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I find it difficult to understand the concept of flux and field.I have searched the internet and refered a few books.From what I understand, generally flux can be defined as "some quantity per unit of some other quantity" E.g: "the force of attraction at a point in space per unit charge placed at that point is called electric flux" , "the rate of heat transfer per unit area is called heat flux in conduction". Please suggest whether I am right or wrong.

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    $\begingroup$ Do you understand the mathematics concept of a vector field? $\endgroup$
    – joojaa
    Nov 9, 2018 at 5:19
  • $\begingroup$ @ joojaa yes I can visualise a vector field (simple vector field). $\endgroup$
    – user17332
    Nov 9, 2018 at 14:31

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From what I understand, generally flux can be defined as "some quantity per unit of some other quantity".

That's about right. The some other quantity is often one of particular interest Typically, it is a surface, or a facet of a surface, used in control volume analysis. And there is a flux operator. It returns a scalar quantity from an vector operation. In the case of fluids, the fluid advects the quantity of interest. So you have a description of the density of some thing of interest at time t (a scalar field) and a velocity field of the fluid (a vector field), and a reference surface S. The Flux operator tallies the rate at which stuff is advecting through the surface.

Of course, when dealing with pure field equations as in Electromagnetism, there is no need for an advective ether. Flux pertains to the vector field itself with respect to some surface of interest.

We often have a system of equations that all involve this same control volume, and which have multiple conservation laws, which relate to a potential field. The flux idea is very useful in writing these equations in a compact form. For example, the equation below can represent a system of three conservation equations. $F$ being the flux function.

$$\frac{d}{dt} \iint_{\Omega}\mathbf q\ dx\ dy\ + \int_{\partial \Omega} F(\mathbf q)\cdot\mathbf n\ ds = \iint_{\Omega} {\psi}\ dx\ dy $$

The equations would be conservation of mass, conservation of x momentum, and conservation of y momentum.

See page 4 of this document to see this in action. - Numerical Approximation of the Nonlinear Shallow Water Equations with Topography and Dry Beds: A Godunov-Type Scheme, by David L. George

There is a nice series of three videos from the Kahn Academy that deals with the flux operator. They play in sequence. The first one is here.

(There should be a badge for learning mathjax. That formula took me 45 minutes.)

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  • $\begingroup$ AFAIK, the control volume equation is an integral over volume and area. For example, energy is $\frac{\partial}{\partial t} \iiint_V\ \rho\ \tilde{e}\ dV + \iint_S \rho\ (\tilde{e} + (p/\rho))\ \vec{v}\bullet\vec{n}\ dA = \dot{q} - \dot{w}$. LaTeX is easier IMO. $\endgroup$ Nov 25, 2018 at 16:58
  • $\begingroup$ Same equation. In the example I gave, nothing varies with z, so it was integrated out in the form of a constant. Your second integral is the flux term. It has a scalar times a velocity vector field dotted to a surface normal (unit) vector. Qdot and Wdot are source terms, so this is the "balance" version of the conservation equation. $\endgroup$
    – Phil Sweet
    Nov 25, 2018 at 17:20
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The two disciplines use the term flux in different ways.

Electricity

Field is the distribution of a potential (NOT potential energy) along a line. It has units V/m. It is alternatively the distribution of a force per unit charge. It has units J / C m = N m / C m = N / C. Multiply this latter by charge and you see how you obtain the force on a charge at any point in an electric field.

Flux is the net amount of the electrical field that is penetrating perpendicularly to a unit area. It has the units V m = N m$^2$ / C.

Mass, Momentum, or Heat Transfer

Flux has units that can be considered from one of two perspectives.

Flux can be written as the amount of mass, momentum, or heat moving through a certain area in a certain time. The units are (amount / m$^2$ s). So, specifically for each, you find

mass: kg / m$^2$ s ... momentum: (kg/m s) / m$^2$ s ... heat: J / m$^2$ s

Alternatively, flux can be written as the concentration of mass, momentum, or heat moving at a certain velocity. The units are (amount / m$^3$)(m / s). The translations follow from above.

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