Transcript Equipotential Lines

-Potential Energy of Multiple
Charges
-Finding the Electric Field from
the Electric Potential
AP Physics C
Mrs. Coyle
The Potential Energy of
Point Charges

Consider two point charges, q1 and q2, separated
by a distance r. The electric potential energy is

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This is the energy of the system, not the
energy of just q1 or q2.
Note that the potential energy of two charged
particles approaches zero as r  
Assume the sphere is a point charge.
Apply conservation of energy.
Ki + Ui = Kf + Uf
Ans: 1.86 x 107 m/s
Potential Energy,U, of Multiple
Charges
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If the two charges are the same sign, U is
positive and work must be done to bring the
charges together
If the two charges have opposite signs, U is
negative and work is done to keep the
charges apart
U with Multiple Charges, final
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If there are more than
two charges, then find
U for each pair of
charges and add them
For three charges:
 q1q2 q1q3 q2q3 
U  ke 



r
r
r
13
23 
 12

The result is
independent of the
order of the charges
Finding E From V

Assume, to start, that E has only an x
component
dV
Ex  
dx

Similar statements would apply to the y and z
components
Equipotential surfaces must always be
perpendicular to the electric field lines passing
through them

E and V for an Infinite Sheet of
Charge
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The equipotential lines
are the dashed blue
lines
The electric field lines
are the brown lines
The equipotential lines
are everywhere
perpendicular to the
field lines
E and V for a Point Charge

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
The equipotential lines
are the dashed blue
lines
The electric field lines
are the brown lines
The equipotential lines
are everywhere
perpendicular to the
field lines
E and V for a Dipole

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The equipotential lines
are the dashed blue
lines
The electric field lines
are the brown lines
The equipotential lines
are everywhere
perpendicular to the
field lines
Equipotential Lines
Simulation of Field with Equipotential Lines
http://glencoe.mcgrawhill.com/sites/0078458137/student_view0/chapter21/elec
tric_fields_applet.html
Electric Field from Potential,
General
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In general, the electric potential is a function
of all three dimensions
Given V (x, y, z) you can find Ex, Ey and Ez as
partial derivatives
V
Ex  
x
V
Ey  
y
V
Ez  
z
Why are equipotentials always
perpendicular to the electric field
lines?
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When a test charge has a displacement, ds,
along an equipotential surface dV= 0
dV= -E·ds=0
So E must be perpendicular to to the
displacement along the equipotential surface.
Note that no work is done to move a test
charge along an equipotential surface.
Ex 25.4 Electric Potential and Electric
Field Due to a Dipole
Ex 25.4 Electric Potential and Electric
Field Due to a Dipole
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An electric dipole consists of two charges of
equal magnitude and opposite sign separated
by a distance 2a. The dipole is along the xaxis and is centered at the origin.
a) Calculate the electric potential at P.
b) Calculate V ans Ex at a point far from the
dipole.
c)Calculate V and Ex if point P is located
anywhere between the two charges.
a) Calculate the electric potential at P.
b) Calculate V ans Ex at a point far
from the dipole.
c)Calculate V and Ex if point P is
located anywhere between the two
charges.
If point P is located to the left of the
negative charge, what would be the
potential?
Quick Quiz 25.8
In a certain region of space, the electric potential is zero everywhere
along the x axis. From this we can conclude that the x component of the
electric field in this region is
(a) zero
(b) in the x direction
(c) in the –x direction.
Quick Quiz 25.8
Answer: (a). If the potential is constant (zero in this case), its derivative
along this direction is zero.
Quick Quiz 25.9
Answer: (b). If the electric field is zero, there is no change in the electric
potential and it must be constant. This constant value could be zero but
does not have to be zero.
Quick Quiz 25.9
In a certain region of space, the electric field is zero. From this we can
conclude that the electric potential in this region is
(a) zero
(b) constant
(c) positive
(d) negative