Fysik 6

Elektrodynamik
John Niclasen

Contents:

D. J. Griffiths, Introduktion to Elektrodynamics, Third Edition, Prentise-Hall, 1999.

Symboler

Symbol	Forklaring
\bar A	Magnetisk vektor potential
\bar B	Magnetisk felt
\bar D	Elektrisk 'displacement'
\bar E	Elektrisk felt
\bar F	Kraft
\bar H	'Auxiliary' felt (Magnetisk \bar D)
I	Strøm
\bar J	Strømtæthed
\bar K	Strømtæthed
\bar M	Magnetisk polarisation
\bar P	Elektrisk polarisation
q	Ladning
\bar S	Energi flux tæthed
U	Energi
u	Energitæthed
V	Elektrisk scalar potential
\bar v	Hastighed
W	Arbejde
\epsilon	'Permittivity'
\mu	'Permeability'
\rho	Ladningstæthed

1. Maxwell's Equations

Afsnit 7.3, side 321-333, med undtagelse af underafsnit 7.3.4, side 327-328. Vi er gået overfladisk hen over udledningen af Maxwells ligninger i medier i afsnit 7.3.5, og I vil derfor ikke blive stillet til regnskab for udledningen, men det er vigtigt, at I kender resultatet i ligning (7.55).

1.1 Maxwell's Equations

1.1.1 Gauss's law \displaylines{\bar \nabla \cdot \bar E={1\over \epsilon _0} \rho } 1.1.2 No name \displaylines{\bar \nabla \cdot \bar B=0} 1.1.3 Faraday's law \displaylines{\bar \nabla \times \bar E=-{\partial \bar B\over \partial t}} 1.1.4 Ampère's law with Maxwell's correction \displaylines{\bar \nabla \times \bar B=\mu _0\bar J+\mu _0\epsilon _0{\partial \bar E\over \partial t}}

1.1.5 Lorentz force law

\displaylines{\bar F=q(\bar E+\bar v\times \bar B)}

1.1.6 Continuity equation

\displaylines{\bar \nabla \cdot \bar J=-{\partial \rho \over \partial t}}

1.2 Maxwell's Equations in Matter

In terms of free charges and currents: \displaylines{\bar \nabla \cdot \bar D=\rho _f\cr \bar \nabla \cdot \bar B=0\cr \bar \nabla \times \bar E=-{\partial \bar B\over \partial t}\cr \bar \nabla \times \bar H=\bar J_f+{\partial \bar D\over \partial t}\cr }

1.2.1 Bound and free charge

\displaylines{\rho _b=-\bar \nabla \cdot \bar P\cr \rho =\rho _f+\rho _b=\rho _f-\bar \nabla \cdot \bar P\cr }

1.2.2 Bound, polarized and free current

\displaylines{\bar J_b=\bar \nabla \times \bar M\cr \bar J_p={\partial \bar P\over \partial t}\cr \bar J=\bar J_f+\bar J_b+\bar J_p=\bar J_f+\bar \nabla \times \bar M+{\partial \bar P\over \partial t}\cr }

1.2.3 Displacement current

\displaylines{\bar J_d={\partial D\over \partial t}}

1.3 Boundary Conditions

1.3.1 Maxwell's equations in their integral form

Over any closed surface S:

\displaylines{\oint _S\bar D\cdot d\bar a=Q_{f enc}\cr \oint _S\bar B\cdot d\bar a=0\cr }

For any surface S bounded by the closed loop P:

\displaylines{\oint _P\bar E\cdot d\bar l=-{d\over dt}\int _S\bar B\cdot d\bar a\cr \oint _P\bar H\cdot d\bar l=I_{f enc}+{d\over dt}\int _S\bar D\cdot d\bar a\cr }

---

\displaylines{D_1^{\bot }-D_2^{\bot }=\rho _f\cr B_1^{\bot }-B_2^{\bot }=0\cr \bar E_1^{\Vert }-\bar E_2^{\Vert }=0\cr \bar H_1^{\Vert }-\bar H_2^{\Vert }=\bar K_f\times \hat n\cr }

If there is no free charge or free current at the interface:

\displaylines{\epsilon _1E_1^{\bot }-\epsilon _2E_2^{\bot }=0\cr B_1^{\bot }-B_2^{\bot }=0\cr \bar E_1^{\Vert }-\bar E_2^{\Vert }=0\cr {1\over \mu _1} \bar B_1^{\Vert }-{1\over \mu _2} \bar B_2^{\Vert }=0\cr }

2. Charge and Energy

Afsnit 8.1, side 345-349.

