Download Electric Field Notes 1

Lesson #5 – Electric Potential
I.
Review of Concepts From Physics 1224
A.
Key Concept – Force (Vector)
In first semester physics, we showed that we could in theory solve ANY problem
involving a system of particles moving at speeds less than 10% of the speed of
light by determining the net external force acting upon the system and applying
Newton’s Laws.
However, we found that many problems were difficult to solve by this method
because either the individual forces acting upon the system were unknown
(collisions) or because of the vector nature of forces (energy problems).
Thus, we solved many problems using the scalar concepts of energy, work, and
potential energy instead of using the vector force methods directly!!
B.
Potential Energy – U
The negative of the work done by a conservative force as an object is moved
from point A to point B is defined to be the difference in the potential energy of
the object due to that force between point A and point B.
Because potential energy is defined in terms of an integral, only the difference in
potential energy has physical meaning!!!
Thus, when someone talks about the potential energy at some point or potential
energy function, they actually mean the potential energy difference with respect
to some implied zero potential reference point!!!
C.
Gravitational Potential Energy (near the surface of the Earth) – Ug
Using the surface of the earth as the zero gravitational potential energy reference
point, the gravitational potential energy of an object of mass M at a height of h
above the Earth is given by
Note: Gravitational potential energy is NOT a scalar field since its value
depends on the mass of the object and not just the object’s location. This is true
of all types of potential energy. Potential energy is a property of a system of
objects. In this case, the gravitational potential energy belongs to system of
masses composed of the Earth and the object at height h.
C.
Example of a Scalar Field
We can create a scalar function that depends only upon the object’s location
above the Earth dividing the gravitational potential energy function by the mass of
the test object. This new function is a scalar gravitational potential field created
by the Earth.
With this new function, we can easily determine the potential energy stored as
object of mass M is moved from the zero potential energy reference location to a
point at a height of h above the Earth by multiplying our scalar gravitational
potential energy field by the object’s mass.
In other words, we can easily find out how much work we must do to move the
object from the Earth’s surface to a height h. Although this process may seem
unnecessary for the simple problem we are considering here, it is extremely
helpful for problems involving more complicated geometry. Furthermore, we will
see shortly that a special instrument called the voltmeter has been constructed to
calculate the difference in the scalar electric potential between two points in
space.
II.
Electric Potential Energy – UE
A.
For non-time varying electric fields, the electric force on a test charge q is a
conservative force.
B.
Using the definition of potential energy from Physics 1224, we have that the
change in electric potential energy for a test object of charge q as it is moved from
point A to point B is given by the equation:
Note: The electric potential energy is not a scalar field as it depends on the charge
of the test object. The electrical potential energy is a property of a system of
charges (those charges that set up the electric field and the test charge).
III.
Electric Potential Difference (Voltage) - ∆V
A.
Definition – The electric potential difference between two points in space is
defined as the change in the electric potential energy that a test object would
experience as it was moved between the two points divided by the charge on the
test object.
B.
Units – Volts ≡
C.
Electric potential difference depends only on the points in space and not on the
test object. Thus, it is a scalar field created by the charges that created the electric
field.
D.
Electric Potential Difference:
(Scalar)
Electric Field :
(Vector)
Electric potential can be uniquely defined at a point in space only after defining a
reference point of zero electric potential. For a point charge, we will choose r = ∞
as our reference point.
IV.
Calculating the Electric Potential Difference From the Electric Field
A.
If you know the electric field, you can calculate the electric potential difference
between two points using the formula:
PROOF:
B.
EXAMPLE: What is the electric potential difference between point B : (2,3) and
r
N
A : (0,0) for a constant electric field, E =100. iˆ
C
SOLN: Since the electric potential depends only on the starting and ending points, we
can choose any path that we wish to evaluate the integral. In the diagram below,
we split the path into two segments: 1) A to C and 2) C to B.
y
C
A
B
x
C.
Although it is possible to calculate the electric potential given the electric field,
this process is rarely done in practices as it defeats the advantage of avoiding the
vector math associated with electric fields. Instead, we will follow the same
process that we did in PHYS1224 when we calculated the work integral for a few
special forces (gravity, spring, etc) to develop potential energy functions. We
then used the potential energy functions when solving problems.
V.
Electric Potential For a Point Charge
A.
We already know that any problem involving a distribution of charge can be
solved by breaking the charge distribution into a series of point charges and then
either integrating or summing the individual point charge contributions.
B.
The electric potential due to a point charge of charge Q at a distance r from the
point charge is given by
where we have chosen r = ∞ as our zero electric potential reference point.
PROOF: Consider the change in electric potential between a point at a distance r from a
point charge Q and r = ∞. Since the electric potential depends only upon the initial point
( r = ∞ ) and the final point ( r = r ) and not the path taken between the points, we can
chose to evaluate the integral by moving along the radius.
r=∞
x
Q
V(r) − V(∞ ) =
Electric Field For A Point Charge :
Displacement Vector :
r
VI.
Electric Potential For Any Charge Distribution
A.
Discrete Set of Point Charges
The total electric potential for N discrete point charges can be found using our
result from part V. The result is
B.
Continuous Charge Distribution
Using our result from part V and Calculus, we see that the total electric field due
to a continuous charge distribution is
C.
Although the proceeding two results may look similar to our results involving
electric fields, they are mathematically much easier to perform because they
involve scalars and not vectors. (No breaking vectors into components or
dealing with ˆ
r)
D.
Another advantage of our approach is that we can experimentally perform these
sums or integration’s using a voltmeter. The common (negative) lead is placed at
the zero reference point (i.e. large r) and the positive lead gives the electric
potential at any point in space where it is placed.
