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Unformatted text preview: Lesson 4.5 Maxwell’s Equations and Electromagnetic Waves Unit current and unit charge If two parallel wires separated by a distance r carry currents I 1 and I 2 respectively, the interaction between the magnetic fields generated by these currents will cause a force between these wires given by: 1 2 2 4 o I I F r μ π = ⋅ μ , the permeability of free space is defined to be exactly 4 π ×10-7 TmA-1 (Tesla meter per ampere) This gives 4 o μ π an exact value of 10-7 . This was given by Ampere and is used to define the unit of current the ampere (A). One ampere is that current in a long straight wire which exerts a magnetic force of 2 × 10-7 newton per meter of wire on a parallel wire one meter away carrying the same current. This definition of a unit of current also lead us to the definition of a unit of charge. Since current is the rate of flow of charge, the unit of charge must be the amount carried past a fixed point in unit time by unit current. Therefore, our unit of charge -- the coulomb-- is defined by stating that a one amp current in a wire carries one coulomb per second past a fixed point. Maxwell’s Equations We have seen from Gauss’s law that the net electric flux leaving or entering a closed surface enclosing a charge Q is given by: o Q E dA ε ⋅ = ∫ Here ε o is the permittivity of free space and it has a value 8.85 × 10-12 C 2 .N-1 .m-2 This is Maxwell’s first equation. Maxwell’s second equation is a similar statement for the magnetic field. Since there are no free magnetic monopoles, a given space cannot enclose any magnetic monopoles. The amount of magnetic flux entering a closed surface must be equal to the flux leaving it. Therefore the net flux out of the enclosed volume is zero leading to the equation: B dA ⋅ = ∫ The first two Maxwell’s equations, given above, are for integrals of the electric and magnetic fields over closed surfaces enclosing a volume. The third and fourth of Maxwell’s equations are for integrals of electric and magnetic fields around closed curves (taking the component of the field pointing along the curve and calculating the line integral). These line integrals represent the work that would be needed to take a charge around a closed curve in an electric field, and a magnetic monopole (if one existed!) around a closed curve in a magnetic field. The simplest version of Maxwell’s third equation is the electrostatic case: The path integral E dl ⋅ = ∫ However, according to Faraday’s Law of Induction, if a closed circuit has a changing magnetic flux through it, a circulating current will arise, which means that there should be a nonzero voltage around the circuit. The induced voltage is directly proportional to the rate of change of magnetic flux, and its direction is such that it opposes the changing magnetic flux. Using this Maxwell’s third law can be amended as: B d d E dl B dA dt dt Φ ⋅ = - ⋅ = - ∫ ∫ Here the area integrated over on the right hand side spans the path (or circuit) on the left hand...
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