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General form of Faraday s Law

Induced emf and Electric Fields. An electric field is created in the conductor as a result of the changing magnetic fluxEven in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty spaceThis induced electric field is nonconservativeUnlike the electric field produced by stationary chargesThe emf for any closed path can be expressed as the line integral of E.ds over the path.

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General form of Faraday s Law

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    1. General form of Faraday’s Law

    2. Induced emf and Electric Fields An electric field is created in the conductor as a result of the changing magnetic flux Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space This induced electric field is nonconservative Unlike the electric field produced by stationary charges The emf for any closed path can be expressed as the line integral of E.ds over the path

    3. E produced by changing B

    4. Problems with Ampere’s Law

    5. But what if…..

    6. Maxwell’s correction to Ampere’s Law

    7. Maxwell’s Equations

    8. Gauss’s law (electrical): The total electric flux through any closed surface equals the net charge inside that surface divided by eo This relates an electric field to the charge distribution that creates it Gauss’s law (magnetism): The total magnetic flux through any closed surface is zero This says the number of field lines that enter a closed volume must equal the number that leave that volume This implies the magnetic field lines cannot begin or end at any point Isolated magnetic monopoles have not been observed in nature

    9. Faraday’s law of Induction: This describes the creation of an electric field by a changing magnetic flux The law states that the emf, which is the line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path One consequence is the current induced in a conducting loop placed in a time-varying B The Ampere-Maxwell law is a generalization of Ampere’s law It describes the creation of a magnetic field by an electric field and electric currents The line integral of the magnetic field around any closed path is the given sum

    10. The Lorentz Force Law Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be calculated F = qE + qv x B This relationship is called the Lorentz force law Maxwell’s equations, together with this force law, completely describe all classical electromagnetic interactions

    11. Maxwell’s Equation’s in integral form dS = n dA Flux = field integrated over a surface No magnetiic monopoles E .dl is an EMF (volts)dS = n dA Flux = field integrated over a surface No magnetiic monopoles E .dl is an EMF (volts)

    12. Maxwell’s Equation’s in free space (no charge or current) dS = n dA Flux = field integrated over a surface No magnetiic monopoles E .dl is an EMF (volts)dS = n dA Flux = field integrated over a surface No magnetiic monopoles E .dl is an EMF (volts)

    13. Vector Calculus Theorems

    14. Maxwell’s Equation’s In Differential Form

    15. Hertz’s Experiment An induction coil is connected to a transmitter The transmitter consists of two spherical electrodes separated by a narrow gap The discharge between the electrodes exhibits an oscillatory behavior at a very high frequency Sparks were induced across the gap of the receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitter In a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave properties Interference, diffraction, reflection, refraction and polarization He also measured the speed of the radiation

    16. Implication A magnetic field will be produced in empty space if there is a changing electric field. (correction to Ampere) This magnetic field will be changing. (originally there was none!) The changing magnetic field will produce an electric field. (Faraday) This changes the electric field. This produces a new magnetic field. This is a change in the magnetic field.

    17. An antenna

    18. Look at the cross section

    19. Angular Dependence of Intensity This shows the angular dependence of the radiation intensity produced by a dipole antenna The intensity and power radiated are a maximum in a plane that is perpendicular to the antenna and passing through its midpoint The intensity varies as (sin2 ?) / r2

    21. Harmonic Plane Waves

    22. Applying Faraday to radiation

    23. Applying Ampere to radiation

    24. Fields are functions of both position (x) and time (t)

    25. The Trial Solution The simplest solution to the partial differential equations is a sinusoidal wave: E = Emax cos (kx – ?t) B = Bmax cos (kx – ?t) The angular wave number is k = 2p/? ? is the wavelength The angular frequency is ? = 2pƒ ƒ is the wave frequency

    26. The trial solution

    27. The speed of light (or any other electromagnetic radiation)

    29. The electromagnetic spectrum

    31. Another look

    32. Energy in Waves

    33. Poynting Vector Poynting vector points in the direction the wave moves Poynting vector gives the energy passing through a unit area in 1 sec. Units are Watts/m2

    34. Intensity The wave intensity, I, is the time average of S (the Poynting vector) over one or more cycles When the average is taken, the time average of cos2(kx - ?t) = ½ is involved

    36. Radiation Pressure

    37. Pressure and Momentum For a perfectly reflecting surface, p = 2U/c and P = 2S/c For a surface with a reflectivity somewhere between a perfect reflector and a perfect absorber, the momentum delivered to the surface will be somewhere in between U/c and 2U/c For direct sunlight, the radiation pressure is about 5 x 10-6 N/m2

    39. Background for the superior mathematics student!

    40. Harmonic Plane Waves

    41. Phase Velocity - Another View

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