Download The Electric Field

The Electric Field
Once again, in order to try and understand the
concepts of electrostatics, we will make an analogy to
mechanics.
It is convenient to picture a “field” by drawing lines to
indicate the direction of the field at any point.
Filed lines are also called lines of force because they
show the direction of the force exerted on a mass or in our
case, a positive test charge.
Example:
Draw the field lines that illustrate the force of gravity
for a spherical object like the Earth.
In electrostatics, at any point near a positive charge,
the electric field points radially away from the charge.
Now, since we know how to illustrate the electric
field, what exactly is it defined as?
The idea is that one charge produces an electric field
everywhere in space that exerts a force on all other charges.
How is it measured?
Assume the following distribution of charges:
If we now introduce a charge q0 in the system, in
general, the charge q0 will change the original distribution.
However, if q0 is small enough so that its effect is
negligible, we can use it to test the field of other charges.
Thus, it is called the test charge.
The net force exerted on q0:
 Is the vector sum of the forces exerted on q0 by the
other charges.
 Proportional to q0.
So, then using our definition of the electric field, it is
defined as the net force on a positive test charge q0, divided
by q0.
Writing this down algebraically:
The electric field:
 Has SI units of N/C.
 Is a vector sum so the law of superposition is valid.
 Is analogous to the gravitational field of the Earth.
We can now write the force as:
Example:
When a 5nC test charge is placed at a certain point, it
experiences a force of 2x10-4 N in the x direction. What is
the electric field at that point?
Back to Field Lines
Rules for illustrating field lines:
1. Electric field lines begin on positive charges and
end on negative charges or at infinity.
2. Lines are drawn symmetrically leaving or entering a
charge.
3. The number of field lines is proportional to the
charge.
4. The density of lines is proportional to the magnitude
of the field.
5. At large distances from a system, the field lines are
equally spaced and radial as if they came from a
point charge equal to the net charge of the system.
6. No two field lines can cross.
Examples:
Motion of particles in an electric field
Now that we have an understanding of the electric
field, we can discuss how particles will move in one.
Recall Newton’s second law:
Therefore, in the presence of an electric field a particle
will undergo an acceleration.
Example:
What is going to happen to a negatively charged
particle moving parallel to an electric field?
Perpendicular?
Positive?
Dipoles
A dipole is a system of 2 oppositely charged particles:
We define a dipole moment as:
Most atoms are neutral, however in the presence of an
E field, a dipole moment will be induced.
Think about it in the following manner:
In some molecules, like water, there is a permanent
dipole moment.
E-Fields will exert a torque on dipoles so they will
align with the field. The magnitude of the torque will be:
Continuous Charge Distributions
Often, we do not encounter point charges in everyday
life. However, we will encounter a continuous set of
charges closely arranged in space. Thus, the concept of a
continuous charge distribution.
Before we begin, we must introduce a few quantities:
Examples of continuous charge distributions are:






Insulating sphere
Insulating or conducting shell
Thin wire
Infinite sheet
Insulating ring
Insulating disk
Q. What is the E-Field for each type of distribution?
Gauss’s Law
Gauss’s Law is a mathematical description of the
qualitative description of the electric field using electric
field lines.
It related the electric field on a closed surface to the
net charge within the surface. We can define a quantity
that describes the number of field lines passing through a
certain area. This is called the electric flux:
E = EA
If the surface is not perpendicular to the field, the flux
through it must be less than the above. In fact it will be
reduced by a quantity of cos :
E = EA cos 


In the above, however, we did not consider the fact the
electric field may vary along the surface. Therefore, we are
only measuring the flux over a small area. To account for
the total electric flux we have to integrate over the entire
surface. Therefore:
E =
E dA
This is a surface integral and we are often interested in
integrating over a closed surface, which divides the space
into an inside and outside region. The surface of a sphere
is a closed surface.
To find an equation for Gauss’s Law, let’s consider a point
charge at the center of a sphere of radius r. The electric
field over the entire surface is a constant so that can come
out of our surface integral. The surface integral of a sphere
is just the surface area of the sphere. Therefore, the electric
flux is given as:
E = (kq/r2) 4r2 = 4kq
where q is the enclosed charge inside the sphere. Using the
equality k = 1/4o we find:
E = q/o
Expanding this argument, we see that the net electric
flux through any surface is the same. In general, we can
expand the argument, that if we can enclose a charge in
some symmetrical way, the net flux will be proportional to
the enclosed charge. Therefore, a quantitative expression
for Gauss’s law is:
E =
E dA = q/o
If there is no charge within the surface chosen, the
electric field is zero!
E = The Electric Field
dA = Surface area of a “Gaussian surface”
q = The enclosed charge
 = The permitivity of free space
A “Gaussian surface” is a surface the will enclose the
charge in a symmetrical way. We will discuss 3 kinds:
 Sphere
 Pillbox
 Cylinder
Example:
Enclose a point charge with a sphere:
Example:
Enclose an infinite sheet with a “pillbox”
Example:
Enclose an infinite wire with a cylinder