![]() Indeed, the flux through any surface enclosing a single-point charge is independent of the shape or size of the surface. There we found that the flux was independent of the size of the surface the same result holds true for a spherical surface. The radius of the sphere cancels out of the result for We came to the same conclusion as when we considered rectangular closed surfaces of two different sizes enclosing a point charge. Where is the total surface area of the sphere: The total flux through the sphere is The flux through a surface portion is Hence the integral becomesĪt any point on the sphere of radius r, the electric field has the same magnitude Therefore can be taken outside the integral, which becomes Solution: The surface is not flat, and the electric field is not uniform, so to calculate the electric flux, we must use the integral equation or the general definition,īecause the sphere is centered on the point charge, at any point on the spherical surface, is directed out of the sphere perpendicular to the surface and is, therefore, in the same direction as The positive direction for both n and E is outward. If the area is not planar, then the evaluation of the flux generally requires an area integral considering the angle between and will be continually changing. These statements collectively and qualitatively describe Gauss’s Law, and they hold true for other configurations of charges and closed surfaces of any shape.
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