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AP Physics C E & MElectric Potential
20151204
www.njctl.org
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WorkIn previous chapters we discussed that work is the integral of force. If an object moves from point a to b the work is represented by:
This can also be represented in terms of the potential energy:
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Work done in a Uniform Electric Field
point a
point b
d
The path which the particle follows through the uniform electric field is independent of the path it takes.
To understand this, think back to gravitational potential energy. Two Hikers of equal mass walk up a hill. Hiker A takes the shortest path to get to the top of the hill and Hiker B takes the longest route to the top. In the end which has a greater potential energy?
The answer is they have the same potential energy.
H
Hiker A
Hiker B
Both hikers will have a potential of mgh
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The force is constant in a uniform magnetic field so we can use:
We discussed in previous chapters that:
and
When we sub in E into the force equation we get:
Therefore the work done in moving a particle through a uniform electric field is:
or
Work done in a Uniform Electric Field
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1 Two charges of charge +Q and 2Q are separated by a distance d. The work required to move a test charge of magnitude +Q from point A to point B which is midway between the two charges is
A dependent on the path
B inversely proportional to the distance
C dependent on the displacement
D zero
E more information is required
d+Q Q
A
B
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2 A negative charge of coulombs is palced in an upward directed uniform electric field of . When the charge is moved 2 meters upward, the work done by the electric force on the charge is
A 1.8 x 104 J
B 1.8 x 104 J
C 1.8 x 1016 J
D 1.8 x 1016 J
E 3.6 x 104 J
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We said before that an uniform electric field is similar to a case under the force of gravity. When we have an object of mass m, and move it up and down we change the amount of potential energy it has. When we go against the force of gravity the potential energy increases and when we allow gravity to take over it falls to a lower potential.
This holds true in a uniform electric field. When we oppose the force acting on the particle within the field we will increase the particles potential energy and if we allow it to move in the direction of the force it falls to a lower potential.
+
EE
F
F
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Uniform Electric Field
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Electric Potential Energy of Two Point Charges
For electric potential between two charges we will have to deal with two different cases. The first deals with a test charge moving away from another charge linearly and the second case is when the test charge moves radially away from the other charge.
Case 1:
+Q
r
p1
p2
rarb
Since the force due to this system is not constant we have to use the integral representation of work
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Now if we deal with the more general case of moving the test charge radially away from the other charge we will find that the potential energy is not dependent on the path we take.
+Q
dldrFrom the sketch we see that:
E
Electric Potential Energy of Two Point Charges
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Electric Potential Energy with several point charges
Three charges are arranged around point p. A test charge has a charge of qo and is placed at this point, how do we find the potential energy of the test charge?
P
q1
q2
q3qo
The potential energy at point P is simply the algebraic sum of each of the potentials due to the other charges.
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q1
q2
q3
In establishing the formations of the charges in the previous example there would also be a potential energy.
The equation for this is also the algebraic sum of the interactions between each of the charges.
In this equation i<j because a charge does not interact with itself, this way the equation takes into account all the interactions between the charges.
Electric Potential Energy with several point charges
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3 Three equal test charges of magnitude are placed at the corners of a square. A test charge of magnitude is placed at the last open corner. What is the electric potential energy of the test charge?
A 29.2 J
B 32.4 J
C 27.8 J
D 16.4 J
E 1 J
Q1
Q2Q3
Test Charge
.5m
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Electric Potential is the potential energy per unit of charge.
If we plug in the equation for the potential energy before we will find:
Like the electric field the electric potential only depends on the charge you are looking at, it is not effect by any other charge like electric potential energy.
If we return to the work definition for a charge we find that the work is equal to the negative change in electric potential.
Electric Potential
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+
E EV decreases
V increases
V decreases
V increases
From these examples we can say that when we move in the direction of the electric field the electric potential decreases and as we move in the opposite direction the electric potential increases.
Electric Potential
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i is the subscript for the charge and the distance of the charge from the point were you are evaluating the net electric potential.
Once again if there is a collection of charges then the net electric potential at some point p is equal to the algebraic sum of each of the charges electric potentials at that point.
P
q1
q2
q3
Once again if there is a collection of charges then the net electric potential at some point p is equal to the algebraic sum of each of the charges electric potentials at that point.
Electric Potential
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If a charge is distributed over a surface such as those in a coulomb's problem we break the overall object into small pieces of charge dq. The net electric potential at a certain point is given by:
(Electric Potential due to a charge distribution)
Before we discussed that the work done by moving a charge through an electric field is given as:
and
Electric Potential
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4 Two charges Q1 and Q2 are placed a distance d apart from each other along a line and have charges +2Q and Q respectively. At how many points is the electric potential zero.
A none
B one
C two
D three
E more information is required
+2Q Q
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5 What is the electric potential at point p? Q1 = 3x1012 C Q2 = 4x1012 C
A .54 volts
B .9 volts
C 1.44 volts
D 1.8 volts
E 2.6 volts
Q1
Q2 P4 cm
3 cm
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Calculating Electric Potential for Charge Distributions (Sphere)
If a solid metal sphere of radius R is given a charge Q. What are the graphs of the electric field and the electric potential for the sphere.
First we have to recognize that since it is a conductor all of the charge will reside on the surface of the sphere. When we apply Gauss's Law we find the electric field inside the sphere to be zero, and the electric field outside of the sphere to be the same for a point charge.
E
RR
V
RR
since E=0, VaVb=0. Therefore Va=Vb and every point within the sphere is at the same electric potential.
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Calculating Electric Potential for Charge Distributions
(Thin Conducting Cylinder)(Electric Field of Charged Cylinder)
+
++
++
++
+
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Calculating Electric Potential for Charge Distributions
(Thin Conducting Cylinder)
To calculate the electric potential for other shapes you simply use a different form of the equation like in Coulomb's Law, where you break it into tiny pieces and add up the charges.
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6 What is the potential energy of a positively charged ring of radius a a distance d away from its center?
A
B
C
D
E
a
d
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Equipotential SurfacesAn equipotential surface is a 3 dimensional structure on which the electric potential is constant.
For a sphere the equipotential surfaces would be spheres, and to sketch the magnitude of each of the surfaces we use contour lines like on a topographic map, a map that shows changes in elevation. When the lines are getting further apart they indicate that the potential is decreasing, when the are evenly spaced the potential is constant, and when they are getting closer together the potential is increasing. This same reasoning can be applied to more complicated shapes, in turn each of there equipotential surfaces would be the same as the actual surface.
+V
V/4V/9 V/16
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7 A test charge of magnitude q moves from point A to point B. What is the work done on the particle?
A 0
B 20q
C 10q
D 40q
E 50q
10 V
20 V
10 V
20 V
30 V
20 V
10 V
A
BC
D
E
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8 A test charge of magnitude q moves from point E to point A. What is the work done on the charge?
A 10q
B 0 C 10q
D 30q
E 60q
10 V
20 V
10 V
20 V
30 V
20 V
10 V
A
BC
D
E
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9 A test charge of magnitude q moves from point D to point C. What is the work done on the test charge?
A 20q
B 20q
C 40q D 30q
E 10q
10 V
20 V
10 V
20 V
30 V
20 V
10 V
A
BC
D
E
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Potential Gradient
(Dot Product)
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The electric potential is the negative gradient of the electric field meaning at each point it indicates the direction in which v decreases most rapidly with a change in its position. Therefore as we explained before E is in the direction in which the potential decreases most rapidly and is perpendicular to the equipotential surface.
Potential Gradient
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10 A charged ring of radius a has a potential energy given by the equation . What is the equation of its electric field?
A
B
C
D
E
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