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Circuits (MTE 120) (Spring 2010)
Akrem El-ghazal
http://pami.uwaterloo.ca/~akrem/
University of Waterloo,
Electrical and Computer Engineering Dep.
Electric Potential (Voltage)
Objectives:
1.
2.
3.
4.
To review some aspects of potential energy
To define the electric potential
To compute the electric potential due to a point charge
To compute the electric potential due to a group of point charges
Potential Energy (Review)
In the figure below, if an object moves from position (i) to position (f), work is done. The work is
defined as:
The change of the potential energy ∆ is:
. ∆ If the initial potential energy ( is greater than the final potential energy ( then the object
lost a potential energy and hence the work is negative. In the other words, if the object is
moving along with the gravitational field the work is negative and the change of the potential
energy is:
If the object is moving against the gravitational field the work is positive.
Circuits (MTE 120) (Spring 2010)
Similarly, if we have a test charge q0 moving with electric field as shown in the figure below.
Electric Potential
Definition: The electric potential difference between two points i and f in an electric field is
equal to the difference in electric potential energy per unit charge between the two points.
J
The unit of the electric potential is C Volt
The Electric Potential Due to a Point Charge
Let us derive the electric potential due to a point charge. Imagine that a positive test charge q0
moves from point i to infinity (f).
2
Circuits (MTE 120) (Spring 2010)
Q
+
R
q0
+
dr
i
Initial Potential Energy Ui
f
Final Potential Energy Uf
F
The work done on q0 by the electric force during the displacement is:
. Then, the total work w is calculated by the following integration:
" . . !
!
Since q0 is moving along with the electric force , the change in the electric potential energy is:
" . !
" cos%0 " " . " !
The difference of electric potential is equal to the difference in electric potential energy per
unit charge:
" !
3
Circuits (MTE 120) (Spring 2010)
As we know from the previous chapter" '
!
'
(
)*
, so:
+
- + ,
!
At ∞ the 0
'
+
1
0 '
+
1
'
+
1
1
1
- + / 0
,
!
To be consistent with the equations derived for the electric field and the electric force, replace
with and R with r:
'
Important Note:
+
The sign of V depends on the sign of Q. For a positive Q , V is positive and for a negative Q , V is
negative.
Calculating the Potential from the Field
The potential difference between two points is the line integral of the electric field along the
any path between the two points.
" . 3
Example:
The figure below shows two points i and f in a uniform electric field" .
4
Circuits (MTE 120) (Spring 2010)
(a)- Find the potential difference by moving a positive test charge q0 from i to f along
the path (i-f) shown in figure (a).
(b)- Find the potential difference by moving a positive test charge q0 from i to f along
the path i-c-f shown in figure (b).
Solution:
(a) Path i-f
" 3 cos%0 " 3
" . 3
" 3
" . 3
(b) Path i-c-f
" 3 "
" cos%45 3 " cos%45 3
" . 3
The integral in the above equation is just the length of the line cf.
" cos%45 .
"
sin %45
As we can see the potential difference between two points does NOT depend on the path
connecting them.
Electric Potential Due to a Group of Point Charges
The net electric potential at a point due to a group of n point charges can be calculated using
the superposition principle.
5
Circuits (MTE 120) (Spring 2010)
9
9
:;
:;
8 ' 8
Note: The sign of the charges must be considered in the above equation. The sum in the above
equation is an algebraic sum NOT a vector like the sum that would be used to calculate the
electric field.
qn
q3
r3
r2
rn
V
q2
r1
q1
Example:
What is the electric potential at p located at the center of the square of point charges shown in
the figure below?
q1= 12nc
q2= -24 nc
d
p
d
d
d
q4= 17 nc
q3= 31 nc
Solution:
6
Circuits (MTE 120) (Spring 2010)
*
1.3
;: ,: <: = >% , @ % , * * 0.919 m
2
2
√2
√2
'x F
; , < =
12x 10KI 24x 10KI 31x 10KI 17x 10KI
@ @ @ G 8.99 x 10I J
@
@
M
; , < =
0.919
0.919
0.919
0.919
N 350 7