EXAMPLE: Calculate the electric potential at point P : (0.00 m, -0.450 m) due to a
6.00 nC point charge at (0.00 m,0.150 m) and a –4.00 nC point charge at (-0.450
m, 0.00 m).
y
6nC
0.15 m
-4nC
x
0.45 m
0.45 m
P
X
Note: Compare this with the work required to determine the electric field at point P.
Example 2: Calculate the electric potential at a distance d to the left of the end of a
uniformly charged thin rod of length L that is lying along the x-axis as shown
below:
dq
A
d
x
r
L
VII.
Calculating the Electric Field From the Electric Potential Difference
If the electric potential is known for a charge distribution then you can find the
electric field by the formula
r
Where E is the electric field
V is the electric potential difference
∇ is the gradient.
In the Cartesian coordinate system, the gradient is given by
∇=
∂ˆ ∂ ˆ ∂ ˆ
i+
j+
k .
∂x
∂y
∂z
It is more complicated in other coordinate systems and you will either need to
look it up in a math handbook or learn the general formula for the gradient for
generalized coordinates whose metric is not one. (see Math Methods for
Physicists and Engineers Web Page or look at the book of the same title by
Arfkin)
You should note that the gradient is an operator. Thus, the position of the
electric potential and the gradient in the equation is IMPORTANT!!
PROOF:
In PHYS1224, we learned that in order for a force to be conservative, you must
be able to write the force as the negative of the gradient of a scalar function
(potential energy function). Thus, we have
Dividing both sides by the charge on the test object, we have that
Since q is a constant, we can bring it inside the gradient operator, thus we have
Now using the definitions for the electric field and electric potential, we have that
EXAMPLE: What is the electric field associated with the electric potential V = 3x2yz 3?
PROBLEM: Given a specified charge distribution, find the acceleration that a test
charge would experience at a specific point in space.
Flow Chart (Electric Field Method)
Given: A charge distribution
Calculate Electric Field
Continuous
Distribution
No
Yes
Calculate the Electric Field
r
E=
∫
charges
kdq
rˆ
r2
Calculate Force on Test Charge
r
r
F=q E
Calculate Force on Test Charge
r
r F
a=
m
r N kQ
E=∑ 2i ˆ
ri
i =1 ri
PROBLEM: Given a specified charge distribution, find the acceleration that a test
charge would experience at a specific point in space.
Flow Chart (Electric Potential Method)
** MORE STEPS but Easier Math **
Given: A charge distribution
Calculate Electric Potential
Continuous
Distribution
N
No
Yes
Calculate the Electric Potential
V=
∫
charges
kdq
r
Calculate Electric Field
r
E=− ∇ V
Calculate Force on Test Charge
r
r
F=q E
Calculate Force on Test Charge
r
r F
a=
m
V=∑
i =1
kQ i
ri
VII.
Equipotential Surfaces
A.
Definition – An equipotential surface is defined as a surface consisting of a
continuous distribution of points having the same electric potential (voltage).
D
B
C
A
VBA =
VCA =
VDA =
B.
From the definition of electric potential, we know that an equipotential surface is
also a constant potential energy surface!!!
Thus, in the same way we can walk on the third floor of a building without being
hurt by gravity, we can place our hand on the Van de Graaff generator and
become charged without being shocked!!! If we step out of a window on the
third floor, then gravity will do work on our mass as we move between two
surfaces of different gravitational potential energy. If we touch something or
someone who is not at the same electric potential as the Van de Graaff then the
electric field will do work on the electric charges in our body as they move
between the two surfaces of different electric potential. (i.e. You get shocked!!)
** If you are having trouble following this discussion then review the material on
the “work energy theorem” and the “conservation of mechanical energy
requirement” from PHYS1224 **
Electrical Engineering Safety Tip –
Electrical engineers and electronic technicians often work on devices that are
capable of severely shocking or even killing a person. In order to reduce the
possibility of this hazard, they often tie the common (negative) lead to a single
reference point on the circuit called “ground.” The electric potential at any point
in the circuit is then measured by touching the positive lead of the voltmeter to the
desired point while keeping the other hand in your pocket or behind your back.
This reduces the chance of you reaching across the circuit and accidentally
touching with your hands two points that greatly differ in electric potential.
Although the point chosen as the ground can theoretically be any convenient point
in the circuit, it is usually specified in the schematics when one is troubleshooting.
C.
Electric field is perpindicular to an equipotential surface at every point on the
surface.
PROOF:
Consider two neighboring points on an equipotential surface. The difference in
electric potential between the two points must be zero so we have that
D.
Electric field lines point in the direction of DECREASING electric potential.
EXAMPLE 1: Draw the electric field lines and equipotential surfaces for the positive
point charge Q shown below:
+Q
EXAMPLE 2: Draw the electric field lines and equipotential surfaces for the charge
distribution shown below:
+3Q
-Q
LAB: We will experimentally determine equipotential surfaces and electric field lines
for different charge distributions in lab. Remember to bring graph paper to lab!!!
VIII. Conductors and Electric Potential
A.
All points on the surface of a conductor are at equipotential!
We know from previous work that the electric field is perpendicular to the surface
of a conductor. From our work in the previous section, it therefore follows that the
conductor’s surface is an equipotential surface.
B.
All points INSIDE a conductor are at equipotential!
Surface of Conductor
B
A
Since the electric field inside a conductor is ______________, we have
VBA = VB – VA =
C.
All points in an EMPTY CAVITY inside a conductor are at the SAME electric
potential as the CONDUCTOR!!
Outer Surface of
Conductor
B
A
Cavity
VBA = VB – VA =