![]() ![]() This is an important example in chemistry because it is assumed that the charge distribution in atoms is spherical (the central field approximation), and it is this assumption that permits us to talk about s, p, d orbitals etc., which are only strictly applicable to the H-atom.īecause of the spherical symmetry the field is directed radially outwards and is constant on any sphere with the same centre as the charge distribution. (if the charge density per unit volume is ρ( r) then the amount of charge sandwiched between r and r + d r is 4π r 2 ρ( r) d r). From Gauss's lawĪs if the charge were concentrated at the origin.Ĭharge q is spread out with a spherically symmetrical density ρ( r) such that ![]() To calculate the field at r a, construct a larger sphere enclosing the shell (black). ![]() By symmetry the field is perpendicular to the shell at every point, so the electric flux on any concentric sphere is EA. The surface area of the shell is 4π a 2 and the charge density is therefore q/4π a 2. The field is constant and perpendicular to the surfaceĬharge q is spread out uniformly over the surface of a spherical shell of radius a (red). You do not need to worry about the details of this, because the only examples we are interested in are those where either (i) If it is zero, the field is parallel to the surface. If the integrand is positive the field is pointing outwards, and if it is negative the field is pointing inwards. The integrand is a scalar product representing the component of the electric field perpendicular to the surface. This denotes an integral over the closed surface S, for example a spherical shell of radius r.Į denotes the vectorial electric field, and n is a unit vector pointing out from the enclosing surface and perpendicular to it at each point. In this law each part can be interpreted very simply, even though it is entirely unfamiliar to chemistry undergraduates. Gauss was one of the greatest mathematicians of the 19th century, and a proper analysis would use all the tools of vector calculus. The mathematics behind this are beyond the scope of this undergraduate course. It applies to any charge distribution and any closed surface. Gauss's law states that the electric flux through any closed surface is equal to the charge enclosed by the permittivity. This result is an example of Gauss's law, which is a much more general law. This quantity is called the electric flux, and denoted Φ E depends only on the charge enclosed by the sphere. The field times the surface area of the sphere is The field lines emanate radially from the charge and so are perpendicular to the sphere. The section on Gauss's law is more advanced.Ĭonsider a spherical surface (blue) with a point charge at its centre (red). Notably the surface area of a sphere, cylinder etc. Level 1 (gold) - this material has some prerequisites that are covered in the first year mathematics for chemists course. ![]()
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