# April 15. Physics 272. Spring Prof. Philip von Doetinchem

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1 Physics 272 April 15 Spring Prof. Philip von Doetinchem Phys272 - Spring 14 - von Doetinchem - 291

2 Lunar eclipse Phys272 - Spring 14 - von Doetinchem - 292

3 Hertzian dipole Source: Change of electric field over time Phys272 - Spring 14 - von Doetinchem - 293

4 Plane electromagnetic waves Electromagnetic waves are transverse waves with an electric and a magnetic component similar to waves on a string Make the following assumption: Electric and magnetic field configuration with wave-like behavior Electric field has only a y component Magnetic field has only a z component Both move in x direction with velocity c We have to test if this assumption is consistent with Maxwell's equation Phys272 - Spring 14 - von Doetinchem - 294

5 Plane electromagnetic waves Why perpendicular? Faraday's law: Integration path perpendicular to electric field: no contribution Magnetic flux change: Magnetic field component in z direction is crucial to comply to Faraday's law Phys272 - Spring 14 - von Doetinchem - 295

6 Plane electromagnetic waves Why perpendicular? In a very similar way: use Ampere's law (no conduction current): Electric and magnetic fields must be perpendicular to fulfill Faraday's and Ampere's law Definite propagation velocity: Phys272 - Spring 14 - von Doetinchem - 296

7 Key properties of electromagnetic waves Electromagnetic waves fulfill the general wave equation for both electric and magnetic fields (not discussed here): Wave is transverse: Electric field and magnetic field are perpendicular to the direction of the wave Magnitudes of E and B are related by the propagation velocity: E=cB In vacuum (or if medium is not changing) the electromagnetic wave is traveling at a constant velocity Electromagnetic waves do not need a medium Simple plane waves can be generalized E and B fields can be superposed For each superposition the simple principles of the plane electromagnetic waves apply Wave packets or sinusoidal waves fulfill Maxwell's equations Phys272 - Spring 14 - von Doetinchem - 297

8 Key properties of electromagnetic waves Electromagnetic waves fulfill the general wave equation for both electric and magnetic fields (not discussed here): Wave is transverse: Electric field and magnetic field are perpendicular to the direction of the wave Magnitudes of E and B are related by the propagation velocity: E=cB In vacuum (or if medium is not changing) the electromagnetic wave is traveling at a constant velocity Electromagnetic waves do not need a medium Simple plane waves can be generalized E and B fields can be superposed For each superposition the simple principles of the plane electromagnetic waves apply Wave packets or sinusoidal waves fulfill Maxwell's equations Phys272 - Spring 14 - von Doetinchem - 298

9 Sinusoidal electromagnetic waves At any instant the fields are uniform over any plane perpendicular to the direction of propagation with speed c The entire pattern travels in the direction of propagation Electric and magnetic field are still transverse to the propagation direction at any instant In a small region of space at great distance from source electromagnetic waves can be treated as plane waves Velocity, wave length, and frequency are related: Phys272 - Spring 14 - von Doetinchem - 299

10 Fields of a sinusoidal waves Electric and magnetic fields oscillate in phase The fields are at the same time at maximum,zero, and minimum Vector product of electric field and magnetic field always points in the propagation direction: Electromagnetic wave can be described by wave function depending on location and time k is the wave number: 2 / Phys272 - Spring 14 - von Doetinchem - 300

11 Fields of a sinusoidal waves The electric field and magnetic field amplitudes are related by: The sign in front of t denotes the direction of the wave: Negative sign: positive x direction Positive sign: negative y direction Phys272 - Spring 14 - von Doetinchem - 301

12 Why are magnetic and electric field in phase? Assume arbitrary phase angle: Relation between electric and magnetic field using Faraday's law: Phys272 - Spring 14 - von Doetinchem - 302

13 Why are magnetic and electric field in phase? Relation between electric and magnetic field using Faraday's law: Phys272 - Spring 14 - von Doetinchem - 303

14 Why are magnetic and electric field in phase? Use our approach for the wave function assuming a phase angle: To make these equations equal at all times we have to require Phys272 - Spring 14 - von Doetinchem - 304

15 Electromagnetic waves in matter The wave speed changes in matter (we assume non-conducting dielectric) We have to use the dielectric constant and the permittivity of the material: Electromagnetic waves can never travel faster than the speed of light in vacuum Definition of index of refraction n of a material ( optics) Careful: dielectric constant depends on frequency for high frequencies materials cannot be polarized as fast as for constant electric fields reduces dielectric constant Phys272 - Spring 14 - von Doetinchem - 305

16 Energy and momentum in electromagnetic waves Electromagnetic waves transport energy: Microwave ovens, radio transmitters, lasers for eye surgery, Combined energy density of electric and magnetic components: In vacuum electric field and magnetic field carry half of the total energy density Phys272 - Spring 14 - von Doetinchem - 310

