Physics 4

Electromagnetism
John Niclasen

Contents:

1. Vector Analysis
1.1 Vector Algebra
1.2 Differential Calculus
1.3 Integral Calculus
1.4 Curvilinear Coordinates
2. Electrostatics
2.1 Coulomb's Law
2.2 The Electric Field
2.3 Continuous Charge Distributions
2.4 Flux
2.5 Gauss's Law
2.6 The Divergence of E
2.7 The Curl of E
2.8 Electric Potential
2.9 Work and Energy in Electrostatics
2.10 Conductors
3. Special Techniques
3.1 Laplace's Equation
3.2 The Method of Images
3.3 Multipole Expansion
4. Electric Fields in Matter
4.1 Polarization
4.2 The Field of a Polarized Object
4.3 The Electric Displacement
4.4 Linear Dielectrics
5. Magnetostatics
5.1 The Lorentz Force Law
5.2 The Biot-Savart Law
5.3 The Divergence and Curl of B
5.4 Magnetic Vector Potential
6. Magnetic Fields in Matter
6.1 Magnetization
6.2 The Field of a Magnetized Object
6.3 The Auxiliary Field H
6.4 Linear and Nonlinear Media
7. Electrodynamics
7.1 Electromotive Force
7.2 Electromagnetic Induction
7.3 Maxwell's Equations

Symbols

Symbol

Meaning

\bar A

Vector potential in magnetostatics

\bar B

Magnetic field

C

Capacitance

\bar D

Electric displacement

\bar E

Electric field

\bar F

Force

\bar H

Auxiliary field

I

Current

\bar J

Volume current density

\bar K

Surface current density

\bar m

Magnetic dipole moment

M

Magnetization

N

Torque

\bar p

Dipole moment

\bar P

Polarization

q, Q

Electric charge

r

Radius (distance)

U

Energy

\bar v

Velocity

V

Electric potential

W

Work (energy)

\alpha

Atomic polarizability

\chi _e

Electric susceptibility

\chi _m

Magnetic susceptibility

\epsilon _0

Permittivity of free space, 8.85\times 10^{-12}C^2N^{-1}m^{-2}

\epsilon

Permittivity of the material, \epsilon \equiv \epsilon _0(1+\chi _e)

\epsilon _r

Relative permittivity, or dielectric constant, \epsilon _r\equiv 1+\chi _e

\mu _0

Permeability of free space, 4\pi \times 10^{-7}N A^{-2}

\mu

Permeability of the material, \mu =\mu _0(1+\chi _m)

\Phi

Flux

dl

Element of length (1D)

da

Element of area (2D)

d\tau

Element of volume (3D)

\lambda

Charge per unit length (1D)

\sigma

Charge per unit area (2D)

\rho

Charge per unit volume (3D)

\rho _b

Bound charge

\rho _f

Free charge

1. Vector Analysis

1.1 Vector Algebra

1.1.1 Dot Product

\displaylines{\bar A\cdot \bar B\equiv A B \cos  \theta \cr \bar A\cdot \bar B=a_x b_x+a_y b_y+a_z b_z\cr \bar A\cdot \bar B=\bar B\cdot \bar A\cr }



1.1.2 Cross Product

\displaylines{\bar A\times \bar B\equiv A B \sin  \theta  \bar n\cr \bar A\times \bar B=\pmatrix{a_y b_z-a_z b_y\cr a_z b_x-a_x b_z\cr a_x b_y-a_y b_x\cr }\cr (\bar B\times \bar A)=-(\bar A\times \bar B)\cr \bar A\times \bar A=0\cr }

