Module 4
Energy and Potential
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COMPLEX NUMBERS AND QUADRATIC EQUATIONS, Exercises of Engineering Mathematics
This file contains many complicated examples of quadratic and complex equations with a detailed solution and explanation.
Typology: Exercises
2022/2023
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Module 4
Energy and Potential
4.1 Energy to move a point
charge through a Field
• Force on Q due to an electric field
• Differential work done by an external
source moving Q
• Work required to move a charge a finite
distance
F
E
QE
dW QE dL
Example
E x y( )
y
x
2
Q 2 A
.
.
1
B
1
0
1
Path:^ x
2
y
2
1 z 1
Calculate the work to cary the charge from point B to point A.
W Q
B 0
A 0
E x y( ) x 0
d Q
B 1
A 1
E x y( ) y 1
d Q
B 2
A 2
E x y( ) z 2
d
Plug path in for x and y in E(x,y)
W Q
B 0
A 0
E 0 1 x x
2
0
d Q
B 1
A 1
E 1 y y
2
0
1
d Q
B 2
A 2
E 0 0( ) z 2
d W 0.
Example
- Same amount of work with a different path
- Line integrals are path independent
E x y( )
y
x
2
Q 2 A
.
.
1
B
1
0
1
Path:^ y 3 ( x 1 ) z 1
(straight line)
Calculate the work to cary the charge from point B to point A.
W Q
B 0
A 0
E x y( ) x 0
d Q
B 1
A 1
E x y( ) y 1
d Q
B 2
A 2
E x y( ) z 2
d
Plug path in for x and y in E(x,y)
W Q
B 0
A 0
E 0[ 3 ( x 1 )] x 0
d Q
B 1
A 1
E y
y
3
1 0
1
d Q
B 2
A 2
E 0 0( ) z 2
d W 0.
4.3 Potential
• Measure potential difference between a
point and something which has zero
potential “ground”
V
AB
V
A
V
B
Example – D4.
E x y( z)
6x
2
6y
4
a) Find Vmn
M
2
6
1
N
3
3
2
V MN
N 0
M 0
6x x
2
d
N 1
M 1
6y y
d
N 2
M 2
4 z
d V MN
139
b) Find Vm if V=0 at Q(4,-2,-35)
Q
4
2
35
V M
Q 0
M 0
6x x
2
d
Q 1
M 1
6y y
d
Q 2
M 2
4 z
d V M
120
c) Find Vn if V=2 at P(1, 2, -4)
P
1
2
4
V N
P 0
N 0
6x x
2
d
P 1
N 1
6y y
d
P 2
N 2
4 z
d 2 V N
19
Example – D4.
Q 15 10
9 P 1
2
3
1
0
8.85 10
12
Q is located at the origin
a) Find V1 if V=0 at (6,5,4)
P 0
6
5
4
V 1
Q
4 0
1
P 1
1
P 0
V 1
20.
b) Find V1 if V=0 at infinity
V 1
Q
4 0
1
P 1
V 1
36.
c) Find V1 if V=5 at (2,0,4)
P 5
2
0
4
V 1
Q
4 0
1
P 1
1
P 5
5 V 1
10.
Potential field of single point
charge
Q
A
|r - r1|
Move A
from infinity
V r( )
Q
r r
Potential due to n point charges
Continue adding charges
V r( )
Q
4 0
r r
Q
4 0
r r
....
Qn
4 0
r r n
V r( )
n
m
Qm
r r
m
Potential as point charges become
infinite
Volume of charge
Line of charge
Surface of charge
V r( ) v prime
v
r prime
4 0
r r prime
d
V r( ) L prime
L
r prime
4 0
r r prime
d
V r( ) S prime
S
r prime
4 0
r r prime
d
Conservative field
No work is done (energy is conserved) around a
closed path
KVL is an application of this
Potential gradient Relationship between
potential and electric field intensity
Two characteristics of relationship:
- The magnitude of the electric field intensity is given
by the maximum value of the rate of change of potential
with distance
- This maximum value is obtained when the direction
of E is opposite to the direction in which the potential is
increasing the most rapidly
V = -
d E dL
Gradients in different coordinate
systems
The following equations are found on page 104 and
inside the back cover of the text:
Cartesian
Cylindrical
Spherical
gradV
V
x
a
x
V
y
a
y
V
z
a
z
gradV
V
a
V
a
V
z
a
z
gradV
V
r
a r
1
r
V
a
1
r sin ^
V
a