17 Electromagnetic energy flow and the Poynting vector Energy is required to establish electric and magnetic fields electromagnetic waves transport energy from one region to the other Definition: energy transferred per unit time per unit cross-sectional area or power per unit area In a certain time the wave moves in space with the propagation velocity energy in a particular region: Phys272 - Spring 14 - von Doetinchem - 311

18 Electromagnetic energy flow and the Poynting vector Poynting vector is a vector quantity: it points in the direction of the propagation of the wave and depends on time SI unit: 1 W/m2 John Henry Poynting ( ) Total energy flow per unit time: Phys272 - Spring 14 - von Doetinchem - 312

19 Electromagnetic energy flow and the Poynting vector Frequency of electromagnetic waves is fast average Poynting vector value (intensity) For our sinusoidal wave: zero on average The intensity for a sinusoidal wave is exactly half of the maximum value: This is what we sense when looking at, e.g., light from the sun variations are too fast to be noticeable for us Phys272 - Spring 14 - von Doetinchem - 313

20 Energy in a sinusoidal wave A radio station on the earth's surface emits a sinusoidal wave with average total power of 50kW Assume that transmitter radiates equally in all directions Area of hemisphere: Intensity: Phys272 - Spring 14 - von Doetinchem - 314

21 Energy in a sinusoidal wave A radio station on the earth's surface emits a sinusoidal wave with average total power of 50kW Assume that transmitter radiates equally in all directions Electric field amplitude: Magnetic field amplitude: magnetic field associated with electromagnetic wave is extremely small compared to what we saw before it is easier to have device sensitive to the electric field Phys272 - Spring 14 - von Doetinchem - 315

22 Electromagnetic momentum flow and radiation pressure Electromagnetic waves also carry momentum ( deeper explanation requires quantum physics) Electromagnetic wave's momentum is transferred to a surface Transferred momentum per time equals the force on the surface Sun and stars create radiation pressure that effects the surrounding material star forming regions Massive Star Forming Region DR21 in Infrared Credit: A. Marston (ESTEC/ESA) et al., JPL, Caltech, NASA Phys272 - Spring 14 - von Doetinchem - 316

23 Solar sail Solar sails try to make use out of the radiation pressure as a propulsion system Usage of sails big as football fields should catch the radiation pressure by the sun Concept is proven, but not yet part of real spacecrafts Source: Phys272 - Spring 14 - von Doetinchem - 317

24 Standing electromagnetic waves Electromagnetic waves can be reflected on surfaces Dielectrics or conductors can serve as reflectors Superposition principle of electric and magnetic fields also applies to electromagnetic waves Superposition of incident and reflected wave forms a standing wave Phys272 - Spring 14 - von Doetinchem - 320

25 Reminder: equipotentials and conductors Electric force is conservative it is not possible to do work on a test charge like that: Electric field lines are perpendicular to surface of conductor. Phys272 - Spring 14 - von Doetinchem - 321

26 Standing electromagnetic waves Electric field cannot have a net component parallel to the surface Oscillating current are induced in surface that give rise to additional field that cancels out the electric field of the incident electromagnetic wave This also creates the reflected wave: The sum of incident and reflected wave must be 0 at all times on the surface: Phys272 - Spring 14 - von Doetinchem - 322

27 Standing electromagnetic waves Electric field cannot have a net component parallel to the surface Oscillating current are induced in surface that give rise to additional field that cancels out the electric field of the incident electromagnetic wave This also creates the reflected wave: The sum of incident and reflected wave must be 0 at all times on the surface: Phys272 - Spring 14 - von Doetinchem - 323

28 Standing electromagnetic waves Position and time factorize for electric field What does the magnetic field look like? Faraday's law still applies Integrate: A standing wave that was reflected on a conductor shows a 90deg phase angle between electric and magnetic field Phys272 - Spring 14 - von Doetinchem - 324

29 Standing waves in a cavity Insert a second conducting plane: cavity example: microwave oven On both planes the electric field has to vanish A standing wave is created when the electromagnetic wave wavelength is an integer multiple of /2 Measuring the node positions measurement of wavelength Reflections generally also happen on surfaces of two materials: Part of the wave is transmitted and a part is reflected Phys272 - Spring 14 - von Doetinchem - 325

30 Review Maxwell's equations predict the existence of electromagnetic waves that propagate at the speed of light Electromagnetic waves are transverse: In matter the wave speed is reduced Electric and magnetic fields are perpendicular to propagation direction Electromagnetic wave cannot travel faster than the speed of light. The poynting vector describes the energy flow rate. The averaged value is called the intensity Nodal planes occur for standing electromagnetic waves Phys272 - Spring 14 - von Doetinchem - 327

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