell's equations predict:

--  A charged particle produces an electric field.  This electric field exerts a force on other charged particles.  Positive charges accelerate in the direction of the field and negative charges accelerate in a direction opposite to the direction of the field.
--  A moving charged particle produces a magnetic field.  This magnetic field exerts a force on other moving charges.  The force on these charges is always perpendicular to the direction of their velocity and therefore only changes the direction of the velocity, not the speed.
--  An accelerating charged particle produces an electromagnetic (EM) wave.  Electromagnetic waves are electric and magnetic fields traveling through empty space with the speed of light c = 3*108 m/s.  A charged particle oscillating about an equilibrium position is an accelerating charged particle.  If its frequency of oscillation is f, then it produces an electromagnetic wave with frequency f.  The wavelength λ of this wave in vacuum is given by λ = c/f.  Electromagnetic waves transport energy through space.  This energy can be delivered to charged particles a large distance away from the source.

Accelerating charges produce changing electric and magnetic fields.  Changing electric fields produce magnetic fields and changing magnetic fields produce electric fields.  This interplay between induced electric and magnetic fields leads to propagating electromagnetic waves.  Electromagnetic waves can propagate through free space.  In classical physics visible light is an electromagnetic wave with a frequency in the range of ~4.3*1014 Hz to ~7.5*1014 Hz.

Visible light makes up just a small part of the full electromagnetic spectrum.  Electromagnetic waves with shorter wavelengths and higher frequencies include ultraviolet light, X-rays, and gamma rays.  Electromagnetic waves with longer wavelengths and lower frequencies include infrared light, microwaves, and radio and television waves.

Type of Radiation   Frequency Range (Hz)   Wavelength Range  
gamma-rays 1020 - 1024 < 10-12 m
x-rays 1017 - 1020 1 nm - 1 pm
ultraviolet 1015 - 1017 400 nm - 1 nm
visible 4*1014 - 7.5*1014 750 nm - 400 nm
near-infrared 1*1014 - 4*1014 2.5 μm - 750 nm
infrared 1013 - 1014 25 μm - 2.5 μm
microwaves 3*1011 - 1013 1 mm - 25 μm
radio waves < 3*1011 > 1 mm

image

In vacuum, where there are no charge and current densities, Maxwell's equations lead to the homogeneous wave equation for E and B in three dimensions.
∇2E - μ0ε0∂2E/∂t2 = 0,  ∇2B - μ0ε0∂2B/∂t2 = 0 , with μ0ε0 = 1/c2.
This equation has infinitely many solution.  But we often concentrate on sinusoidal plane waves solutions of the form
E(r,t) = Re(E0ei(k∙r - ωt)),  B(r,t) = Re(B0ei(k∙r - ωt)), with ω/k = λf = c.
Here k is the wave number, k =|k|= 2π/λ, and ω = 2π/T = 2πf is the angular frequency of the wave.
k is the wave vector.  k points in the direction of propagation of the wave.

The wave equation is a linear equation, so the real and the imaginary part of a solution do not mix, and it is often easier to work with exponential functions than with trigonometric functions.  The solutions expressed in terms of trigonometric functions are
E(r,t) = |E0|cos(k∙r - ωt + φ),  B(r,t) = |B0|cos(k∙r - ωt + φ).
E0 and B0 are complex numbers, E0 = |E0|eiφ,  B0 = |B0|eiφ,  φ is called the phase constant.

Maxwell's equations imply that EM waves in free space are transverse waves.
We have  E ⊥ B,  E ⊥ k,   B ⊥  k,  B = (k/k)×E/c.
The magnetic field of the electromagnetic wave is perpendicular to the electric field and has magnitude B = E/c in free space.  For electromagnetic waves in vacuum  E and B are always perpendicular to each other and perpendicular to the direction of propagation.  The direction of propagation is the direction of E×B.

Because the wave equation is a linear equation, any solution, (spherical waves, wave pulses, etc) can be written as a linear superposition of plane waves with properly chosen directions of propagation, amplitudes, angular frequencies and phase constants, as long as ω/k = c.

Problem:

An electromagnetic wave in vacuum has an electric field amplitude of Emax = 220 V/m.  Calculate the amplitude Bmax of the corresponding magnetic field.

Solution:

Problem:

The eye is most sensitive to light having a wavelength of 5.5*10-7 m, which is in the green-yellow region of the electromagnetic spectrum.  What is the frequency of this light?

Solution:

Problem:

imageA plane electromagnetic wave is propagating through space.  In some plane at some time t the fields are uniform and oriented as shown.  What is the direction of propagation of the plane wave?

Solution:

Plane wave solutions have unphysical properties.  They are of infinite length and the wave fronts have infinite area.  Everywhere in an infinite plane normal to the direction of propagation E(t) and B(t) are the same.  So for a sinusoidal plane wave propagating along the z-direction we have
E(r,t) = |E0|cos(kz - ωt + φ),
i.e. E only varies along the z-direction, and not along any direction in the xy plane.

Click here for an attempt to visualize the electric field of a plane wave propagating along the z-direction in three dimensions.

There are many situations where waves exist that are experimentally indistinguishable from plane waves.  This is the case when the length of the wave and the width area of the wave front is large compared to the wavelength of the wave.

AI Study Tip:
Use AI to identify "conceptual gaps." If you understand the math but not the physical "why," ask the AI to explain the bridge between the wave model and the ray model.

Example Prompt:  'I am studying Module 1 of a Modern Optics course.  I understand Maxwell’s equations in vacuum, but I’m struggling to visualize how 'rays' emerge from wave equations.  Act as a Socratic tutor.  Don't give me the full answer immediately.  Instead, ask me a series of questions to help me derive the relationship between wave fronts and light rays myself.'