Maxwell's Equations Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory treatment of the subject, except perhaps as summary relationships. These basic equations of electricity and magnetism can be used as a starting point for advanced courses, but are usually first encountered as unifying equations after the study of electrical and magnetic phenomena. 1. Gauss' Law for Electricity The electric flux out of any closed surface is proportional to the total charge enclosed within the surface. The integral form of Gauss' Law finds application in calculating electric fields around charged objects. Integral form: (Both E and dA are vectors) ∮E·dA = q/ε₀ = 4πkq Differential form ▽·E = ρ/ε₀ = 4πkρ 2. Gauss' Law for Magnetism The net magnetic flux out of any closed surface is zero. This amounts to a statement about the sources of magnetic field. For a magnetic dipole, any closed surface the magnetic flux directed inward toward the south pole will equal the flux outward from the north pole. The net flux will always be zero for dipole sources. If there were a magnetic monopole source, this would give a non-zero area integral. The divergence of a vector field is proportional to the point source density, so the form of Gauss' law for magnetic fields is then a statement that there are no magnetic monopoles. Integral form: (Both B and dA are vectors) ∮B·dA = 0 Differential form ▽·B = 0 3. Faraday's Law of Induction The line integral of the electric field around a closed loop is equal to the negative of the rate of change of the magnetic flux through the area enclosed by the loop. This line integral is equal to the generated voltage or emf in the loop, so Faraday's law is the basis for electric generators. It also forms the basis for inductors and transformers. Integral form: (Both E and ds are vectors) ∮E·ds = – dΦʙ/dt Differential form ▽×B = –∂B/∂t 4. Ampere's Law In the case of static electric field, the line integral of the magnetic field around a closed loop is proportional to the electric current flowing through the loop. This is useful for the calculation of magnetic field for simple geometries. Integral form: (E, ds, dA are vectors) ∮B·ds = (μ₀i) + (1/c²)(d/dt)(∫ E·dA) Differential form ▽×B = (4πkJ/c²) + (1/c²)(∂E/∂t) ▽×B = (J/ε₀c²) + (1/c²)(∂E/∂t) Symbols Used: E = Electric field (electric field intensity) in volt per meter   or newton pe coulomb B = Magnetic field (Magnetic field density) in Tesla or Weber per square meter D = Electric displacement field (electrid flux density) in coulombs per squre meter or   Newton per volt-meter H = Magnetic field strength in amps per meter ▽· = the divergence operator, units are per meter ▽× = the curl operator, units are per meter ∂/∂t = partial derivative with respect to time, units are per second dA = differential vector element of surface area A, with infinitesimally   small magnitude and direction normal to surface S. Units are square meter ds = differential vector element of path length tangential to the path/curve   unit is meters ε₀ = permittivity of free space, in Farads per meter μ₀ = permeability of free space, in henries per meter ρ = total charge density in coulombs per cubic meter Φʙ = Magnetic flux through surface B, in Webers or volt-seconds i = electric current in amps J = current density in amps per square meter c = speed of light in m/s k = 1/(4πε₀) c = 1/√(μ₀ε₀) Home Area, Volume Atomic Mass Black Body Radiation Boolean Algebra Calculus Capacitor Center of Mass Carnot Cycle Charge Chemistry   Elements   Reactions Circuits Complex numbers Constants Curves, lines deciBell Density Electronics Elements Flow in fluids Fourier's Law Gases Gravitation Greek Alphabet Horizon Distance Interest Magnetics Math   Trig Math, complex Maxwell's Eq's Motion Newton's Laws Octal/Hex Codes Orbital Mechanics Particles Parts, Analog IC   Digital IC   Discrete Pendulum Planets Pressure Prime Numbers Questions Radiation Refraction Relativistic Motion Resistance, Resistivity Rotation Series SI (metric) prefixes Skin Effect Specific Heat Springs Stellar magnitude Thermal Thermal Conductivity Thermal Expansion Thermodynamics Trigonometry Units, Conversions Vectors Volume, Area Water Wave Motion Wire, Cu   Al   metric Young's Modulus