Physics I
Class 21
Thinking About
Electric Fields and
Electric Potential
Rev. 02-Apr-04 GB
21-1
Class #19
Take-Away Concepts (Review)
1.
Electric field from point charge sources:
(Total field is the superposition of point source fields.)

1 qi
E
( r̂i )
2
4   0 ri
2.
Force on a charge in an electric field:
3.
4.
Electric field points away from + source charges.
Electric field points toward – source charges.


F  qE
21-2
The Electric Field of a Point
Charge (as a Source)
The electric field is a vector field, meaning at each point in
space the electric field has a magnitude and a direction. We
show that by drawing arrows at representative points in the
correct directions with lengths proportional to the magnitudes.
Away from positive
Toward negative
+
-
Just because we don’t draw an electric field vector at
a point doesn’t mean there is no electric field there.
21-3
Class #20
Take-Away Concepts (Review)
Physicist General’s Health Warning:
Confusing electric potential (V) with electric potential energy
(U) will be hazardous to your score on the next exam.
1.
Electric potential (V) and electric potential energy (U):
2.
Electric potential from point source charges:
U  qV
V
3.
4.
5.
1 qi
4   0 ri
Relationship of electric field and electric potential.
Electric field lines and equipotential lines.
Calculating change in kinetic energy using electric potential.
21-4
Why do we solve problems like:
“Where is the Net E Field = 0?”
There are several problems in electromagnetic design where the
objective is to find a location where the electric (or magnetic) field
has specified properties, like being zero, constant, or similar.
Examples range from electrostatic traps (for atoms, ions, or antimatter) to magnetic resonance imaging systems.
We pose problems like this in Physics I to give you the chance to
exercise your knowledge of electric fields and electric potential.
You need to keep clear in your mind which is a vector, which is a
scalar, and how contributions add or subtract in different regions.
Since both the electric field and electric potential due to a finite
collection of point charges go to zero at infinite distance from the
charges, that is usually not an interesting answer.
21-5
How to Solve Problems Asking
“Where is Net Electric Field = 0?”
+X
Region I
Region II
+ charge
Region III
- charge
Many things to consider:
 The number of regions (N+1) is one more than number of charges (N).
 The total electric field is the sum of the individual fields from each charge.
(Principle of Superposition)
 In each region, the direction of electric field from each charge is constant in that region.
(Away from +, toward –.)
 The magnitude of electric field depends on the value of charge and inverse distance squared
to the charge.
 In any region, if you get close enough to one of the charges at the end points, the inverse
distance squared dependence will make the field from that charge much larger than the
fields from all of the other charges.
 If you get far enough away from all the charges in Region I or Region N+1, the inverse
distance term is about the same for all points, so the relative sizes of the electric fields from
each charge will be determined by the sizes of the charges.
 In a region where the total electric field is zero at some location, it will point in one direction
at one end of the region and the other direction somewhere else in the region – the middle or
the opposite end.
21-6
How to Solve Problems Asking
“Where is Net Electric Field = 0?”
+X
Region I
Region II
Region III
+7 C
-4 C
Region I:
E field from +7C direction is to the left.
E field from –4C direction is to the right.
Every point is closer to +7C than to –4C, so net field is always to the left.
No points where E = 0.
Region II:
E field from +7C direction is to the right.
E field from –4C direction is to the right.
Net field is always to the right.
No points where E = 0.
Region III:
E field from +7C direction is to the right.
E field from –4C direction is to the left.
Close to –4C, net E field is to the left.
Far from –4C, net E field is to the right because 7 > 4 and far enough away the inverse
distance squared will be about the same for both.
There is one point where E = 0.
21-7
How to Solve Problems Asking
“Where is Net Electric Field = 0?”
+X
Region I
Region II
+7 C
d
Region III
-4 C
x-d
x
1 7
1
4

