Flux (meaning flow in Latin), is a concept used to conceptualise Electric Fields as a pseudo-fluid, in the sense that it ‘flows’ into negative charges and out of positive charges. Hence, we call it electric flux.

Definition

The electric flux at a surface, , can be defined as the Dot Product of the electric field and the area perpendicular to the field. It is represented by capital Phi:

Formula - 1

Electric Flux on an Area

\Phi_{E} = \vec{E} \cdot \vec{A} = EA\cos(\theta)

##### Formula - 2 >[!formula] Electric Flux on a Surface > >$$ \Phi_{E} = \oint \vec{E} \cdot \text{d}{\vec{A}}
Terms
  • = Electrical flux over the surface, in or
  • = The electric field influencing , in ()
  • = The area vector#todo of . It is normal to the surface and has a magnitude equal to the area of
  • = The angle between the and
  • represents the Surface Integral of the surface , and is a way of ensuring that we can calculate the flux on even if it’s curved, as long as it is a closed surface.#todo

Implications

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Note that due to the nature of the Dot Product, is a maximum when i.e. and a minimum when i.e. when

Basically, if flux is greatest when the surface ‘cuts through’ the field, and zero when it’s passing through.

On complex surfaces

If the surface is more complex than a simple shape, we can use an infinitesimal parts of (i.e. ) to obtain the flux:

#tosee Vector Calculus? This in turn can be integrated:

The integral itself may look scary, unless we have one of these two scenarios:

  • The electric field is parallel to the entire surface. In this case, the net flux is always 0. It does not matter if is a non-uniform field!
  • The electric field is perpendicular to the entire surface and it is uniform everywhere. In this case, the net flux is always

Both these useful scenarios are abused by using Gauss’ Law

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Through a neutral object

In a 3D object that is neutral (i.e. does not hold any charge of it’s own), the net flux is always zero. For example, take the cube below, with area vectors

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The same principle applies to any volume.

Through a charged object

See Gauss’ Law