2.1 The Continuity Equation

\displaylines{{\partial \rho \over \partial t}=-\bar \nabla \cdot \bar J}

2.2 Poynting's Theorem

The work done on the charges by the electromagnetic force is equal to the decrease in energy stored in the field, less the energy that flowed out through the surface. \displaylines{{dW\over dt}=-{d\over dt}\int _V{1\over 2}\left (\epsilon _0E^2+{1\over \mu _0}B^2\right )d\tau -{1\over \mu _0}\oint _S(\bar E\times \bar B)\cdot d\bar a}

Poynting vector \displaylines{\bar S\equiv {1\over \mu _0}(\bar E\times \bar B)}

2.2.1 Compact form of Poynting's Theorem

\displaylines{{dW\over dt}=-{dU_{em}\over dt}-\oint _S\bar S\cdot d\bar a}

---

\displaylines{u_{em}={1\over 2}\left (\epsilon _0E^2+{1\over \mu _0}B^2\right )\cr {\partial \over \partial t}(u_{mech}+u_{em})=-\bar \nabla \cdot \bar S\cr }

3. Electromagnetic Waves

3.1 Electromagnetic Waves in Vacuum

9.2-9.3.2, side 375-386. Kommentarerne om impuls og tryk i lign (9.58), (9.59), (9.62) og (9.64) er ikke pensum.

3.1.1 The Wave Equation for E and B

\displaylines{\nabla ^2\bar E=\mu _0\epsilon _0{\partial ^2\bar E\over \partial t^2}\cr \nabla ^2\bar B=\mu _0\epsilon _0{\partial ^2\bar B\over \partial t^2}\cr v={1\over \sqrt {\epsilon _0\mu _0}}=3.00\times 10^8 m/s=c\cr }

3.1.2 Monochromatic Plane Waves

\displaylines{\widetilde E(z,t)=\widetilde E_0 e^{i(kz-\omega t)}\cr \widetilde B(z,t)=\widetilde B_0 e^{i(kz-\omega t)}\cr }

The actual (real) fields:

\displaylines{\bar E(z,t)=E_0 \cos (kz-\omega t+\delta )\hat x\cr \bar B(z,t)={1\over c} E_0 \cos (kz-\omega t+\delta )\hat y\cr }

In any direction:

\displaylines{\widetilde E(\bar r,t)=\widetilde E_0e^{i(\bar k\cdot \bar r-\omega t)} \hat n\cr \widetilde B(\bar r,t)={1\over c} \widetilde E_0e^{i(\bar k\cdot \bar r-\omega t)}(\hat k\times \hat n)={1\over c}\hat k\times \widetilde E\cr }

, where \hat n is the polarization vector.

\displaylines{\hat n\cdot \hat k=0}

The actual (real) fields:

\displaylines{\bar E(\bar r,t)=E_0 \cos (\bar k\cdot \bar r-\omega t+\delta ) \hat n\cr \bar B(\bar r,t)={1\over c} E_0 \cos (\bar k\cdot \bar r-\omega t+\delta ) (\hat k\times \hat n)\cr }

3.1.3 Energy and Momentum in Electromagnetic Waves

Se s. 380-382

3.2 Electromagnetic Waves in Matter

Se s. 382-386

3.3 Absorption and Dispersion

Afsnit 9.4.1 og 9.4.2, side 392-398.

Se s. 392-398

4. Potentials and Fields

4.1 The Potential Formulation

Afsnit 10.1 og 10.2.1, side 416-426.

4.1.1 Magnetic vector potential A

\displaylines{\bar B=\bar \nabla \times \bar A}

4.1.2 Electric scalar potential V

\displaylines{\bar E=-\bar \nabla V-{\partial \bar A\over \partial t}}

4.1.3 Gauge Transformations

\displaylines{\bar A'=\bar A+\bar \nabla \lambda \cr V'=V-{\partial \lambda \over \partial t}\cr }

4.1.4 Coulomb Gauge

\displaylines{\bar \nabla \cdot \bar A=0\cr \nabla ^2V=-{1\over \epsilon _0} \rho \cr }

Solution:

\displaylines{V(\bar r,t)={1\over 4\pi \epsilon _0}\int {\rho (\bar r',t)\over r}d\tau '}

Differential equation for \bar A in the Coulomb gauge:

\displaylines{\nabla ^2\bar A-\mu _0\epsilon _0{\partial ^2\bar A\over \partial t^2}=-\mu _0\bar J+\mu _0\epsilon _0 \bar \nabla \left ({\partial V\over \partial t}\right )}

4.1.5 Lorentz Gauge

\displaylines{\bar \nabla \cdot \bar A=-\mu _0\epsilon _0{\partial V\over \partial t}\cr \nabla ^2\bar A-\mu _0\epsilon _0{\partial ^2\bar A\over \partial t^2}=-\mu _0\bar J\cr }

