sjdlecture21

Electric ChargesTwo basic charges
Positive and Negative
1.6 x 10-19 Coulombs
Experiments have shown that
Like signed charges repel each other
Unlike signed charges attract each other
For an isolated system, the net charge of the system remains constant
Charge Conservation
Fall 2008
Lecture 1-*
Physics 231
Conductors
Materials, such as metals, that allow the free movement of charges
Insulators
Materials, such as rubber and glass, that don’t allow the free movement of charges
Fall 2008
Lecture 1-*
Physics 231
Coulomb’s Law
Coulomb found that the electric force between two charged objects is
Proportional to the product of the charges on the objects, and
Inversely proportional to the separation of the objects squared
k being a proportionality constant, having a value of 8.988 x 109 Nm2/c2
Fall 2008
Lecture 1-*
Physics 231
Electric Force
This gives the force on charged object 2 due to charged object 1
The direction of the force is either parallel or antiparallel to this unit vector depending upon the relative signs of the charges
As with all forces, the electric force is a Vector
So we rewrite Coulomb’s Law as
is a unit vector pointing from object 1 to object 2
q2
q1
Fall 2008
Lecture 1-*
Physics 231
Electric Force
The force acting on each charged object has the same magnitude -
but acting in opposite directions
(Newton’s Third Law)
Fall 2008
Lecture 1-*
Physics 231
Example 1
A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12 directly above Q1.
When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at a distance d23 (< d12) directly above Q3.
For 1a and 1b which is the correct answer
1a: A) The charge of Q3 has the same sign of the charge of Q1
B) The charge of Q3 has the opposite sign as the charge of Q1
C) Cannot determine the relative signs of the charges of Q3 & Q1
1b: A) The magnitude of charge Q3 < the magnitude of charge Q1
B) The magnitude of charge Q3 > the magnitude of charge Q1
C) Cannot determine relative magnitudes of charges of Q3 & Q1
Q2
Q1
Q2
g
d12
d23
Q3
Fall 2008
Lecture 1-*
Physics 231
To be in equilibrium, the total force on Q2 must be zero.
The only other known force acting on Q2 is its weight.
Therefore, in both cases, the electrical force on Q2 must be directed upward to cancel its weight.
Therefore, the sign of Q3 must be the SAME as the sign of Q1
1a: A) The charge of Q3 has the same sign of the charge of Q1
B) The charge of Q3 has the opposite sign as the charge of Q1
C) Cannot determine the relative signs of the charges of Q3 & Q1
Example 1
Fall 2008
Lecture 1-*
Physics 231
The electrical force on Q2 must be the same in both cases … it just cancels the weight of Q2
Since d23 < d12 , the charge of Q3 must be SMALLER than the charge of Q1 so that the total electrical force can be the same!!
1b: A) The magnitude of charge Q3 < the magnitude of charge Q1
B) The magnitude of charge Q3 > the magnitude of charge Q1
C) Cannot determine relative magnitudes of charges of Q3 & Q1
Example 1
What is the net force if both charges are present?
The net force is given by the Superposition Principle
q
q1
q2
If q1 were the only other charge, we would know the force on q due to q1 -
If q2 were the only other charge, we would know the force on q due to q2 -
Fall 2008
Lecture 1-*
Physics 231
Superposition of Forces
If there are more than two charged objects interacting with each other
The net force on any one of the charged objects is
The vector sum of the individual Coulomb forces on that charged object
Fall 2008
Lecture 1-*
Physics 231
Example Two
What is the force acting on qo?
qo, q1, and q2 are all point charges where qo = -1mC, q1 = 3mC, and q2 = 4mC
What are F0x and F0y ?
x (cm)
y (cm)
4
3
2
1
qo
q2
q1
q
Fall 2008
Lecture 1-*
Physics 231
x (cm)
y (cm)
4
3
2
1
qo
q2
q1
At an angle given by
The magnitude of is
4
3
2
1
qo
q2
q1
Note on constants
k is in reality defined in terms of a more fundamental constant, known as the permittivity of free space.
Fall 2008
Lecture 1-*
Physics 231
Electric Field
Action at a Distance
The electric force can be thought of as being mediated by an electric field.
Fall 2008
Lecture 1-*
Physics 231
What is a Field?