0
4   0 x 2 4   0 ( x  d) 2
7
4

x 2 ( x  d) 2
7
2

x x d
7 ( x  d)  2 x
x
7
d  4.097 d
72
21-8
How to Solve Problems Asking
“Where is Electric Potential = 0?”
+X
Region I
Region II
+ charge
Region III
- charge
This is similar to finding zero electric field, but V is a scalar, not vector!
 The total electric potential is the sum of the individual potentials from each charge.
(Principle of Superposition)
 The sign of an individual contribution is always the same as the sign of charge.
 The magnitude of potential depends on the value of charge and inverse distance to the charge –
not squared in this case.
 If you get close enough to one of the charges, the inverse distance dependence will make the
potential from that charge much larger (in magnitude) than the potentials from all of the other
charges.
 If you get far enough away from all the charges, the inverse distance term is about the same for
all, so the relative sizes of the electric potentials from each charge will be determined by the
relative sizes of the charges.
 On any line or curve segment, not passing through a charge, where the electric potential is
negative at one end and positive at the other, there will be at least one point on that segment
where the electric potential is zero.
 The electric potential can be zero at a point not at infinity only if there are at least two charges in
the problem, one positive and one negative. In that case, there will be infinitely many points not
at infinity where the electric potential is zero.
21-9
How to Solve Problems Asking
“Where is Electric Potential = 0?”
+X
Region I
Region II
+7 C
Region III
-4 C
Region I:
Every point is closer to +7C than to –4C, so net V is
always positive.
No points where V = 0.
Region II:
On the line from +7C to –4C, V switches sign.
There is a point where V = 0.
Region III:
Close to –4C, net V is negative.
Far from –4C, net V is positive.
There is a point where V = 0.
21-10
How to Solve Problems Asking
“Where is Electric Potential = 0?”
+X
Region I
Region II
+7 C
Region III
-4 C
Region II
1 7
1
4

0
4   0 x 4   0 (d  x )
Region III
1 7
1
4

0
4   0 x 4   0 ( x  d)
7
4

x (d  x )
7 (d  x )  4 x
7
x
d  0.6364 d
74
7
4

x ( x  d)
7( x  d )  4 x
7
x
d  2.333 d
74
21-11
Class #21
Take-Away Concepts
To find points where E = 0, consider the following:
1. Consider N+1 regions for N charges on a line.
2. Superposition Principle: Add E from each charge.
3. E points away from + charge, toward – charge.
4. E depends on charge value and inverse distance squared.
5. Very close to a charge, E from that charge dominates.
6. Far away right or left, all distances are about the same.
7. Look for reversal of total E direction in a region.
Finding where V = 0 is similar but easier, since V is a scalar.
Look for regions where V changes sign.
21-12
Class #21
Problems of the Day
___1. In the example problem on slides 7 & 8, suppose we change
the +7 C charge to –14 C and change the –4 C charge to +8 C.
What effect will that have on the point where the electric field is
zero?
A.
B.
C.
D.
E.
F.
There will be no change.
The point will move to the right.
The point will move to the left but stay in Region III.
The point will switch into Region II.
The point will switch into Region I.
There will be no point where the electric field is zero.
21-13
Class #21
Problems of the Day
2. In the diagram below, the red charge is at position X = 0 and
the blue charge is at X = d. Find the location of zero electric field.
If there is no location, explain why.
Region I
Region II
+5 C
d
Region III
+3 C
21-14
Class #21
Problems of the Day
3. In the diagram below, the red charge is at position X = 0 and
the blue charge is at X = d. Find a location of zero electric
potential not at infinity, assuming electric potential is zero at
infinity. If there is no location, explain why.
Region I
Region II
+5 C
d
Region III
+3 C
21-15
Activity #21
Electric Fields II
(Pencil and Paper Activity)
Objective of the Activity:
1.
2.
3.
Think about the relationship of force and field.
Finding points where E = 0.
Introduction to electric dipoles.
21-16