Electromagnetism

Ampere's law can be proven with Biot Savart Law. It has an addition that doesn't mean much when the electric field is produced by a static charge.

Coulomb's law describes that charges produce electric fields, influencing charges

Integrating that electric field over a loop equals zero.

Biot Svart's law describes that moving charges produce magnetic fields, influencing moving charges.

There is a problem! Somehow, changing magnetic fields causes electrons to feel a force! It's something we haven't talked about: Faraday's Induction

Faraday's Induction

Faraday's law states that changing magnetic field produces a circulating electric field.

$\varepsilon =\oint _{C}{\vec {E}}_{n}\cdot d{\vec {l}}=-{\frac {d\Phi _{B}}{dt}}$

It's not really surprising. Magnetic field is produced from a current, it makes sense that some quantity produces an electric field!

Motion EMF

Faraday's law also works when magnetic field is fixed but the loop is changing

$\oint _{C}{\vec {v}}\times {\vec {B}}d{\vec {l}}=-{\frac {d}{dt}}\iint _{S}{\vec {B}}\cdot d{\vec {s}}$

The proof comes from

$\oiint _{S}{\vec {B}}\cdot d{\vec {s}}=0$

Negative Sign

It makes sense: if the produced electric field is in the positive direction, it will create a positive feedback loop that makes more and more magnetic and electric fields.

Maxwell's Equation

1. Integration of total electric field over closed surface is charge over permittivity. There is no modification since the induced electric field is circulating
2. Integration of total electric field over closed loop is opposite of magnetic flux change
3. An addendum is added to Ampere's Law. This is Maxwell's prediction, confirmed by Hertz.
4. One can calculate force on a charge depending on its total electric field, magnetic field, and velocity

Parallels

A line current produces a magnetic field just like a line magnetic flux produces an electric field. A circular current creates a magnetic field just like a magnetic flux loop produces an electric field!

Solenoids

The magnetic field on the axis and inside the solenoid is pretty much the same.