Field lines directed into a closed surface are considered negative; those directed out of a closed surface are positive. If there is no net charge within a closed surface, every field line directed into the surface continues through the interior and is directed outward elsewhere on the surface.

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The relationship between potential and field (E) is a differential: electric field is the gradient of potential (V) in the x direction. This can be represented as: Ex=−dVdx E x = − dV dx . Thus, as the test charge is moved in the x direction, the rate of the its change in potential is the value of the electric field.

Remember our convention for flux orientation: positive means flux is leaving, negative means flux is entering. In this example, water is falling downward, or entering the tube. This means the top surface has negative flux (it appears to be siphoning up water).

Find the flux of the vector field in the negative z direction through the part of the surface z=g(x,y)=16-x^2-y^2 that lies above the xy plane (see the figure below). Hence, the flux through the surface in the downward z direction is -128*pi cubic units per unit time.

The net electric flux is zero through any closed surface surrounding a zero net charge. Therefore, if we know the net flux across a closed surface, then we know the net charge enclosed.

Gauss’s law of electricity, which is included within the fundamental laws of electromagnetism, states that the electric flux through a closed surface with electric charge inside, just depends on the net charge enclosed in the surface and does not depend the shape or size of the surface.