A Field is something that can be defined anywhere in space
A field represents some physical quantity
(e.g., temperature, wind speed, force)
It can be a scalar field (e.g., Temperature field)
It can be a vector field (e.g., Electric field)
It can be a “tensor” field (e.g., Space-time curvature)
Fall 2008
Lecture 1-*
Physics 231
A Scalar Field
A scalar field is a map of a quantity that has only a magnitude, such as temperature
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A Vector Field
A vector field is a map of a quantity that is a vector, a quantity having both magnitude and direction, such as wind
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Fall 2008
Lecture 1-*
Physics 231
Electric Field
We say that when a charged object is put at a point in space,
The charged object sets up an Electric Field throughout the space surrounding the charged object
It is this field that then exerts a force on another charged object
Fall 2008
Lecture 1-*
Physics 231
Electric Field
the electric field is also a vector
If there is an electric force acting on an object having a charge qo, then the electric field at that point is given by
(with the sign of q0 included)
Fall 2008
Lecture 1-*
Physics 231
Electric Field
The force on a positively charged object is in the same direction as the electric field at that point,
While the force on a negative test charge is in the opposite direction as the electric field at the point
Fall 2008
Lecture 1-*
Physics 231
Electric Field
A positive charge sets up an electric field pointing away from the charge
A negative charge sets up an electric field pointing towards the charge
Fall 2008
Lecture 1-*
Physics 231
Electric Field
The electric field of a point charge can then be shown to be given by
Earlier we saw that the force on a charged object is given by
The term in parentheses remains the same if we change the charge on the object at the point in question
The quantity in the parentheses can be thought of as the electric field at the point where the test object is placed
Fall 2008
Lecture 1-*
Physics 231
Electric Field
As with the electric force, if there are several charged objects, the net electric field at a given point is given by the vector sum of the individual electric fields
Fall 2008
Lecture 1-*
Physics 231
Electric Field
If we have a continuous charge distribution the summation becomes an integral
Fall 2008
Lecture 1-*
Physics 231
2) Choose variables for integration carefully.
3) Check limiting conditions for appropriate result
Fall 2008
Lecture 1-*
Physics 231
Electric Field
Fall 2008
Lecture 1-*
Physics 231
Two equal, but opposite charges are placed on the x axis. The positive charge is placed at x = -5 m and the negative charge is placed at x = +5m as shown in the figure above.
Example 3
1) What is the direction of the electric field at point A?
a) up b) down c) left d) right e) zero
2) What is the direction of the electric field at point B?
a) up b) down c) left d) right e) zero
Fall 2008
Lecture 1-*
Physics 231
Example 4
Two charges, Q1 and Q2, fixed along the x-axis as
shown produce an electric field, E, at a point
(x,y) = (0,d) which is directed along the negative
y-axis.
(a) Both charges Q1 and Q2 are positive
(b) Both charges Q1 and Q2 are negative
(c) The charges Q1 and Q2 have opposite signs
Q2
Q1
(c)
E
Q2
Q1
(b)
E
Q2
Q1
x
y
E
d
Q2
Q1
(a)
E
Electric Field Lines
Possible to map out the electric field in a region of space
An imaginary line that at any given point has its tangent being in the direction of the electric field at that point
The spacing, density, of lines is related to the magnitude of the electric field at that point
Fall 2008
Lecture 1-*
Physics 231
At any given point, there can be only one field line
The electric field has a unique direction at any given point
Fall 2008
Lecture 1-*
Physics 231
Electric Dipole
An electric dipole is a pair of point charges having equal magnitude but opposite sign that are separated by a distance d.
Two questions concerning dipoles:
1) What are the forces and torques acting on a dipole when placed in an external electric field?
2) What does the electric field of a dipole look like?
Fall 2008
Lecture 1-*
Physics 231
Given a uniform external field
Then since the charges are of equal magnitude, the force on each charge has the same value
However the forces are in opposite directions!
Therefore the net force on the dipole is
Fnet = 0
Fall 2008
Lecture 1-*
Physics 231
Torque on a Dipole
The individual forces acting on the dipole may not necessarily be acting along the same line.
If this is the case, then there will be a torque acting on the dipole, causing the dipole to rotate.
Fall 2008
Lecture 1-*
Physics 231
The torque is then given by t = qE dsinf
d is a vector pointing from the negative charge to the positive charge
Fall 2008
Lecture 1-*
Physics 231
Given a dipole in an external field:
Dipole will rotate due to torque
Electric field will do work
The work done is the negative of the change in potential energy of the dipole
The potential energy can be shown to be
Fall 2008
Lecture 1-*
Physics 231
2
2
1
r
q
q
k
F

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