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Φ = E · n ^ 2 A 2 .

Check out this video to observe what happens to the flux as the area changes in size and angle, or the electric field changes in strength.

Area vector

For discussing the flux of a vector field, it is helpful to introduce an area vector A . This allows us to write the last equation in a more compact form. What should the magnitude of the area vector be? What should the direction of the area vector be? What are the implications of how you answer the previous question?

The area vector    of a flat surface of area A has the following magnitude and direction:

  • Magnitude is equal to area ( A )
  • Direction is along the normal to the surface ( n ^ ); that is, perpendicular to the surface.

Since the normal to a flat surface can point in either direction from the surface, the direction of the area vector of an open surface needs to be chosen, as shown in [link] .

Figure shows two horizontal planes labeled A. The first has two arrows pointing up from the plane. The longer is labeled vector A and the shorter is labeled n hat. The second plane has the same two arrows pointing down from the plane.
The direction of the area vector of an open surface needs to be chosen; it could be either of the two cases displayed here. The area vector of a part of a closed surface is defined to point from the inside of the closed space to the outside. This rule gives a unique direction.

Since n ^ is a unit normal to a surface, it has two possible directions at every point on that surface ( [link] (a)). For an open surface, we can use either direction, as long as we are consistent over the entire surface. Part (c) of the figure shows several cases.

Figure a shows a curved rectangular surface. Two arrows originate from a point at its center and point in opposite directions. They are both perpendicular to the surface. They are labeled n hat 1 and n hat 2. Figure b shows a 3 dimensional surface shaped somewhat like a light bulb. There are five arrows labeled n hat, which originate from various points on the surface and point outward, perpendicular to the surface. Figure c shows three rectangular surfaces labeled S1, S2 and S3. Two arrows labeled n hat are perpendicular to S1 and point in opposite directions. Three arrows labeled n hat are perpendicular to S2, one pointing in a direction opposite to the other two. There are three arrows perpendicular to S3. All point outward from the same side of the surface.
(a) Two potential normal vectors arise at every point on a surface. (b) The outward normal is used to calculate the flux through a closed surface. (c) Only S 3 has been given a consistent set of normal vectors that allows us to define the flux through the surface.

However, if a surface is closed, then the surface encloses a volume. In that case, the direction of the normal vector at any point on the surface points from the inside to the outside. On a closed surface such as that of [link] (b), n ^ is chosen to be the outward normal at every point, to be consistent with the sign convention for electric charge.

Electric flux

Now that we have defined the area vector of a surface, we can define the electric flux of a uniform electric field through a flat area as the scalar product of the electric field and the area vector, as defined in Products of Vectors :

Φ = E · A ( uniform E , flat surface ) .

[link] shows the electric field of an oppositely charged, parallel-plate system and an imaginary box between the plates. The electric field between the plates is uniform and points from the positive plate toward the negative plate. A calculation of the flux of this field through various faces of the box shows that the net flux through the box is zero. Why does the flux cancel out here?

A cube ABCDKFGH is shown in the center. A diagonal plane is shown within it from KF to BC. The top surface of the cube, FGHK has a plane labeled minus q slightly above it and parallel to it. Similarly, another plane is labeled plus q is shown slightly below the bottom surface of the cube, parallel to it. Small red arrows are shown pointing upwards from the bottom plane, pointing up to the bottom surface of the cube, pointing up from the top surface of the cube and pointing up to the top plane. These are labeled vector E.
Electric flux through a cube, placed between two charged plates. Electric flux through the bottom face ( ABCD ) is negative, because E is in the opposite direction to the normal to the surface. The electric flux through the top face ( FGHK ) is positive, because the electric field and the normal are in the same direction. The electric flux through the other faces is zero, since the electric field is perpendicular to the normal vectors of those faces. The net electric flux through the cube is the sum of fluxes through the six faces. Here, the net flux through the cube is equal to zero. The magnitude of the flux through rectangle BCKF is equal to the magnitudes of the flux through both the top and bottom faces.
Practice Key Terms 3

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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