1.1.3 Triple Producs

\displaylines{\bar A\cdot (\bar B\times \bar C)=\bar B\cdot (\bar C\times \bar A)=\bar C\cdot (\bar A\times \bar B)\cr \bar A\cdot (\bar B\times \bar C)=(\bar A\times \bar B)\cdot \bar C\cr \bar A\times (\bar B\times \bar C)=\bar B(\bar A\cdot \bar C)-\bar C(\bar A\cdot \bar B)\cr (\bar A\times \bar B)\times \bar C=-\bar C\times (\bar A\times \bar B)=-\bar A(\bar B\cdot \bar C)+\bar B(\bar A\cdot \bar C)\cr (\bar A\times \bar B)\cdot (\bar C\times \bar D)=(\bar A\cdot \bar C)(\bar B\cdot \bar D)-(\bar A\cdot \bar D)(\bar B\cdot \bar C)\cr \bar A\times (\bar B\times (\bar C\times \bar D))=\bar B(\bar A\cdot (\bar C\times \bar D))-(\bar A\cdot \bar B)(\bar C\times \bar D)\cr }

1.2 Differential Calculus

1.2.1 Divergence

\displaylines{\bar \nabla \cdot \bar v={\partial v_x\over \partial x}+{\partial v_y\over \partial y}+{\partial v_z\over \partial z}}

1.2.2 Curl

\displaylines{\bar \nabla \times \bar v=\hat x\left ({\partial v_z\over \partial y}-{\partial v_y\over \partial z}\right )+\hat y\left ({\partial v_x\over \partial z}-{\partial v_z\over \partial x}\right )+\hat z\left ({\partial v_y\over \partial x}-{\partial v_x\over \partial y}\right )}

1.3 Integral Calculus

1.3.1 Fundamental Theorem for Gradients

Along a path:

\displaylines{\int _{\bar a}^{\bar b}=(\bar \nabla T)\cdot d\bar l=T(\bar b)-T(\bar a)}

1.3.2 Fundamental Theorem for Divergences

Gauss's theorem, Green's theorem or the divergence theorem

\displaylines{\int _V(\bar \nabla \cdot \bar v)d\tau =\oint _S\bar v\cdot d\bar a}

1.3.3 Fundamental Theorem for Curls

Stoke' Theorem

\displaylines{\int _S(\bar \nabla \times \bar v)\cdot d\bar a=\oint _P\bar v\cdot d\bar l}

1.4 Curvilinear Coordinates

1.4.1 Spherical Polar Coordinates

\displaylines{\bar A=A_r \hat r+A_{\theta } \hat \theta +A_{\phi } \hat \phi \cr \hat r=\sin  \theta  \cos  \phi  \hat x+\sin  \theta  \sin  \phi  \hat y+\cos  \theta  \hat z\cr \hat \theta =\cos  \theta  \cos  \phi  \hat x+\cos  \theta  \sin  \phi  \hat y-\sin  \theta  \hat z\cr \hat \phi =-\sin  \phi  \hat x+\cos  \phi  \hat y\cr }

Along a path (line):

\displaylines{d\bar l=dr \hat r+r d\theta  \hat \theta +r \sin  \theta  d\phi  \hat \phi }

Over the surface of a sphere:

\displaylines{d\bar a_1=dl_{\theta } dl_{\phi } \hat r=r^2 \sin  \theta  d\theta  d\phi  \hat r}

In the xy-plane with constant \theta :

\displaylines{d\bar a_2=dl_r dl_{\phi } \hat \theta =r dr d\phi  \hat \theta }

Volume:

\displaylines{d\tau =dl_r dl_{\theta } dl_{\phi }=r^2 \sin  \theta  dr d\theta  d\phi }

1.4.2 Cylindrical Coordinates

\displaylines{x=s \cos  \phi \quad ,\quad y=s \sin  \phi \quad ,\quad z=z\cr \hat s=\cos  \phi  \hat x+\sin  \phi  \hat y\cr \hat \phi =-\sin  \phi  \hat x+\cos  \phi  \hat y\cr \hat z=\hat z\cr dl_s=ds\quad ,\quad dl_{\phi }=s d\phi \quad ,\quad dl_z=dz\cr }

Along a path (line):

