CHAPTER 23
ELECTRIC FIELDS
Properties of Electric Charges
EM interactions involve particles that have electric charge.
• There are two kinds of charge: positive (+), negative (−)
• Like charges repel one another.
• Unlike chages attract one another.
• All materials are formed of charged particles: electron, proton, neutron
• Electrically neutral means, the object has equal amount of positive and
negative charges. (no net charge  neutral)
Properties of Electric Charges
• Electric charge is always conserved (not lost or created)
2 (-) charges are transferred to plastic from wool.
Total amount of charge does not change.
• Only negative charges can move from one object to another.
Charge is Quantized!
Electric charge exists as discrete packets.
Smallest electric charge  e = 1.6 x 10-19 Coulomb (C)
The charge of 1 electron –e = – 1.6 x 10-19 C
The charge of 1 proton +e = + 1.6 x 10-19 C
For an object with charge q
q = N e where N is an integer always. N = ± 1, ± 2, ± 3, …
1.3 e charge does NOT exist!
Conductors and Insulators
Materials are classified in terms of their ability to conduct electrical charge.
Conductors, are materials in which charges move freely.
Metals are good conductors: copper, aluminum, silver etc.
When a small region of a metal is charged, the charge is distributed over its
entire surface very quickly.
Insulators, are materials that do not transport charge.
Glass, rubber, wood, ceramic etc.
When insulators are charged by rubbing, only a small area is charged and
charge does not move to other regions of the material.
Grounding and Induction
Earth is an infinite sink to which electrons can easily migrate. Only
negative charges move!
Induction is a method for charging a conductor.
Neutral conducting
sphere
Polarized by
a negative rod
Grounding moves
some electrons
Positively charged
Conducting sphere
Coulomb’s Law: Electrostatic force between charges?
Experiments show that, the electric force is;
• Proportional to the product of charges, q1q2
• Inversely prop. to the square of separation, 1/r2
• Directed along the line joining the two,
• Attractive for opposite charges, repulsive for same charges.
magnitude of the force
Coulomb constant
permittivity of free space
Coulomb’s Law: Electrostatic force is a vector!
Vector form:
𝑟
𝑟=
𝑟
is the unit vector in 𝑟 direction.
If there are more than two charged particles:
Electrostatic forces on q1,
q2 attracts q1: 𝐹12
q3 repels q1: 𝐹13
q4 repels q1: 𝐹14
vector addition!
magnitude
Example:
For a net zero force on q3, two forces should be equal in magnitude
Electric Field: 𝐸
How does q1 know the presence of q2 and exert a force on it?
A particle with electric charge sets up an electric field at its surround.
The electric field is strong near the particle, and becomes weaker for
larger distance.
Electric force is actually related to this electric field.
𝐹 = 𝑞𝑜 𝐸
Electric field is a vector!
The unit of E.field is Newton/Coulomb
Electric Field:
q0 is a positive test charge that is put to a point hypothetically, to find the
E. Field at that point.
E. field exists if the test charge experiences a force!
E. field is there, regardless of whether or not q0 exists!
E. Field lines are always outward from a positive charge and inward for a
negative charge. They extend to infinity.
Electric Field Lines
• E. Field vector is always tangent to the field lines.
• E. Field lines does not intersect with each other.
• The number of drawn lines, shows the strength of the field.
Number of lines per unit area on plane A is bigger, therefore 𝐸 field is
larger.
Electric Field Lines
• The lines begin on a positive charge and terminate on a negative
charge.
• The number of lines drawn leaving a positive charge or approaching a
negative charge is proportional to the magnitude of the charge.
Electric Field of point charges:
Coulomb’s law:
Electric Field (vector):
E. Field (magnitude):
Coulomb constant
electric permittivity
of free space
The Superposition of Electric Fields:
At any point P, the total electric field due to a group of source charges
equals the vector sum of the electric fields of all the charges.
Example:
First find the magnitudes of the electric fields at P
due to each charge:
Example:
Then put a positive test charge at point P and find the
directions and do the vector addition:
Example:
E1 and E2 are paralel add them first.
Then the magnitudes, E1 + E2 and E3 are the same.
y-components cancel each other.
Add the x-components:
Example:
Remember the field lines around a dipole
Example:
Y-components cancel each other, add the x-components
For very far points y>>a;
Valid for any
very distant point
E.Field of a continous charge distribution
The electric field at P due to one charge element ∆q is;
The total electric field at P due to all elements;
When you divide to infinitely many elements ∆qi  0
Now how do we write dq element?
Uniformly Distributed Line Charge
If a charge Q is uniformly distributed along a line of length L,
the linear charge density λ (lamda) is defined by;
the unit is Coulombs per meter
integral limits should define
the position of the charge distr.
Uniformly Distributed Surface Charge
If a charge Q is uniformly distributed on a surface
of area A, the surface charge density σ (sigma) is
defined by;
the unit is Coulombs per square meter
Uniformly Distributed Volume Charge
If a charge Q is uniformly distributed throughout a
volume V, the volume charge density ρ (rho) is defined
by;
the unit is Coulombs per cubic meter
Example:
Charged Rod
dx is the length of one small segment with charge dq
The field dE at P due to one small segment is
The total field at P due to
all segments of the rod
limits define the
position of the rod.
Example:
Charged Rod
Now Let’s do the integral;
(note: a writen as d below)
Example:
Charged Ring
For one small segment with charge dq:
Perpendicular components cancel because of symmetry around x-axis
Example:
Charged Ring
Example:
Charged Ring
E.Field of a ring along its axis
If point P is very near to the center x<<a
If x=0 then E=0 (center)
If point P is very far away x>>a
Like a point charge!
Example: A rod of charge –Q is a 120° circular arc
with radius a. Find the electic field at the center.
y-components cancel because of symmetry
There are two variables, θ and s. Eliminate one
variable to integrate it! Arc length:
Example: A rod of charge –Q is a 120° circular arc
with radius a. Find the electic field at the center.
Not finished, write λ in terms of the charge Q
Example:
Charged Disk
Disk  a set of rings with same center
A ring has an area dA, charge dq, and E.field dEx
from ring example
Write the dq element with surface charge density σ
Now that is the dA area element?
Example:
Charged Disk
Integral limits: different radii of the rings from 0 to R
Example:
Charged Disk
For near field approximation R>>x,
R  ∞ or x  0
So at very near points, the disk is like an infinite plate
and the Electric field is constant.
important!!
Infinite plate produces uniform Electric Field!
Two infinite plates with opposite charges: Capacitors (a circuit element)
Motion of Charged Particles in a uniform E. Field
A point charge q with mass m is placed in an external E. field E, the electric
force exerted on the charge is;
uniform E  constant acceleration
If the charge is positive, its acceleration is in the direction of the E.field.
If the charge is negative, its acceleration is in the direction opposite to the
direction of the E. field.
Example:
uniform E  constant acceleration
We can use the kinematic equations for constant acceleration.
Example: A proton is released from rest near +σ plate. What speed will
it strike to -σ plate. What will be the transit time? σ = 1x10-6 C/m2 given.
Deflection of an electron: (neglect gravity here)
After the electron leaves the E. field, the electric force
vanishes and the electron continues to move in a straight
line in the direction its velocity.
Example: