Introduction to Electrodynamics/Chapter 7

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Overview

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Chapter 7 consolidates what we have learned in electrostatics (Chapters 3 and 4) and magnetostatics Chapters 5 and 6) into universal equations that cover moving and stationary charges --- electrodynamics.

Sections

Key Findings

  • The rules of electrostatics and magnetostatics pretty much apply in electrodynamics. The new rule is that changes to a field behave like a "current" for the other field, generating a curl.
  • E is not necessarily 0 in a conductor.
  • Current density <math>\vec{J}\;</math> is proportional to the electric field in a conductor.
    • <math>\sigma\;</math> is the conductivity of a material, how rapidly charge will move through the material given a force.
    • <math>\rho = 1 / \sigma\;</math> is the resistivity of a material, how poorly the charge moves through the material given a force.
    • Perfect conductors have <math>\sigma = \infty\;</math>.
    • Insulators have very, very large resistivities.
    • <math>\vec{J} = \sigma \vec{f}\;</math> (force per unit charge)
    • <math>\vec{J} = \sigma (\vec{E} + \vec{v} \times \vec{B})\;</math> (ignoring all other forces)
    • <math>\vec{J} = \sigma \vec{E}\;</math> (The real Ohm's Law) (magnetism usually doesn't matter.)
  • Ohm's Law: <math>V = I R\;</math>
    • Not really a law
    • Surprising it applies at all. Isn't <math>\sum \vec{F} = m \vec{a}\;</math>?
    • The resistance <math>R\;</math> is a function of the properties of the material and the geometry. It is in ohms <math>1 \Omega = 1 V / 1 A\;</math>
  • For steady currents within a material of uniform conductivity: <math>\nabla \cdot \vec{E} = {1 \over \sigma} \nabla \cdot \vec{J} = 0\;</math>. The charge density is therefore 0.
    • Just like for conductors in electrostatics!
  • For metal conductors, conductivity is so large that we can assume it is a perfect conductor with little consequence.
  • Joule heating law: <math>P = VI = IR^2\;</math>
    • <math>P = \sigma \int E^2\ d\tau\;</math>
  • Current is constant through a circuit since any charge build up will generate an electric field to disperse it.
    • Total electrostatic force <math>\vec{f} = \vec{f_s} + \vec{E}\;</math>
    • <math>\vec{f_s}\;</math> is the source force: battery, photoelectric cell, etc, whatever will push charge SOMEWHERE in the circuit.
    • <math>\int \vec{f_s} \cdot d\vec{l} = \mathcal{E}\;</math>, the electromotive force (emf).
  • Loops moving through a current generate an electromotive force:
    • <math>\vec{\Phi} = \int \vec{B}\cdot d\vec{a}</math> is the magnetic flux through a loop
    • <math>\vec{\mathcal{E}} = - {d \vec{\Phi} \over dt}\;</math> (flux rule)
  • Faraday's Law:
    • <math>\oint \vec{E}\cdot d\vec{l} = - {d \vec{\Phi} \over dt}\;</math> (integral form)
    • <math>\nabla \times \vec{E} = - {\partial \vec{B} \over \partial t}\;</math> (differential form)
    • <math>\vec{E} = - {\partial \vec{A} \over \partial t}\;</math> (Magnetic vector potential form)
    • Magnetic flux for electric fields is like a current for magnetic fields.
    • Lenz's Law: Current flows to preserve magnetic field.
  • quasistatic: Not quite electrostatic or magnetostatic, but the deviation is so slight that it is close enough to use those equations.
  • Neumann formula: Mutual inductance <math>M = {\mu_0 \over 4 \pi} \oint\oint {d\vec{l_1} \cdot d\vec{l_2} \over \vec{\mathcal{r}}}\;</math>
    • Purely geometric.
    • <math>M_{21} = M_{12}\;</math>
    • <math>\vec{\Phi_a} = M\vec{I_b}\;</math>
  • Changing currents induce an emf on themselves.
    • <math>\vec{\Phi} = L\vec{I}\;</math>
    • <math>\mathcal{E} = -L{dI \over dt}\;</math>
    • L is measured in Henries (H). <math>1 H = 1 V s / A\;</math>
    • Sometimes the self-induced EMF is called the back EMF.
  • <math>W = {1 \over 2} LI^2\;</math>
  • <math>W = {1 \over 2 \mu_0} \int_\text{all space} B^2\ d\tau\;</math>
    • Magnetic fields do no work, but a changing magnetic field creates an electric field that does the work.
  • Maxwell's Equations
    1. <math>\nabla \cdot \vec{E} = {1 \over \epsilon_0}\rho\;</math>
    2. <math>\nabla \cdot \vec{B} = 0\;</math> (no magnetic monopoles)
    3. <math>\nabla \times \vec{E} = -{\partial \vec{B} \over \partial t}\;</math>
    4. <math>\nabla \times \vec{B} = \mu_0\vec{J}\;</math>
    • What about an Amperian Loop around half a capacitor? No current, yet there IS a magnetic field.
  • <math>\nabla \times \vec{B} = \mu_0\vec{J} + \mu_0 \epsilon_0 {\partial \vec{E} \over \partial t}\;</math>
    • Just as a changing B field creates a curly E, so does a changing E field create a curly B.
    • Changing E is a current for B just like Changing B is a current for E.
    • Displacement current <math>\vec{J_d} = \epsilon_0 {\partial \vec{E} \over \partial t}\;</math>
    • <math>\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \int \left({\partial \vec{E} \over \partial t}\right) \cdot d\vec{a}\;</math>
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