\displaylines{d\bar l=ds \hat s+s d\phi  \hat \phi +dz \hat z}

Volume:

\displaylines{d\tau =s ds d\phi  dz}

2. Electrostatics

2.1 Coulomb's Law

\displaylines{\bar F={1\over 4\pi \epsilon _0} {q Q\over r^2}\hat r}

2.2 The Electric Field

\displaylines{\bar F=Q\bar E\cr \bar E(\bar r)\equiv {1\over 4\pi \epsilon _0}\sum _{i=1}^n{q_i\over r_i^2}\hat r_i\cr }

2.3 Continuous Charge Distributions

\displaylines{\bar E(\bar r)\equiv {1\over 4\pi \epsilon _0}\int {1\over r^2}\hat r dq\cr }
\displaylines{dq\to \lambda  dl'\sim \sigma  da'\sim \rho  d\tau '}

2.3.1 Line charge

\displaylines{\bar E(\bar r)\equiv {1\over 4\pi \epsilon _0}\int _P{\lambda (\bar r')\over r^2}\hat r dl'\cr }

2.3.2 Surface charge

\displaylines{\bar E(\bar r)\equiv {1\over 4\pi \epsilon _0}\int _S{\sigma (\bar r')\over r^2}\hat r da'\cr }

2.3.3 Volume charge

\displaylines{\bar E(\bar r)\equiv {1\over 4\pi \epsilon _0}\int _V{\rho (\bar r')\over r^2}\hat r d\tau '\cr }

2.4 Flux

2.4.1 Point charge

\displaylines{\bar E(\bar r)\equiv {1\over 4\pi \epsilon _0}{q\over r^2}\hat r\cr }

Flux of \bar E through a surface S:

\displaylines{\Phi _E\equiv \int _S\bar E\cdot d\bar a}

2.5 Gauss's Law

2.5.1 Integral form

\displaylines{\oint _S\bar E\cdot d\bar a={1\over \epsilon _0}Q_{enc}}

2.5.2 Differential form

\displaylines{\bar \nabla \cdot \bar E={1\over \epsilon _0}\rho }

2.6 The Divergence of E

\displaylines{\int _V\bar \nabla \cdot \bar E d\tau =\oint _S\bar E\cdot d\bar a={1\over \epsilon _0}\int _V\rho  d\tau ={1\over \epsilon _0}Q_{enc}\cr }

2.7 The Curl of E

By applying Stokes' theorem:

\displaylines{\oint \bar E\cdot d\bar l=0\cr \Rightarrow \quad \bar \nabla \times \bar E=0\cr }

2.8 Electric Potential

\displaylines{V(\bar r)\equiv -\int _O^{\bar r}\bar E\cdot d\bar l\cr \int _{\bar a}^{\bar b}(\bar \nabla  V)\cdot d\bar l=-\int _{\bar a}^{\bar b}\bar E\cdot d\bar l\cr \Rightarrow \quad \bar E=-\bar \nabla  V\cr }

2.8.1 Poisson's Equation

\displaylines{\nabla ^2V=-{\rho \over \epsilon _0}}

2.8.2 Laplace's Equation

\displaylines{\rho =0\cr \Rightarrow \quad \nabla ^2V=0\cr }

2.8.3 Line Charge

\displaylines{V(\bar r)={1\over 4\pi \epsilon _0}\int {\lambda (\bar r')\over r}dl'\cr }

2.8.4 Surface Charge

\displaylines{V(\bar r)={1\over 4\pi \epsilon _0}\int {\sigma (\bar r')\over r}da'\cr }

2.8.5 Volume Charge

\displaylines{V(\bar r)={1\over 4\pi \epsilon _0}\int {\rho (\bar r')\over r}d\tau '\cr }

