PHYS 1008 Electromagnetic Waves

The Link between Electricity and Magnetism Objectives: by the end of this you will be able to understand the relationship between electricity and magnetism be able to relate wavelength and frequency for waves Understand what an electromagnetic wave is Say how energy is carried in EM waves Explain the EM spectrum understand how we can communicate with EM waves

Electricity and Magnetism

• There are obvious analogies
• Static charges ⇒ electric field ⇒ forces on static (or moving) charge
• Moving charges ⇒ magnetic field ⇒ forces on moving charges

• Also a coincidence: Electrostatic force is defined via
  \color{red}{ F_E = \frac{1}{{4\pi \varepsilon _0 }}\frac{{q_1 q_2 }}{{r^2 }}}
• and the magnetic force via
  \color{red}{ F_M = \frac{{\mu _0 }}{{2\pi }}\frac{{I_1 I_2 L}}{r}}
• with
  \color{red}{ \varepsilon _0 = \frac{1}{{4\pi \times 9 \times 10^9 }},\mu _0 = 4\pi \times 10^{ - 7} }
• What is \color{red}{\frac{1}{{\varepsilon _0 \mu _0 }}}
• What does this quantity mean? i.e. what are the units?
• \color{red}{\left[ {\frac{{F_E }}{{F_M }}} \right] = 1}
• so \color{red}{\left[ {\frac{1}{{\varepsilon _0 \mu _0 }}} \right] = \frac{{\left[ Q \right]^2 \left[ L \right]^2 }}{{\left[ I \right]^2 }}}
• \color{red}{\left[ {\frac{1}{{\varepsilon _0 \mu _0 }}} \right] = \frac{{\left[ Q \right]^2 \left[ L \right]^2 }}{{\left[ I \right]^2 }} = \left[ L \right]^2 \left[ T \right]^{ - 2} }
• i.e. this is a (velocity)². What is it?
• The connection is very profound: electricity and magnetism are just two aspects of a more general theory called electromagnetism.

Classical Physics

We now know every equation needed for classical physics!

• Newton's Second law

  F = ma  
  

• Universal Gravitation\color{red}{F_G = \frac{{Gm_1 m_2 }}{{r^2 }}}
• Coulomb & Lorentz Forces \color{red}{F = qE + qvB}
• Coulomb's Law \color{red}{\vec E\left( {\vec r} \right) = \frac{{kq_1 }}{{r^2 }}\hat r}
• Actually we use Gauss' Law, which follows from Coulomb's Law and is more general \color{red}{\int {\vec E.d\vec A = \frac{q}{{\varepsilon _0 }}} }
• no mag. charges.\color{red}{\int {\vec B.d\vec A = 0} }
• Faraday's Law \color{red}{V = \int {\vec E.d\vec s = - \frac{{d\varphi }}{{dt}}} }
• Ampere's Law which follows from Biot-Savart and includes a modification due to Maxwell.\color{red}{\int {\vec B.d\vec s = \mu _0 I + } \varepsilon _0 \mu _0 \frac{{d\varphi }}{{dt}}}
• Note
  • E, B are usually functions of time and space E(r,t)
  • Solving them is harder than writing them down.......

Electromagnetic Radiation

• The simplest way to visualize this is to imagine the field lines from a charge

• which is suddenly moved producing a "kink" in the lines of force, which travels with a certain vel.

• Repeating the process would give

• Reality is a bit more complicated: need an oscillating dipole

• which can give rise to a continuous wave.

it is the combination that produces EM radiation. Note that the mag field B ⊥ E ⊥ v

Speed of Light

All EM radiation travels with the same speed in vacuum c~2.997x10⁸ ms^-1.

Obviously an important quantity to measure.

• Originally assumed to be infinite (although Leonardo da Vinci tried to measure it).

• First measurement due to Romer, who noticed the predicted time of the eclipses of the satellites of Jupiter was wrong. What difference should he have found (Rearth = 1.5x108 km, RJupiter = 7.8x108km)

• first good measurements used rotating mirrors (Michelson).

• Much more sophisticated methods now.
• Note that in a dielectric material, ε₀ ⇒ Kε₀ where K is the dielectric constant. speed of light is slower (by K^-1/2) in a transparent material.

1. 10^-4 s
2. 2x10^-4 s
3. 2x10^-7 s
4. 10^-7 s

• What would the frequency of rotation of the octagonal (8-sided) mirror need to be so that the next face was in place when the light returned?
  1. 5 kHz
  2. 2x10^-4 Hz
  3. 625 Hz
  4. 3927 Hz

Electromagnetic Spectrum

Light is part of the whole electromagnetic spectrum

c = λf

• The "energy' in the above diagram gets explained later.

• We "see" only one octave.

• Why do we see so little of the spectrum?

• Answer lies in the transparency of the atmosphere

• Rule of thumb is that you need an antenna about as big as the wavelength to produce radiation: e.g. to produce radio waves, need a tower, to produce light need an atom. However we can detect most of the ranges.

Communication

• (e.g. TV, radio) depends on this. A signal typically is a modulation of a carrier wave:
• e.g. Morse Code (Signal is just on or off)
• e.g AM (Amplitude Modulation)

  has sound at (e.g.) 1 kHz used to vary the amplitude of a wave at 800 kHz.

• Pick it up in antenna
• Use resonant circuit to select only 800 kHz signal (probably at a level of ~ μW)

• Rectify it

• Filter it (to get rid of the 800 kHz

• Amplify it

FM (Frequency Modulation)

differs by having the frequency of the signal modified. THis is much less prone to distortion, hence sound quality is better

Polarization

• We have introduced polarization of transverse waves: this of course applies to light

• Polaroid acts as a "slot" to let through radiation in only one poln. state

• Hence two successive sheets of polaroid at right angles will eliminate all light

• if a ray with some polarization at an angle θ passes through a filter with a vertical polarization axis then the projection of E is E₀cos(θ). However, \color{red}{I \propto E_y ^2 = E_0^2 \cos ^2 \left( \theta \right)} so the intensity is reduced by cos²(θ).

• If the original light is unpolarized, the intensity is given by the average value
  \color{red}{ \left\langle {\cos ^2 \left( \theta \right)} \right\rangle = \frac{1}{2}\left\langle {1 + \cos \left( {2\theta } \right)} \right\rangle = \frac{1}{2}}

• Hence if light passes through a succession of filters, it will lose a factor cos²(θ) for each one.

• Note: this assumes that the filter is perfect: in fact some absorption will always occur.

1. 3/4
2. 3/8
3. 1/2
4. 1/8

• Depends on angle of incidence if tan(θ) = n₂/n₁ = n (if outer medium is air) then reflected ray is 100% polarized

• What is this for water (n = 1.33)?
• Technology of handling light is "Geometric Optics"