Gauss's Law

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This is a topic from Higher Physics 1B

Contents

Introduction

Gauss’s law is an expression of the general relationship between the net electric flux through a closed surface (Gaussian surface) and the charge enclosed by the surface. In order to understand this law one must first define and examine the term ‘flux’

Electric Flux

In general, ‘flux’ refers to the rate of flow per area, and so electric flux refers to the rate of flow of electric field lines through a given area. Thus electric flux is the product of the magnitude of the electric field and the surface area perpendicular to it, as given by the equation:

ΦE = EA

Where ΦE represents electric flux in Voltmeters (Vm) or Newton-metres-squared-per-coulomb (Nm2C-1)

E represents the electric field (NC-1)
A represents perpendicular area (m2)

Eflux.gif

  • The perpendicular area is given by the product of area and the cosine of the angle between the surface and the normal to the field lines. This means that flux is zero when the surface is parallel to the field lin (as Cos90° = 0)
  • For more complicated and non-uniform surfaces, the net flux is calculated as the sum of the fluxes through small areas of the surface ΔA. The method of summing these fluxes is called surface integration, and uses the slightly different integration symbol:
  • For a closed surface an electric field line leaving the surface causes positive flux, while field lines entering the surface cause negative flux
  • Thus the net flux is proportional to the total number of field lines leaving the object minus the total number of field lines entering the object

Statement and Equation

Gauss’s law states that the net electric flux through a closed surface surrounding an inclosed charge is proportional to the magnitude of the charge, and is given by

ΦE = q / εo and given that
ΦE = E.dA it shows that
E.dA = q / εo

Where ΦE represents electric flux in Voltmeters (Vm) or Newton-metres-squared-per-coulomb (Nm2C-1)

Q represents the inclosed charge (C)
E represents the electric field (NC-1)
A represents perpendicular area (m2)
εo represents the permittivity of free space (8.8542 x 10-12 C2N-1m-2)

Selecting a Gaussian Surface

  • Gauss’s law states that regardless of which hypothetical Gaussian surface is chosen to calculate the flux the result is the same, and so the equation is independent of the shape of the surface provided it is closed
  • For the purpose of making calculations easier though one should pick surfaces where sides are either parallel to the field lines (so flux is zero) or perpendicular the field lines (flux is given by E dA).
  • For a point charge or spherical charged object the ideal surface for calculations is a larger sphere, as every point on its surface is perpendicular to field lines
  • For a line of charge or wire the ideal surface for calculations is a cylinder with the line at its axis such that the two circular faces are parallel to field lines and the curved face is perpendicular to field lines
  • For a charged plane the ideal surface for calculations is a cylinder with its circular faces parallel to the plane such that the two circular faces are perpendicular to field lines and the curved face is parallel to field lines

Derivation

This derivation is included for the purpose of bettering one’s understanding of Gauss’s Law and is not necessarily examinable. Choosing a sphere as the Gaussian surface for the reasons mentioned above,

ΦE = E.dA ....................... (By definition)
= EA ..................................... (Due to the properties of the sphere)
= E(4π r2)
= (keq / r2) (4π r2)
= ke4π q
= q/εo .................................... (By the definition of εo)

Electrostatic Equilibrium

Electrostatic equilibrium is when there is no net motion of charge within a conductor. A conductor in electrostatic equilibrium has the following properties:

  1. The electric field is zero everywhere within the conductor, regardless of whether is it solid or hollow
  2. If the conductor is isolated, any charge it carries resides on the surface
  3. If the conductor is charged, the electric field at a point just outside and perpendicular to its field lines has magnitude σ/εo
  4. If the conductor is irregularly shaped the surface charge density is greatest at locations where the radius of curvature is the smallest (to over-simplify, the ‘pointy bits’) as seen in the figure below
  5. The surface of the conductor has a constant Electric Potential and as such is an equipotent surface

Screen Shot 2012-07-30 at 6.14.54 PM.png

The first two properties have simple justifications which may assist in learning but are not necessarily examinable:

Property 1 – If the internal electric field of a conductor were not zero then the free electrons inside would undergo a force by Coulomb’s law. They would then accelerate, undergoing a net motion, which would mean the conductor is not in electrostatic equilibrium
Property 2 – Imagine a Gaussian surface extremely close to the charged conductor’s surface. There is no net flux through the Gaussian surface, thus there is no enclosed charge. Therefore the charge must reside outside the Gaussian surface, on the surface of the conductor itself

The remaining properties do have justifications which can be found in the Textbook.

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