Differential equation for V:

\displaylines{\nabla ^2V-\mu _0\epsilon _0{\partial ^2V\over \partial t^2}=-{1\over \epsilon _0} \rho }

4.1.6 d'Alembertian

Se s. 422

4.2 Continuous Distributions

4.2.1 Retarded Potentials

\displaylines{t_r\equiv t-{r\over c}\cr V(\bar r,t)={1\over 4\pi \epsilon _0}\int {\rho (\bar r',t_r)\over r}d\tau '\cr \bar A(\bar r,t)={\mu _0\over 4\pi }\int {\bar J(\bar r',t_r)\over r}d\tau '\cr }

5. Radiation

5.1 Dipole Radiation

Afsnit 11.1.1 og 11.1.2 side 443-449.

5.1.1 What is Radiation

The total power passing out through a surface is the integral of the Poynting vector:

\displaylines{P(r)=\oint \bar S\cdot d\bar a={1\over \mu _0}\oint (\bar E\times \bar B)\cdot d\bar a}

The power radiated is the limit of this quantity as r goes to infinity:

\displaylines{P_{rad}\equiv \lim _{r\to \infty }P(r)}

5.1.2 Electric Dipole Radiation

\displaylines{V(r,\theta ,t)=-{p_0\omega \over 4\pi \epsilon _0c}\left ({\cos  \theta \over r}\right )\sin \lbrack \omega (t-r/c)\rbrack \cr \bar A(r,\theta ,t)=-{\mu _0p_0\omega \over 4\pi r} \sin \lbrack \omega (t-r/c)\rbrack \hat z\cr \bar E=-\bar \nabla V-{\partial \bar A\over \partial t}=-{\mu _0p_0\omega ^2\over 4\pi }\left ({\sin  \theta \over r}\right )\cos \lbrack \omega (t-r/c)\rbrack \hat \theta \cr \bar B=\bar \nabla \times \bar A=-{\mu _0p_0\omega ^2\over 4\pi c}\left ({\sin  \theta \over r}\right )\cos \lbrack \omega (t-r/c)\rbrack \hat \phi \cr }

The energy radiated by an oscillating electric dipole is determined by the Poynting vector:

\displaylines{\bar S={1\over \mu _0}(\bar E\times \bar B)={\mu _0\over c}\left \{{p_0\omega ^2\over 4\pi }\left ({\sin  \theta \over r}\right )\cos \lbrack \omega (t-r/c)\rbrack \right \}^2\hat r}

The intensity is obtained by averaging (in time) over a complete cycle:

\displaylines{\langle \bar S\rangle =\left ({\mu _0p_0^2\omega ^4\over 32\pi ^2c}\right ){\sin ^2\theta \over r^2}\hat r}

The total power radiated is found by integrating \langle \bar S\rangle over a sphere of radius r:

\displaylines{\langle P\rangle =\int \langle \bar S\rangle \cdot d\bar a={\mu _0p_0^2\omega ^4\over 32\pi ^2c}\int {\sin ^2\theta \over r^2}r^2\sin  \theta  d\theta  d\phi ={\mu _0p_0^2\omega ^4\over 12\pi c}}

6. Electromagnetic waves in wires

Fra noterne "More on waves" s. 11-16

6.1 Coaxial cable

6.1.1 Capacitance per unit length

\displaylines{c={2\pi \epsilon \over \ln (b/a)}}

, where a and b are the inner and outer radius.

6.1.2 Inductance per unit length

\displaylines{l={\mu \over 2\pi }\ln (b/a)}

6.2 Telegrapher's equations

\displaylines{{\partial V\over \partial t}=-{1\over c} {\partial I\over \partial x}\cr {\partial I\over \partial t}=-{1\over l} {\partial V\over \partial x}\cr {\partial ^2V\over \partial t^2}={1\over lc} {\partial ^2V\over \partial x^2}\cr {\partial ^2I\over \partial t^2}={1\over lc} {\partial ^2I\over \partial x^2}\cr v={1\over \sqrt {lc}}\cr }

6.3 Waves traveling down the wire

\displaylines{{\partial I\over \partial t}=-v{\partial I\over \partial x}\cr {\partial V\over \partial x}=vl{\partial I\over \partial x}\cr }

6.3.1 Characteristic impedance

\displaylines{Z_0=vl=\sqrt {{l\over c}}\cr \Rightarrow \quad V=Z_0I\cr }

For a coaxial cable:

\displaylines{Z_0={1\over 2\pi }\sqrt {{\mu \over \epsilon }} \ln \left ({b\over a}\right )}