Physics Formulae/Electromagnetism Formulae

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Electromagnetism.

Maxwell's Equations
Below is the set in the differential and integral forms, each form is found to be equivalant by use of vector calculus. There are many ways to formulate the laws using scalar/vector potentails, tensors, geometric algebra, and numerous variations using different field vectors for the electric and magnetic fields.

The Field Vectors

Central to electromagnetism are the electric and magnetic field vectors. Often for free space (vacumm) only the familiar E and B fields need to be used; but for matter extra field vectors D, P, H, and M must be used to account for the electric and magnetic dipole incluences throughout the media (see below for mathematical definitions).




 * The magnetic field vectors are related by:
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 * $$\mathbf{B} = \mu_0 \left ( \mathbf{H} + \mathbf{M} \right )\,\!$$
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Interpretation of the Field Vectors

Intuitivley;

— the E and B (electric and magnetic flux densities) fields are the easiest to interpret; field strength is propotional to the amount of flux though cross sections of surface area, i.e. strength as a cross-section density.

— the P and M (electric polarization and magnetization respectivley) fields are related to the net polarization of the dipole moments thoughout the medium, i.e. how well they respond to an external field, and how the orientation of the dipoles can retain (or not) the field they set up in response to the external field.

— the D and H (electric displacement and magnetic intensity field) fields are the least clear to understand physically; they are introduced for convenient thoretical simplifications, but one could imagine they relate to the strength of the field along the flux lines, strength as a linear density along flux lines.

Hypothical Magnetic Monopoles

— As far as is known, there are no magnetic monopoles in nature, though some theories predict they could exist.

— The approach to introduce monopoles in equations is to define a magnetic pole strength, magnetic charge, or monopole charge (all synonomous), treating poles analogously to the electric charges.

— One pole would be north N (numerically positive by convention), the other south S (numerically negative). There are two units which can be used from the SI system for pole strength.

— Pole srength can be quantified into densities, currents and current densities, as electric charge is in the previous table, exactly in the same way.

Maxwell's Equations would become one of the columns in the table below, at least theoretically. Subscripts e are electric charge quantities; subscripts m are magnetic charge quantities.

They are consistent if no magnetic monopoles, since monopole quantities are then zero and the equations reduce to the original form of Maxwell's equations.

Pre-Maxwell Laws
These laws are not fundamental anymore, since they can be derived from Maxwell's Equations. Coulomb's Law can be found from Gauss' Law (electrostatic form) and the Biot-Savart Law can be deduced from Ampere's Law (magnetostatic form). Lenz' Law and Faraday's Law can be incorperated into the Maxwell-Faraday equation. Nonetheless they are still very effective for simple calculations, especially for highly symmetrical problems.

Electric Quantities
Electric Charge and Current

Electric Fields

Electrostatic Fields
Common corolaries from Couloumb's and Gauss' Law (in turn corolaries of Maxwell's Equations) for uniform charge distributions are summarized in the table below.

For non-uniform fields and electric dipole moments, the electrostatic torque and potential energy are:


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 * $$U=\int \left ( \mathbf{E}\cdot\mathrm{d}\mathbf{p}+\mathbf{p}\cdot\mathrm{d}\mathbf{E} \right ) \,\!$$

$$\boldsymbol{\tau}=\int \left ( \mathrm{d}\mathbf{p}\times\mathbf{E}+\mathbf{p}\times\mathrm{d}\mathbf{E} \right )\,\!$$
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Electric Potential and Electric Field


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 * $$ \Delta V = -\int_{r_1}^{r_1} \mathbf{E} \cdot d\mathbf{r}\,\!$$

$$\nabla V = -\mathbf{E}\,\!$$
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Magnetic Forces
Force on a Moving Charge


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 * $$\mathbf{F}= q \mathbf{v} \times \mathbf{B}\,\!$$
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Force on a Current-Carrying Conductor


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 * $$\mathbf{F}= I \mathbf{l} \times \mathbf{B}\,\!$$
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Magnetostatic Fields
Common corolaries from the Biot-Savart Law and Ampere's Law (again corolaries of Maxwell's Equations) for steady (constant) current-carrying configerations are summarized in the table below.

For these types of current configerations, the magnetic field is easily evaluated using the Biot-Savart Law, containing the vector $$\mathrm{d}\mathbf{l}\times\mathbf{r}\,\!$$, which is also the direction of the magnetic field at the point evaluated.

For conveinence in the results below, let $$\mathbf{b} = \mathrm{d}\mathbf{l}\times\mathbf{r}\,\!$$ be a unit binormal vector to $$\mathbf{l}\,\!$$ and $$\mathbf{r}\,\!$$, so that

$$\mathbf{\hat{b}} = \frac{ \mathrm{d}\mathbf{l}\times\mathbf{r} }{ \left | \mathrm{d}\mathbf{l}\times\mathbf{r} \right | } \,\!$$

then $$\mathbf{\hat{b}}\,\!$$ also the unit vector for the direction of the magnetic field at the point evaluated.

For non-uniform fields and magnetic moments, the magnetic potential energy and torque are:


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 * $$U=\int \left ( \mathbf{B}\cdot\mathrm{d}\mathbf{m}+\mathbf{m}\cdot\mathrm{d}\mathbf{B} \right ) \,\!$$

$$\boldsymbol{\tau}=\int \left ( \mathrm{d}\mathbf{m}\times\mathbf{B}+\mathbf{m}\times\mathrm{d}\mathbf{B} \right )\,\!$$
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