2.9 Work and Energy in Electrostatics

\displaylines{W=\int _{\bar a}^{\bar b}\bar F\cdot d\bar l=-Q\int _{\bar a}^{\bar b}\bar E\cdot d\bar l=Q\lbrack V(\bar b)-V(\bar a)\rbrack \cr W=Q V(\bar r)\cr }

2.9.1 The Energy of a Point Charge Distribution

\displaylines{W={1\over 2}\sum _{i=1}^nq_i V(\bar r_i)\cr }

2.9.2 The Energy of a Continuous Charge Distribution

\displaylines{W={1\over 2}\int \rho  V d\tau }

Alternative:

\displaylines{W={\epsilon _0\over 2}\int _{allspace}E^2 d\tau }

2.10 Conductors

  • \bar E=0 inside a conductor
  • \rho =0 inside a conductor
  • Any net charge resides on the surface
  • A conductor is an equipotential
  • \bar E is perpendicular to the surface, just outside a conductor

2.10.1 Surface Charge and the Force on a Conductor

\displaylines{\bar E={\sigma \over \epsilon _0}\hat n\cr \bar f={1\over 2\epsilon _0}\sigma ^2\hat n\cr }

This amounts to an outward electrostatic pressure:

\displaylines{P={\epsilon _0\over 2}E^2}

2.10.2 Capacitors

\displaylines{C\equiv {Q\over V}}

C is measured in farads (F).

For a parallel-plate capacitor:

\displaylines{V={Q\over A\epsilon _0}d\cr C={A\epsilon _0\over d}\cr }

, where A is the areal and d is the distance between the plates.

For a charged capacitor:

\displaylines{W={1\over 2}CV^2}

2.10.3 Child-Langmuir Law

The current flowing between a Cathode and an Anode:

\displaylines{I=K V_0^{3/2}}

3. Special Techniques

3.1 Laplace's Equation

\displaylines{\nabla ^2V=0}

Written out in Cartesian coordinates:

\displaylines{{\partial ^2V\over \partial x^2}+{\partial ^2V\over \partial y^2}+{\partial ^2V\over \partial z^2}=0}

3.1.1 1D

\displaylines{{\partial ^2V\over \partial x^2}=0}

Solution:

\displaylines{V(x)=mx+b}

3.1.2 2D

\displaylines{{\partial ^2V\over \partial x^2}+{\partial ^2V\over \partial y^2}=0}

No general solution.

3.1.3 3D

See Griffiths p. 114!

3.2 The Method of Images

An infinite grounded conducting plate acts like a mirror charge.

3.3 Multipole Expansion

\displaylines{V(\bar r)={1\over 4\pi \epsilon _0}\sum _{n=0}^{\infty }{1\over r^{(n+1)}}\int (r')^nP_n(\cos \theta ')\rho (\bar r')d\tau '\cr }

, where P_n is Legendre Polynomials:

\displaylines{P_0(x)=1\cr P_1(x)=x\cr P_2(x)=(3x^2-1)/2\cr P_3(x)=(5x^3-3x)/2\cr P_4(x)=(35x^4-30x^2+3)/8\cr P_5(x)=(63x^5-70x^3+15x)/8\cr }

3.3.1 Dipole Moment

\displaylines{\bar p\equiv \int \bar r'\rho (\bar r')d\tau '}

3.3.2 Dipole Contribution to the Potential

\displaylines{V_{dip}(\bar r)={1\over 4\pi \epsilon _0} {\bar p\cdot \hat r\over r^2}}

3.3.3 The Electric Field of a Dipole

\displaylines{E_{dip}(r,\theta )={p\over 4\pi \epsilon _0r^3}(2\cos \theta \hat r+\sin \theta \hat \theta )}

3.3.4 For a "pure" Dipole

\displaylines{E_{dip}(\bar r)={1\over 4\pi \epsilon _0} {1\over r^3} \lbrack 3(\bar p\cdot \hat r)\hat r-\bar p\rbrack }

4. Electric Fields in Matter

4.1 Polarization

4.1.1 Induced Dipoles

\displaylines{\bar p=\alpha \bar E}

, where \alpha is atomic polarizability. See Griffiths p. 161!

