Electrostatics

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This is a topic from Higher Physics 1B

Contents

Introduction

Electrostatics is the study of stationary electric charges and fields (as opposed to moving charges and currents)

Properties of Electric Charges

The electromagnetic force between charged particles is a fundamental phenomenon within nature, and the charges that generate these forces have the following properties:

  • The symbol for charge is q or Q
  • Charges come in two varieties, positive and negative
  • The unit of electric charge is the Coulomb, denoted the symbol C
  • Electric charge is quantized into ‘packets’ of energy equal to the fundamental unit of charge, denoted the symbol 'e' where e = 1.6 x 10^-19 C as follows:
Q = ne
Where n is a positive integer (natural number)
  • Charge is quantized because there exist particles which carry the smallest magnitude of charge (the fundamental unit of charge) such that all quantities of charge must be a multiple of this unit
  • Those particles are protons and electrons which carry a charge of ‘+e’ and ‘-e’ respectively
  • The ratio of protons to electrons present at a given point within a material determines whether that point has a positive, negative or neutral charge

Conductors and Insulators

  • Conductors are materials in which some electrons are ‘free electrons’, meaning that some electrons are not bound to the atoms but can move relatively freely through the material. When a conductor is charged in one area the charge distributes itself throughout the whole conductor
  • Insulators on the other hand have the opposite properties, whereby the electrons move with great difficulty through the material, such that if one region is charged the charge remains isolated to that region. This means that the surface of an insulator can be charged whole its interior can remain unaffected

Coulomb’s Law

Definition

This law was discovered experimentally (not mathematically derived) by Charles-Augustin de Coulomb by measuring the magnitudes of electric forces between charged spheres in a controlled environment. It states that the force was proportional to the product of the charges and inversely proportional to the square of the distance between the spheres’ centres of mass, as follows:

Fe = keq1q2 / r2

Where Fe represents force in Newtons (N)

ke represents Coulomb’s constant, equal to 8.9875 x 109 in Newton-metres-squared-per-coulomb-squared (Nm2C-2)
q1 and q2 represent the two charges involved in Coulombs (C)
r represents the distance between the centres of mass of the two charged particles in metres (m)

Note the similar structure of this equation to Newton’s Law of Gravitation covered in Physics 1A. The two masses are replaced by two charges, and of course the constant is different. This similarity may assist in conceptualizing and memorizing this formula

Permittivity of Free Space

Coulomb’s constant is related to another useful constant, εo, known as the Permittivity of Free Space by the following equation:

εo = 1/4πke

Where εo = 8.8542 x 10-12 C2N-1m-2

Notes for Application

Given that forces are vector quantities (as opposed to scalar quantities) the direction must be considered when using the equation:

  • With similar signs for the charges the product q1q2 is positive and the force is positive (repulsive)
  • With opposite signs for the charges the product q1q2 is negative and the force is negative (attractive)
  • The resultant force of various electrical forces is equal to the vector sum of those forces
  • Electrical forces obey Newtons Third Law such that F12 = - F21

Electric Fields

Definition

An electric field is said to exist in the region of space around a charged object, known as the source charge. When another charged object, known as the test charge, enters the field it undergoes a force. Thus an electric field is defined as the electric force on the test charge per unit charge:

E = Fe / qo → Fe = Eqo

Where E represents the electric field (NC-1)

qo represents the test charge

The electric field vector at a point in space is defined as the electric force acting upon a positive test charge - the fact that the test charge is assumed positive will be important when it comes to drawing and interpreting field lines in exam and homework questions

Charge Density

A charge can be distributed in one, two or three dimensions - on a line, over an area or throughout a volume respectively.

  • If the total charge ‘q’ is distributed evenly over a line of length ‘l’ then the charge density ‘λ’ is defined by:
λ = q/l
  • If the total charge ‘q’ is distributed evenly over a surface of area ‘A’ then the charge density ‘σ’ is defined by:
σ = q/A
  • If the total charge ‘q’ is distributed evenly throughout an object of volume ‘V’ then the charge density ‘ρ’ is defined by:
ρ = q/V

Deriving the Electric Field of a Uniformly Charged Object

  • Recalling that the charge of an object results from a surplus of protons or neutrons throughout the object or on its surface, an object’s electric field is really the resultant vector sum of the electric fields of its subatomic particles
  • It is impractical to consider electric fields on such a microscopic scale however, and so in order to solve problems involving charged objects on a more reasonable scale we sum the electric fields due to small charges, Δq
  • This approach is only valid for objects of uniform charge, as in the following example involving a rod thin enough considered to be a line in one dimension:

DerivationElectrostatics1.gif

  • A charge Δq at a point on the rod in the figure above exerts a force on a test charge qo as follows:
Fe = keΔqqo / r2 (Coulomb’s Law)
  • Substituting into the definition of an electric field gives:
E = Fe/qo
= keΔq / r2
  • Summing the electric fields caused by these small charges Δq can gives the electric field of the whole rod:
E ≈ ke Σi Δqi / r2
  • Taking the limit as the size of these charges tends towards zero gives an exact solution:
Screen Shot 2012-07-30 at 5.14.34 PM.png
  • The next step in calculating the electric field at a distance ‘b’ from the rod along its axis is to find an expression for dq in terms of distance. This is done using the equations for charge density, in this case for a charge distributed evenly on a line:
λ = q/L
q = λL
dq = λdL (remember that λ is a constant, but distance (x or L) is a variable)
Screen Shot 2012-07-30 at 5.24.10 PM.png
  • If one wanted to check the validity of this result, they need only consider what would happen as the rod’s length L got shorter and shorter until it approached zero. Then the derived result above would give the equation for the electric field about a point particle, which is what the rod would have become

The Field Within a Charged Object

For a closed surface of uniform charge density throughout its volume the electric field is zero at the centre, increasing in a linear fashion until it reaches a maximum at the objects surface, after which is decays by the inverse square law. The reason the electric field is zero is that the forces resulting from the surrounding charges cancel each other out, such that the net force is zero. As the test charge moves away from the centre towards the surface there is more charge on one side than the other, and so there is a net electric field. This is expressed in the figure below: Screen Shot 2012-07-30 at 5.33.40 PM.png

Electric Field Lines

Electric field lines are a pictorial tool used to express the relative direction and strength of the electric field at a given point.

  • The trend of the line expresses the direction
  • The concentration of lines in a given area expresses the strength
  • No two electric field lines can cross one another

Some illustrative and common examples of field lines are below:

Plates.jpeg

Imgres-1.jpeg

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