Maxwell’s Equations
Maxwell’s equations for vacuum
- These equations form the foundation of electromagnetic theory.
- These equations in electromagnetism have the same importance as the Newton’s law of motion in mechanics.
- Maxwell’s equations generate the wave equations that predict the existence of electromagnetic waves propagate with the speed of light.
Gauss’s law of electrostatics (Maxwell’s first equation)
- According to Gauss’s law of electrostatics the total electric flux through a closed surface is 1/ε0 times the total charge enclosed by the surface.
- If ρ is volume charge density, then the charge enclosed by the surface is
- Gauss’ law
- From divergence theorem
- This is differential form of Gauss’s law of electrostatics and is also known as Maxwell’s first equation.
Gauss’s law of magnetostatics (Maxwell’s second equation)
- According to it Gauss’s law of magnetostatics the total magnetic flux through a closed surface is always zero.
- If B is magnetic induction, then
- This equation is Maxwell’s second equation.
- Maxwell’s second equation indicates that the magnetic induction B is a solenoidal field.
- The flux leaving the element is same as entering in the field or the source of magnetic induction do not exist.
- No isolated magnetic pole or magnetic monopoles exist.
Faraday law of electromagnetic induction (Maxwell’s third equation)
- According to Faraday law of E.M.I.
- When the magnetic flux linked with the circuit changes an induced e.m.f. is produced.
- The induced e.m.f. is directly proportional to the rate of change of magnetic flux.
- From Stokes’s theorem
- This is differential form of Faraday law of electromagnetic induction and is also known as Maxwell’s third equation.
Modified form of Ampere’s law (Maxwell’s fourth equation)
- According to Ampere’s law the line integral of magnetic induction around any closed path is μ0 times the total current passing through the surface bounding the closed path.
- From Stokes’s theorem
…(1)
…(2)
Therefore from equations (1) and (2), we get
- This is modified form of Ampere’s law and is also known as Maxwell’s fourth equation.
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