4.1.2 Alignment of Polar Molecules

A dipole \bar p=q\bar d in a uniform field \bar E experiences a torque

\displaylines{\bar N=\bar p\times \bar E}

For a "perfect" dipole of infinitesimal length:

\displaylines{\bar F=(\bar p\cdot \bar \nabla )\bar E}

Energy of an ideal dipole:

\displaylines{U=-\bar p\cdot \bar E}

4.1.3 Polarization

\bar P {\equiv } dipole moment per unit volume, which is called polarization.

4.2 The Field of a Polarized Object

4.2.1 Bound Charges

\displaylines{\sigma _b=\bar P\cdot \hat n\cr \rho _b\equiv -\bar \nabla \cdot \bar P\cr V(\bar r)={1\over 4\pi \epsilon _0}\oint _S{\sigma _b\over r}da'+{1\over 4\pi \epsilon _0}\int _V{\rho _b\over r}d\tau '\cr }

4.3 The Electric Displacement

4.3.1 Gauss's Law in the Presence of Dielectrics

\displaylines{\bar D\equiv \epsilon _0\bar E+\bar P\cr \Rightarrow \quad \bar \nabla \bar D=\rho _f\cr }

In integral form:

\displaylines{\oint \bar D\cdot d\bar a=Q_{f_{enc}}}

4.4 Linear Dielectrics

4.4.1 Susceptibility, Permittivity, Dielectric Constant

\displaylines{\bar P=\epsilon _0\chi _e\bar E\cr \bar D=\epsilon \bar E\cr }

, where

\displaylines{\epsilon \equiv \epsilon _0(1+\chi _e)}

Relative permittivity, or dielectric constant:

\displaylines{\epsilon _r\equiv 1+\chi _e={\epsilon \over \epsilon _0}}

See Griffiths p. 180!

4.4.2 Energy in Dielectric Systems

\displaylines{W={1\over 2}CV^2\cr C=\epsilon _rC_{vac}\cr W={1\over 2}\int \bar D\cdot \bar Ed\tau \cr W_{tot}=W_{free}+W_{bound}+W_{spring}\cr }

4.4.3 Forces on Dielectrics

A dielectric material in a parallel-plate capacitor:

\displaylines{F=-{dW\over dx}\cr F={1\over 2}V^2{dC\over dx}\cr }

5. Magnetostatics

5.1 The Lorentz Force Law

\displaylines{\bar F_{mag}=Q(\bar v\times \bar B)}

In the presence of both electric and magnetic fields:

\displaylines{\bar F=Q\lbrack \bar E+(\bar v\times \bar B)\rbrack }

Work

Magnetic forces do no work.

\displaylines{dW_{mag}=\bar F_{mag}\cdot d\bar l=Q(\bar v\times \bar B)\cdot \bar vdt=0}

5.1.1 Currents

Amperes (A):

\displaylines{1 A=1 C/s}
\displaylines{\bar F_{mag}=\int I(d\bar l\times \bar B)}

For constant current, I:

\displaylines{\bar F_{mag}=I\int (d\bar l\times \bar B)}

5.1.2 Surface Current Density

\displaylines{\bar K\equiv {d\bar I\over dl_{\bot }}\cr \bar K=\sigma \bar v\cr }

The magnetic force on a surface current

\displaylines{\bar F_{mag}=\int (\bar v\times \bar B)\sigma  da=\int (\bar K\times \bar B)da}

5.1.3 Volume current density

\displaylines{\bar J\equiv {d\bar I\over da_{\bot }}\cr \bar J=\rho \bar v\cr }

The magnetic force on a volume current

\displaylines{\bar F_{mag}=\int (\bar v\times \bar B)\rho  d\tau =\int (\bar J\times \bar B)d\tau }

