Notebook

Electromagnetic Fields

Chapter 3: Electric Flux

Flux

Flux, also known as flow, describes the quantity which passes through a surface or substance. In electromagnetism, flux is defined as the surface integral of the vector field components which are perpendicular to the surface at each point.

Electric Flux

In the case of electric flux, the flux $\psi$ depends on the number and the magnitude of electric field lines $\vec{E}$ going perpendicularly through a normally perpendicular surface and the surface area $\vec{S}$. In other words, electric flux is 'tubes' of electric field. $\psi = \vec{E}.\vec{S}=E.S.cos(\theta)$. To generalize for any surface, $\psi = \iint \vec{E}.d\vec{S}$ (Note that it is a double integral because it is an area, not a line).

Flow

Gauss'Law

For a closed system with a symmetrical Gaussian surface, $\psi = \oint_S \vec{E}.d\vec{S} = \cfrac{q}{4.\pi.r^2.\varepsilon}{4.\pi.r^2}=\cfrac{q}{\varepsilon}$. This can be interpreted as: The total flux over a given Gaussian surface is equal to the total charge inside divided by the medium permittivity.

A Gaussian surface is a symmetrical closed surface such as cyllinder or sphere.

Electric Flux Density is a measure of the electric field independent of the material. It is defined as $D=\varepsilon.E=\cfrac{q}{4.\pi.r^2}=\cfrac{q}{A}$

Using this relationship, a further relation can be made - $\int_S \vec{D}.{d\vec{S}}=q_{enc}$. The law is summarized with the expression - The electric flux passing through any closed surface is equal to the total charge enclosed

$C=q/V = (\int D dS)/V = \cfrac{\int D dS}{\int E dl} = \varepsilon . \cfrac{\int dS}{\int dl}$

For a Gaussian surface, the capacitance can easily be calculated. For example, for a parallel plate capacitor with plate distance $d$ and plate area $A$, $C=\varepsilon. A/d$

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