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Electric Potential
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Electric Potential
• The change in potential energy is the
negative of the work done by the field.
 
U  W   q0 E ds
E
B
B
A
B 

U
VB  VA  V 
   E ds
A
q0
1V = 1J/C
1eV=1.6×10-19J
A
Voltage from a
Sphere
• Ex. What is the electric
potential (voltage)
between the surface of a
sphere with radius 1m
and a point 0.5 m from
the surface if the sphere
has a charge of +4mC?
B
+
A
+ + +
+
+
+
+
+ + +
+
+
+
+
+
Different Fields
• Uniform field
 
V  E  d
• Point Charge
1 1
VB  VA  kq  
 rB rA 
• If starting location (reference) is a infinity,
then
kq
VB 
rB
Superposition
• Using infinity as a reference point, we can
find the net voltage from multiple charges.
qi
V  k
i ri
• Notice that we are adding scalars, not
vectors.
• For the
diagram to the
right what is
the voltage at
the origin
where 0 volts
is at infinity?
+6 mC
3 mC
+6 mC
E Field from
Electric Potential
• The rate at which voltage changes is related
to the strength of the electric field.
dV
Ex  
dx
• Considering all three dimensions we get

 ˆ dV ˆ dV ˆ dV
E  V   i
j
k
dy
dz
 dx



Find the E Field
• What is the electric field at (3m, 2m) for the
following voltage function?
V ( x, y)  x  5xy  3 y
2
2
• Taking the appropriate derivatives gives an
electric field of

E ( x, y )  (2 x  5 y )iˆ  (5 x  6 y ) ˆj
• Evaluating
 that field at (3m,2m) gives
E (3,2)  16iˆ  27 j N / C
FYI
• Using Maple we can
visualize the electric
field and equipotential
lines.
V ( x, y)  x 2  5xy  3 y 2

E ( x, y )  (2 x  5 y )iˆ  (5 x  6 y ) ˆj
Equipotential
Surfaces
• Surfaces which are at the same voltage level.
• Equipotential surfaces intersect electric field lines
perpendicularly.
• The surface of a charged conducting object is at the same
electric potential.
• Any point within a conductor is at the same potential as its
surface.
Continuous Charge
• If charge is distributed over an object, then
dq
V  k
r
• Ex. An electron is placed 5 m on axis from
a uniformly charged ring. The ring is 0.03
m in radius and has a charge per length of 3
mC/m. What is the speed of the electron as
it passes through the loop?
Lightning
• Air ionizes with an
electric field of 75,000
volts per inch.
• Lightning takes the
easiest path from its
perspective.
• Rapidly expanding air
forms a shock wave
called thunder.