5.1.4 The Continuity Equation

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

5.2 The Biot-Savart Law

5.2.1 Steady Currents

\displaylines{\bar \nabla \cdot \bar J=0}

5.2.2 The Magnetic Field of a Steady Current

The Biot-Savart law:

\displaylines{\bar B(\bar r)={\mu _0\over 4\pi }\int {\bar I\times \hat r\over r^2}dl'={\mu _0\over 4\pi }I\int {d\bar l'\times \hat r\over r^2}\cr }

5.2.3 Tesla

\displaylines{1 T=1 N/(A\cdot m)}

5.2.4 Surface Current

\displaylines{\bar B(\bar r)={\mu _0\over 4\pi }\int {\bar K(\bar r')\times \hat r\over r^2}da'}

5.2.5 Volume Current

\displaylines{\bar B(\bar r)={\mu _0\over 4\pi }\int {\bar J(\bar r')\times \hat r\over r^2}d\tau '}

5.3 The Divergence and Curl of B

5.3.1 Straight-Line Currents

\displaylines{\oint \bar B\cdot d\bar l=\oint {\mu _0I\over 2\pi s}dl={\mu _0I\over 2\pi s}\oint dl=\mu _0I\cr }

In cylindrical coordinates:

\displaylines{\bar B={\mu _0I\over 2\pi s}\hat \phi }
\displaylines{\oint \bar B\cdot d\bar l=\mu _0I_{enc}\cr I_{enc}=\int \bar J\cdot d\bar a\cr \int (\bar \nabla \times \bar B)\cdot d\bar a=\mu _0\int \bar J\cdot d\bar a\cr \bar \nabla \times \bar B=\mu _0\bar J\cr }

5.3.2 The Divergence of B

\displaylines{\bar \nabla \cdot \bar B=0}

5.3.3 The Curl of B

Ampère's law:

\displaylines{\bar \nabla \times \bar B=\mu _0\bar J}

Integral version:

\displaylines{\oint \bar B\cdot d\bar l=\mu _0I_{enc}}

5.4 Magnetic Vector Potential

\displaylines{\bar B=\bar \nabla \times \bar A\cr \bar \nabla \cdot \bar A=0\cr \nabla ^2\bar A=-\mu _0\bar J\cr \bar A(\bar r)={\mu _0\over 4\pi }\int {\bar J(\bar r')\over r}d\tau '\cr }

5.4.1 Line Current

\displaylines{\bar A={\mu _0\over 4\pi }\int {\bar I\over r}dl'={\mu _0I\over 4\pi }\int {1\over r}d\bar l'}

5.4.2 Volume Current

\displaylines{\bar A={\mu _0\over 4\pi }\int {\bar K\over r}da'}

5.4.3 Multipole Expansion of the Vector Potential

\displaylines{\bar A_{dip}(\bar r)={\mu _0\over 4\pi } {\bar m\times \hat r\over r^2}}

, where \bar m is the magnetic dipole moment:

\displaylines{\bar m\equiv I\int d\bar a=I\bar a}

5.4.4 Magnetic field of a dipole

\displaylines{\bar B_{dip}(\bar r)={\mu _0\over 4\pi } {1\over r^3}\lbrack 3(\bar m\cdot \hat r)\hat r-\bar m\rbrack }

6. Magnetic Fields in Matter

6.1 Magnetization

6.1.1 Torques and Forces on Magnetic Dipoles

\displaylines{\bar N=\bar m\times \bar B\cr \bar F=\bar \nabla (\bar m\cdot \bar B)\cr }

6.1.2 Magnetization

M{\equiv } magnetic dipole moment per unit volume, also called magnetization.

