Ampère’s circuital law and its equivalent forms
– Ampère’s original circuital law states that an electric current creates a magnetic field around it.
– The magnetic field lines encircle the current-carrying wire and lie in a plane perpendicular to the wire.
– Reversing the direction of the current reverses the direction of the magnetic field.
– The strength of the field is directly proportional to the magnitude of the current.
– The original circuital law can be written in integral or differential form.
– It can be expressed using SI units or cgs units.
– The forms can use either the B or H magnetic fields.
– The B and H fields are related by the constitutive equation.
– Other units are possible, but rare.
– The integral form of the original circuital law is a line integral of the magnetic field around a closed curve.
– The curve bounds a surface through which the electric current passes.
– The law relates the circulation of the magnetic field around the path to the current passing through the enclosed surface.
– The line integral can be expressed in terms of the total current or the free current.
– ∇ × is the curl operator.
– There are sign ambiguities associated with the line integral, vector area, and net current passing through the surface.
– These ambiguities are resolved using the right-hand rule.
– There are infinitely many possible surfaces that have the same curve as their border.
– The choice of surface does not matter in the magnetostatic case.
– The most convenient surface is usually chosen for integration.
Free current versus bound current
– Free current refers to the current that passes through a wire or battery.
– Bound current arises in magnetized or polarizable materials.
– Bound current can be due to magnetization or polarization.
– The total current density includes free, magnetization, and polarization currents.
– Bound current is often treated differently from free current for practical reasons.
Issues with the continuity equation for electrical charge
– The divergence of the curl of a vector field must always be zero.
– The original Ampère’s circuital law implies that the current density is solenoidal.
– The continuity equation for electric charge states that the divergence of the current density is equal to the negative time derivative of the charge density.
– In a capacitor circuit, time-varying charge densities exist on the plates, leading to a non-zero divergence of the current density.
– The continuity equation for electric charge is in conflict with the original Ampère’s circuital law.
Issues with the propagation of electromagnetic waves
– The circuital law implies that the magnetic field is irrotational in free space.
– To maintain consistency with the continuity equation for electric charge, the circuital law must include the contribution of displacement current.
– Displacement current is added to the current term in the circuital law to account for the propagation of electromagnetic waves.
– James Clerk Maxwell conceived displacement current as a polarization current in the dielectric material.
– The addition of displacement current allows for the correct prediction of magnetic fields, wave propagation, and conservation of electric charge.
– Displacement current is related to the time rate of change of the electric field.
– In a dielectric, displacement current is also related to the polarization of the individual molecules of the material.
– The displacement current is defined as the time derivative of the electric displacement field.
– The electric displacement field is a combination of the electric constant, relative static permittivity, and polarization density.
– The displacement current has two components: one present everywhere and another associated with the polarization of the dielectric material.
Extending the original law: the Ampère-Maxwell equation and related concepts
– The original circuital law is extended by including the polarization current to improve its applicability.
– The Ampère-Maxwell equation includes the H-field, electric displacement field, and free current density.
– The equation can be expressed in both integral and differential forms.
– The Ampère-Maxwell equation accounts for magnetization current density, conduction current density, and polarization current density.
– The addition of the displacement current resolves the charge continuity issue and allows for wave propagation in free space.
– The formulations of the circuital law in terms of free current and total current are equivalent.
– The equation involving the magnetic field and free current density is equivalent to the equation involving the H-field, electric displacement field, and total current density.
– The proof focuses on the differential forms of the equations.
– The equivalence is sufficient since the differential forms capture the essence of the integral forms.
– The addition of the displacement current in the Ampère-Maxwell equation leads to the correct prediction of electromagnetic wave propagation.
– Related concepts include the Biot-Savart law, displacement current, capacitance, Ampèrian magnetic dipole model, and electromagnetic wave equation. Source: https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law
In classical electromagnetism, Ampère's circuital law (not to be confused with Ampère's force law) relates the circulation of a magnetic field around a closed loop to the electric current passing through the loop.
James Clerk Maxwell (not Ampère) derived it using hydrodynamics in his 1861 published paper "On Physical Lines of Force". In 1865 he generalized the equation to apply to time-varying currents by adding the displacement current term, resulting in the modern form of the law, sometimes called the Ampère–Maxwell law, which is one of Maxwell's equations which form the basis of classical electromagnetism.
