Electromagnetic Duality

Electromagnetic duality is a property of Biot-Savart law.

Multiple corrections are made to Maxwell's equations.

The line integral of magnetic field is verified to be different from the surface integral of the curl of magnetic field because the magnetic field of Biot-Savart law diverges.

Faraday's induction law is examined by inserting a capacitor into the coil loop to measure the voltage. The electric field inside the capacitor is directly proportional to the time derivative of magnetic flux.

An optional capacitor is also attached to the end of the straight segment of electric wire. The time derivative of the electric field inside the capacitor is verified to be proportional to the line integral of the magnetic field from the electric current.

Biot-Savart law describes magnetic field due to the electric current in a conductive wire.

For a long straight wire, the magnetic field is proportional to (I/r). The curl of magnetic field is proportional to (dI/dr). For a constant current, the curl of magnetic field is zero.

Consequently, the surface integral of the curl of magnetic field is zero but the line integral of the magnetic field is not.

Stokes' theorem can not be applied to the magnetic field vector generated by a constant electric current because the magnetic field is not a differentiable vector.

The divergence theorem states that the surface integral of the flux is equal to the volume integral of the divergence of the flux. This is not true if there is singularity in the volume integral.

One example is the electric field flux described by Coulomb's law. The divergence theorem requires a differentiable vector but Coulomb field is not differentiable at the origin.

Consequently, Gauss's flux theorem is not applicable to the divergence of the electric field.