Electromagnetic Field Equation 1: UNIZOR.COM - Physics4Teens - Waves - Field Waves
Channel: Zor Shekhtman on UNIZOR Education
Date: 10/08/2022

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com

Electromagnetic Field Equations 1

This and a few subsequent lectures are about certain laws related to properties of electromagnetic field expressed in a format of equations.
These laws, in their general mathematical form, were formulated in 19th century by a Scottish mathematician and scientist James Maxwell. We will concentrate on their meaning and mathematical expression in some simple cases.

We assume that the concepts of electric and magnetic fields are known by now from the "Electromagnetism" part of this course.

One of the concepts introduced there was the electric field intensity E - a vector defined at each point of space where electric field exists, that is a vector field, acting as a force on any electric charge proportional to its amount of charge.

In particular, if the electric field is produced by a point charge q0 located at point O, all vectors of electric field intensity are radial, directed towards or away from point O depending on its charge (positive or negative). Then on the distance r from O at point P the magnitude of the electric field intensity equals to E=k·q0 /r², it's directed from O to P for positive charge q0 or from P to O for negative q0.
In this formula the constant k is often expressed as k=1/(4πε), where ε is the permittivity of the media between points O and P with ε0 used for permittivity of vacuum and εr=ε/ε0 defined as relative permittivity of the media.

In general, an electric field intensity E (a vector) produces a force Fe (also a vector) on any charge q:
Fe = q·E

The Gauss Law - Electric

At any point of the electric field, where its intensity vector E is defined, we can position an infinitesimal flat piece of surface σ and calculate the electric field flux ΦE(σ) flowing through it.

This flux is defined as a scalar product of two vectors - the vector of electric field intensity E and a normal vector A to a piece of a surface σ proportional in its magnitude to an area of this piece of surface and directed perpendicularly to it:
ΦE(σ) = E·A
where |A| = area(σ) and A⊥σ.

If electric field intensity vector is already perpendicular to a piece of a surface, the flux is just a product of magnitude of this vector and an area of a piece of a surface.

Consider a simple case of a point charge q. Space around it constitutes the electric field and, as mentioned above, the vector of electric field intensity E is radial and its magnitude on a distance r equals to E=k·q/r² where constant k=1/(4πε).

Consider a sphere of radius r with a center at this point charge. Since the magnitude of a vector of electric field intensity is constant on this sphere and the direction of this vector is always radial (which implies that it's perpendicular to a sphere's surface), the total electric field flux through a sphere can be expressed as a product of the magnitude of the electric field intensity by an area of a sphere:
ΦE=[1/(4πε)]·(q/r²)·4πr²=q/ε

We have received a wonderfully simple formula that states that the total flux of electric field through a sphere does not depend on a radius of this sphere, it depends only on amount of charge inside this sphere and the permittivity of a media.

Remarkably, the same result we would be able to obtain, if the surface around a point charge is not exactly spherical, but is any closed surface.
The mathematics of it are a little more involved, and we just state this as a fact.

Moreover, the charge inside a closed surface does not have to be a point charge. Any distribution of charge inside a closed surface will produce the same result - the total flux through a surface is the total amount of charge divided by permittivity of media. It's easy to understand because multiple fields have a property of superposition - total intensity is a vector sum of individual component intensities.

The above is the Gauss Law for electric field.

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