The net flux is the sum of the infinitesimal flux elements over the entire surface. It is positive when the angle between E → i E → i and n ^ n ^ is less than 90 ° 90 ° and negative when the angle is greater than 90 ° 90 °. Then the flux d Φ d Φ through an area dA is given by d Φ = E → Since the elements are infinitesimal, they may be assumed to be planar, and E → i E → i may be taken as constant over any element. In the limit of infinitesimally small patches, they may be considered to have area dA and unit normal n ^ n ^. However, when you use smaller patches, you need more of them to cover the same surface. This estimate of the flux gets better as we decrease the size of the patches. This is similar to the way we treat the surface of Earth as locally flat, even though we know that globally, it is approximately spherical. If we divide a surface S into small patches, then we notice that, as the patches become smaller, they can be approximated by flat surfaces. In general, when field lines leave (or “flow out of”) a closed surface, Φ Φ is positive when they enter (or “flow into”) the surface, Φ Φ is negative.Īny smooth, non-flat surface can be replaced by a collection of tiny, approximately flat surfaces, as shown in Figure 6.8. Therefore, quite generally, electric flux through a closed surface is zero if there are no sources of electric field, whether positive or negative charges, inside the enclosed volume. Therefore, if any electric field line enters the volume of the box, it must also exit somewhere on the surface because there is no charge inside for the lines to land on. The reason is that the sources of the electric field are outside the box. The magnitude of the flux through rectangle BCKF is equal to the magnitudes of the flux through both the top and bottom faces. Here, the net flux through the cube is equal to zero. The net electric flux through the cube is the sum of fluxes through the six faces. The electric flux through the other faces is zero, since the electric field is perpendicular to the normal vectors of those faces. The electric flux through the top face ( FGHK) is positive, because the electric field and the normal are in the same direction. Electric flux through the bottom face ( ABCD) is negative, because E → E → is in the opposite direction to the normal to the surface.
Notice that N ∝ E A 1 N ∝ E A 1 may also be written as N ∝ Φ N ∝ Φ, demonstrating that electric flux is a measure of the number of field lines crossing a surface.įigure 6.7 Electric flux through a cube, placed between two charged plates. Electric flux is a scalar quantity and has an SI unit of newton-meters squared per coulomb ( N We represent the electric flux through an open surface like S 1 S 1 by the symbol Φ Φ. The quantity E A 1 E A 1 is the electric flux through S 1 S 1. If N field lines pass through S 1 S 1, then we know from the definition of electric field lines ( Electric Charges and Fields) that N / A 1 ∝ E, N / A 1 ∝ E, or N ∝ E A 1. To quantify this idea, Figure 6.4(a) shows a planar surface S 1 S 1 of area A 1 A 1 that is perpendicular to the uniform electric field E → = E y ^. Again, flux is a general concept we can also use it to describe the amount of sunlight hitting a solar panel or the amount of energy a telescope receives from a distant star, for example. Similarly, the amount of flow through the hoop depends on the strength of the current and the size of the hoop. As you change the angle of the hoop relative to the direction of the current, more or less of the flow will go through the hoop. The numerical value of the electric flux depends on the magnitudes of the electric field and the area, as well as the relative orientation of the area with respect to the direction of the electric field.Ī macroscopic analogy that might help you imagine this is to put a hula hoop in a flowing river. Figure 6.3 The flux of an electric field through the shaded area captures information about the “number” of electric field lines passing through the area.
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