Lecture 2 - Birminghamepweb2.ph.bham.ac.uk/user/lazzeroni/EM2_2017/Lecture2_EM... · 2017. 1....

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Electromagnetism II Cristina Lazzeroni [email protected] Lecture 2

Transcript of Lecture 2 - Birminghamepweb2.ph.bham.ac.uk/user/lazzeroni/EM2_2017/Lecture2_EM... · 2017. 1....

  • Electromagnetism II

    Cristina [email protected]

    Lecture 2

  • Lecture 2:

    - Maxwell’s only contribution to the laws of EM- Need for displacement current- Maxwell’s equations in free space in differential form- Equation of continuity (conservation of charge)

  • Maxwell’s equations, integral form (from lecture 1)

  • Ampere’s Law is incomplete

    For current distributions involving a high degree of symmetry,instead of the Biot-Savat law, Ampere’s law can be used tocalculate the magnetic field

    Maxwell knew that a change in B-field produces an E-field:

    Can a changing E-field produce a B-field ?

    M4

    M3

  • Induced magnetic fields

    Direct experiment shows thatB field is generated by changing E field both inside the plates and outside

    Capacitor is being charged

    Ampere’s law to a loop inside the plates: conduction current inside the gap is zero !!!Maxwell realized that something called displacement current flows in the gap

  • Maxwell argued as follows:

    Ampere’s Law will work at all times if we add this term to thecurrent density

    Displacementcurrent

  • Maxwell’s Law of induction(induced magnetic field)His sole contribution to the laws of E,B fields but a crucial one

    Ampere-Maxwell Law

    B field can be set up either by a conduction or a displacementcurrent

  • Maxwell’s equations in free space, differential form

  • Gauss’ law in differential form:

    M1

  • Similarly:

  • Faraday’s law in differential form:

    Differentiate at a fixed place wrt time so time derivative can go inside integral

  • Similarly, Ampere-Maxwell’s law in differential form:

  • Lorentz equation:

  • Equation of continuity:

  • Derivation of equation of continuity:

    Conservation of charge: any variation in the total chargewithin a closed surface must be due to charges that flowacross the surface

    For any net charge that leaves Sthere must be an equivalent reduction in Q

    Charge leaving Surface S =change in the amount ofcharge inside the volume bounded by S

  • Differential form of thelaw of conservationof electric charge

    Can also be derived from M1 and M4, taking divergence of M4

  • Can also start from steady state equationUse continuity equation to justify displacement current:

    Take the divergence: (A)

    But from

    (A)must be modified by adding to the right hand sidea quantity that will make the divergence everywhere zero.Start from Gauss’ law and make ρ change with time:

    Therefore the quantity to add is

    Indeed the displacement current

  • Example:

    A parallel plate capacitor with circular plats is being charged

    Derive expressions forthe induced magneticfield at various radii r

  • Example:

    A parallel plate capacitor with circular plats is being charged

    Derive expressions forthe induced magneticfield at various radii r

  • Summary

  • Next Lecture:Electrostatic solutions to Maxwell’s equations

    Recommended readings:Grant+Phillips: 1.4.3, 1.4.4, 4.2.2, 4.5, 4.5.1, 4.5.2 6.14

    Before next lecture please : Redo the examples in lectures 1,2 by yourself Revise vector calculusMemorize Maxwell’s equations in free space

  • Digression: vector calculus

    See also Maths notes from lectures 22,23,24,25 of Term 1