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KEY CONCEPTS
ELECTROSTATICS
1. ELECTRIC CHARGES AND COULOMB’S LAW
There are two different types of electric charges: positive charges and negative
charges. The basic unit of charge in the MKS system is the coulomb, C. On an
atomic level, protons are positively charged, electrons are negatively charged, and
neutrons have no net charge. A general theory of charge is that total charge is
conserved. In other words, the net electric charge produced in any process is zero.
Depending on the behavior of their electrical charges, materials usually fall into
one of two classes, conductors or insulators. Negative electrical charges flow freely in
conductors such as metals (copper, aluminum) and ionic fluids (human body). Charges
do not flow easily in insulators such as glass and plastic (pvc), but tend instead to build
up on their surfaces.
The electrical force between two charges, q1 and q2, separated by a distance r, is
given by Coulomb's law:
F=
1 q1q2
4πε o r 2
Where the term εo is known as the permittivity constant and has a value of 8.85×10-12
C2/N⋅m2. For two positive charges, the force is positive, and the two charges repel each
other. If one charge is positive and the other is negative, the force is negative and the
charges attract each other. If more than two charges are present, the total force acting
on one of the charges is the sum of the forces exerted on that charge by the other
charges.
2. ELECTRIC FIELDS
A charged object sets up a disturbance in the space around itself that is known
as an electric field, E. It is a vector field, since it has both magnitude and direction. The
electric field is associated with a force, F. If a charge, q, is placed in the field, the
electric field is defined as:
E=
F
q
At all points of an electric field of a charged sphere, the field is directed radially outward
from the sphere. A positive charge exerts an electric field directed away from itself,
whereas a negative charge exerts an electric field directed towards itself. The density of
field lines indicates the field strength. By using Coulomb's law, the electric field created
by a charge, q, can be related to the separation distance, r, between it and a test
charge, qo:
KEY CONCEPTS
ELECTROSTATICS
E=
F
1 q
=
r
qo 4πε o r 2
If more than one charge is present, the total electric field is equal to the sum of the
electric fields of the individual charges.
Etotal = E1 + E2 + E3 + … + En = ∑En
A dipole configuration results when two equal but opposite charges are close
together. The strength of the electric field induced by a dipole is given by:
E=
1 2aq
4πε o r 3
The electric field that results from a uniformly distributed ring of charge lies along
the axis of the ring. A small element of charge, dq, along a small section of the ring
induces an electric field, dEcosθ, along the axis. Using Coulomb's law, the magnitude of
the electric field, dE, resulting from an element of charge, dq, is given by:
dE =
1 dq
4πε o r 2
Therefore, the magnitude of the entire electric field, E, along the axis is determined by
integrating all of the small charge elements:
E = ∫ dEcosθ =
1
qx
4πε o (x + a 2 )3/ 2
2
3. ELECTRIC POTENTIAL
The electric potential difference is the work done, WAB, when moving a charge,
q, from point A to point B in an electric field:
VB − V A =
WAB
q
The potential difference is independent of the path taken from point A to point B. Point A
is commonly taken to be at an infinite distance and to have a potential equal to zero.
Thus the electric potential, V, is the energy per unit charge as given by:
V=
W
q
The electric potential is given in units of volts, where one volt is equal to one joule per
coulomb. If the total work done in moving a charge along a path from point A to point B
is zero, then the potential at point A is the same as that at point B. Because the electric
potential along the path does not change, it is called an equipotential surface. For a
point charge, equipotential surfaces are concentric circles around the charge.
The change in potential, ΔV, between two points in a uniform electric field, E,
separated by a distance, d, is given by:
2
KEY CONCEPTS
ELECTROSTATICS
ΔV =
WAB qEd
=
= Ed
q
q
Thus, the magnitude of a uniform electric field, E, is given by:
E=
ΔV
d
with units of volts per meter. For the case of a nonuniform electric field, a more general
expression for the potential difference can be determined by integrating the work done
along the entire path from point A to point B:
B
W
VB − V A = AB =
q
− q ∫ E • dl
A
q
B
= − ∫ E • dl
A
Substituting the electric field for a point charge into this expression gives:
q
q ⎛ 1 1⎞
dr
=
−
2
4πε o ⎜⎝ rB r A ⎟⎠
A 4πε o r
B
VB −VA = − ∫
1
If point A is at infinity, then the general expression for the electric potential, V, at a
distance, r, from a point charge, q, becomes:
V=
q 1
4πε o r
4. GAUSS’S LAW
The electric flux, φE, through a closed surface is the integral of the dot products
of the electric field vector, E, and surface area vector, A:
φE = ∫ E • dA
Gauss's law states that the electric flux, φE, is equal to the charge enclosed by a
surface, Q, divided by the permittivity, εo:
φE = ∫ E • dA =
Q
εo
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