6.2 The Field of a Magnetized Object

\displaylines{\bar J_b=\bar \nabla \times \bar M\cr \bar K_b=\bar M\times \hat n\cr \bar A(\bar r)={\mu _0\over 4\pi }\int _V{\bar J_b(\bar r')\over r}d\tau '+{\mu _0\over 4\pi }\oint _S{\bar K_b(\bar r')\over r}da'\cr }

6.3 The Auxiliary Field H

6.3.1 Ampère's law in Magnetized Materials

\displaylines{\bar H\equiv {1\over \mu _0}\bar B-\bar M\cr \bar \nabla \times \bar H=\bar J_f\cr }

In integral form:

\displaylines{\oint \bar H\cdot d\bar l=I_{f_{enc}}}

6.3.2 A Deceptive Parallel

\displaylines{\bar \nabla \cdot \bar H=-\bar \nabla \cdot \bar M}

6.3.3 Boundary Conditions

\displaylines{H_{above}^{\bot }-H_{below}^{\bot }=-(M_{above}^{\bot }-M_{below}^{\bot })\cr \bar H_{above}^{\Vert }-\bar H_{below}^{\Vert }=\bar K_f\times \hat n\cr B_{above}^{\bot }-B_{below}^{\bot }=0\cr \bar B_{above}^{\Vert }-\bar B_{below}^{\Vert }=\mu _0(\bar K\times \hat n)\cr }

6.4 Linear and Nonlinear Media

\displaylines{\bar M=\chi _m\bar H\cr \bar B=\mu _0(\bar H+\bar M)=\mu _0(1+\chi _m)\bar H\cr \bar B=\mu \bar H\cr \mu \equiv \mu _0(1+\chi _m)\cr }

6.4.1 Energy of a Magnetic Dipole in a Magnetic Field B

\displaylines{U=-\bar m\cdot \bar B}

7. Electrodynamics

7.1 Electromotive Force

7.1.1 Ohm's Law

\displaylines{\bar J=\sigma \bar E\cr V=I R\cr }

7.1.2 Joule Heating Law

\displaylines{P=V I=I^2 R}

7.1.3 Electromotive Force

\displaylines{\varepsilon \equiv \oint \bar f\cdot d\bar l=\oint \bar f_s\cdot d\bar l}

7.1.4 Flux

\displaylines{\Phi \equiv \int \bar B\cdot d\bar a\cr \varepsilon =-{d\Phi \over dt}\cr }

7.2 Electromagnetic Induction

7.2.1 Faraday's Law

A changing magnetic field induces an electric field.

\displaylines{\oint \bar E\cdot d\bar l=-\int {\partial \bar B\over \partial t}\cdot d\bar a\cr \bar \nabla \times \bar E=-{\partial \bar B\over \partial t}\cr }

7.2.2 Energy in Magnetic Fields

\displaylines{W={1\over 2}L I^2\cr W={1\over 2\mu _0}\int _{all^ace}B^2 d\tau \cr }

7.3 Maxwell's Equations

7.3.1 Fixing Ampère's Law

\displaylines{\bar \nabla \times \bar B=\mu _0\bar J+\mu _0\epsilon _0{\partial \bar E\over \partial t}}

Integral form:

\displaylines{\oint \bar B\cdot d\bar l=\mu _0I_{enc}+\mu _0\epsilon _0\int \left ({\partial \bar E\over \partial t}\right )\cdot d\bar a}

A changing electric field induces a magnetic field.

Displacement current:

\displaylines{\bar J_d\equiv \epsilon _0{\partial E\over \partial t}}


7.3.2 Maxwell's Equations

\displaylines{\bar \nabla \cdot \bar E={1\over \epsilon _0}\rho \cr \bar \nabla \cdot \bar B=0\cr \bar \nabla \times \bar E=-{\partial \bar B\over \partial t}\cr \bar \nabla \times \bar B=\mu _0\bar J+\mu _0\epsilon _0{\partial \bar E\over \partial t}\cr }

7.3.3 Maxwell's Equations in Matter

\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 }

7.3.4 Boundary Conditions

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 }

In linear media with no free charge of 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 }

NicomDoc - 28-Jun-2007 - niclasen@fys.ku.dk