Physics · Ch 1 — Electrostatics
Electric Flux
Electric Flux
The number of electric field lines crossing a given area, held perpendicular to the field, is called the electric flux through that area, usually denoted Phi_E. For a flat area A placed in a uniform field E, with the area's outward normal making an angle theta with the field direction, the flux is Phi_E = E A cos(theta) = E . A (treating the area itself as a vector A of magnitude A directed along the outward normal). The SI unit of electric flux is newton metre squared per coulomb (N m^2 C^-1). Because flux involves the dot product of two vectors, it is itself a scalar quantity, but it carries an algebraic sign: it is maximum and positive when the surface directly faces the field (theta = 0, all the field lines pass straight through), decreases as the surface is tilted (multiplied by cos theta), becomes exactly zero when the surface is turned edge-on, parallel to the field (theta = 90 degrees, no field lines cross it at all), and becomes negative if the surface's chosen outward normal points generally against the field direction. For a curved surface sitting in a non-uniform field, the surface must first be divided into a very large number of infinitesimally small, effectively flat area elements dA, each with its own local outward normal; the total flux through the whole surface is the sum (in the limitin …
What this figure shows. A flat area A is drawn inside a region of uniform field lines, with a dashed arrow marked normal to the surface (perpendicular to it) and the angle theta between that normal and the field direction explicitly labelled. The number of field lines threading through the tilted surface is visibly fewer than the number that would pass through the same area held perpendicular to the field, illustrating why the flux formula must include the cos(theta) factor: only the component of area a …
What this figure shows. Three small panels show the same flat surface of area A placed at three different orientations relative to a uniform field E: first held exactly perpendicular to the field lines (theta = 0, every line threads straight through, flux at its maximum value EA), then tilted at some intermediate angle (fewer lines effectively cross, flux reduced by the cos theta factor), and finally turned edge-on so that it lies parallel to the field lines (theta = 90 degrees, no lines cross the surface at all, flux exa …
What this figure shows. An irregularly curved surface is divided into many small patches, each treated as an approximately flat area element dA with its own local outward normal and its own local angle to the (now spatially varying) field E at that patch; the total flux is built up by adding the small contribution E.dA from every patch, in the limit of infinitesimally small patches becoming a surface integral. This generalises the simple flat-surface, uniform-field formula of Figure 1.31 to the fully general case needed before Gauss's law can be applied to any given closed sur …