ELECTRICITY AND MAGNETISM

CoulombÒs Law

The force of attraction or repulsion between two point charges q₁ and q₂ is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.	, where F₁₂ is the force exerted on point charge q₁ by point charge q₂ when they are separated by a distance r₁₂.
The unit vector is directed from q₂ to q₁ along the line between the two charges. Force is in Newtons (N), distance in meters (m) and charge in coulombs (C).
The constant is called the permittivity of free space.
CoulombÒs law and the Principle of Superposition constitute the physical input for electrostatics. The force on any one charge due to a collection of other charges is the vector sum of the forces due to each individual charge. _àà Here F₂=F₂₁+F₂₃+F₂₄
Problem Solving Strategies: 1. Draw a clear diagram of the situation. Be sure to distinguish between the fixed external charges and the charges which the forces must be found. The diagram should contain the coordinate axes for reference. 2. Electric force is a vector quantity; when many forces are present the net force is a vector sum. Example: Consider three point charges q₁ = q₂ = 2.0 mC and q₃ = -3 mC which are placed as shown. Calculate the net force on q₁ and q₃. The force on q₁ is F₁ = F₁₂ + F₁₃. Similarly, F₃ = F₃₁ + F₃₂

Electric Field

Electric field is defined as the electric force per unit charge. The direction of the field is taken to be the direction of the force it would exert on a positive test charge. The electric field is radially outward from a positive charge and radially in toward a negative point charge.

The electric field can be defined by measuring the magnitude and direction of the electric force F on a test charge q₀. A small test charge is used so as not to interfere with the field distribution of the other charges. Thus we define the electric field as

The SI unit of electric field is NC^-1. For a point charge

Field Line Diagrams:

A convenient way to visualize the electric field due to any charge distribution is to draw a field line diagram. At any point the field line has the same direction as the electric field vector. Electric field lines diverge from positive charges and converge into negative charges.

Rules for constructing filed lines:

1. Field lines begin at positive charge and end at negative charge

2. The number of field lines shown diverging from or converging into a point is proportional to the magnitude of the charge.

3. Field lines are spherically symmetric near a point charge

4. If the system has a net charge, the field lines are spherically symmetric at great distances

5. Field lines never cross each other.

The Electric Dipole and their Electric Fields

An electric dipole consists of two charges + q and Öq, of equal magnitude but opposite sign, that are separated by a distance L.

From the diagram, E = E_x i = (E_1x + E_2x)i = 2E_1x i where

Define the product p = qL as the electric dipole moment. We make p = qL a vector by defining L to be directed from Öq to +q. The vector p points from the negative charge to the positive charge. The electric field decreases with r as 1/r³. àFinally
If r >>L, then and	Electric Dipole Field:

Charge Distributions:

The simplest kind of charge distribution is an isolated point charge (i.e., an amount of charge covering such a small region of space that we need not be concerned about its dimensions).

When the finite size of the space occupied by a collection of charges must be considered, it is useful to consider the density of charge. The word density is used is used in three different ways.

(rho)	for the charge per unit volume, the *volume* density	Cm^-3
(sigma)	for the charge per unit area, the *area* density	Cm^-2
(lambda)	for the charge per unit length, the *linear* density	Cm^-1

Find the electric field a distance z above the midpoint of a straight line segment of length 2L which carries a uniform line charge l.

Thus we can write (Model of a transmission line)

It is advantageous to chop the line up into symmetrically placed pairs (at - x).

Here where q is the total charge on the rod. The horizontal components of the two fields cancel. The net field of the pair is

Here , and x runs from 0 to L
How to evaluate this integral
Consider ;	Letà , then àand _.
Substitute these into the integral:

From diagram, à

Hence

Substitute the limits x = 0 and x = L to get the desired answer.

For points far from the line ( z >> L),

this result simplifies to .

The line ÓlooksÔ like a point charge , so the field reduces to that of a point charge

Find the electric field a distance z above one end of a straight line segment of length L, which carries a uniform line charge l.

(Model of a transmission line)

Net electric field E = E_x + E_z



For z >> L and ,
Find the electric field a distance z above the center of a circular loop of radius R, which carries a uniform line charge l. Here, where q is the total charge of the loop Horizontal components cancel, leaving:
But
Find the electric field a distance z above the center of a flat circular disk of radius a, which carries a uniform surface density, . A typical element is a ring of radius r and thickness dr, which has an area . The charge density on the disk is . The charge on the element is:	(Models an electrostatic microphone)
The electric field at P produced by this ring is

Since we have expressed a positive ring thickness as dr, we sum the rings from the center toward the edge. That is, the radius ranges from 0 to a.

Let ,

then du = 2r dr, and

When z << a,

When z >> a, àwhich is the expected result for a point charge located at the origin.

Suppose that, instead of having discrete charges, we have a continuous distribution of charge represented by a charge density ρ(r). Thus, the charge at position vector ${\bf r}'$ is ρ(r^/) d³r^/, where $d^3{\bf r}'$ is the volume element at ${\bf r}'$ . That the electric field generated by this charge distribution is:

$\begin{displaymath} {\bf E}({\bf r}) =\frac{1}{4\pi \epsilon_0} \int \rho({\b... ...r}- {\bf r}' } {\vert{\bf r} - {\bf r}'\vert^3} d^3{\bf r}', \end{displaymath}$

where the volume integral is over all space, or, at least, over all space for which ρ(r^/) is non-zero

ELECTRICITY AND MAGNETISM

CoulombÒs Law

Find the electric field a distance z above the midpoint of a straight line segment of length 2L which carries a uniform line charge l.

Thus we can write (Model of a transmission line)

How to evaluate this integral