Microsoft PowerPoint - chapter23•Potential Energy = U = stored work in a system
•Work = energy put into or taken out of system by forces
•Work done by a (constant) force F :
θcos|||| rFrFW v
r
F
θ
Gravitational Potential Energy Lift a book by hand (Fext) at constant velocity.
F ext
Note that get to define U=0,
typically at the ground.
“internal energy” in Thermo.
Gravitational Potential Energy (cont)
is conserved.
Coulomb force is also a conservative force.
Friction is not a conservative force.
If only conservative forces are acting, then
E Mech
- - - - - - - - - - - - - -
+
Initial position
Final position
Uelec = change in U when moving +q from initial to final position.
fieldextif WWUUU −=+=−=
rEqU v
from initial (i) position to final (f)
position.
Electric Potential Energy Since Coulomb forces are conservative, it means that the change
in potential energy is path independent.
∫ ⋅−= f
i
Electric Potential Energy Positive charge in a constant field
Electric Potential Energy Negative charge in a constant field
Observations • If we need to exert a force to “push” or “pull” against the
field to move the particle to the new position, then U
increases. In other words “we want to move the particle to
the new position” and “the field resists”.
• If we need to exert a force to “hold” the particle so the field
will not move the particle to the new position, then U
decreases. In other words, “the field wants to move the
particle”, and “we resist”.
• Both can be summarized in the following statement:
“If the force exerted by the field opposes the motion of the
particle, the field does negative work and U increases,
otherwise, U decreases”
charges
2
Imagine doing work to move the objects from infinitely far apart
(initial) to the configuration drawn above (final).
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constant over the path!
from infinity to r.
r+Q 1 +Q
Moving Q 2
through the field.
Potential Energy between two point charges
We also need to define the zero point for potential energy.
This is arbitrary, but the convention is U=0 when all charged
objects are infinitely far apart.
r
Potential Energy between two
E. Potential energy vs. distance
E. Potential Energy of a charge
distribution
an integral
r1 r2
distribution
an integral
energy we have to consider all the
fields produced by all the charges qi
on the other(n-1) charges qj
r1 r2
point charges separated by a
distance r.
negative charges? A) Left Plot,
B) Right Plot
energy.
Earlier we found that not only using forces, but also electric fields
was very useful.
2
•Electric Field is the force per unit of charge due to the presence of Q 1
•Electric field from Q 1
is there even if Q 2
is not there.
Electric Field
We find a similarly useful thing with electric potential energy.
r
2
•Electric potential is the electric potential energy per unit of charge due
to the presence of Q 1
•Electric potential from Q 1
is there even if Q 2
is not there.
Electric Potential
+Q 1
a source charge Q 1 .
Units are [Newtons/Coulomb].
a source charge Q 1 .
Units are [Joules/Coulomb] or [Volts].
r r
Electric Potential of a point charge
r
Analogy: Electrical pressure or electrical "height"
Positive charges want to get away from higher voltage towards lower voltage.
Just like a gas wants to move from high to low pressure.
Electric Potential of a charge
distribution
an integral
field produced by the charges qi
Electric potential due to charges qi
∑= i i
∫∫ ⋅−=⋅−== f
i
f
i
constant elevation.
Gravitational potential
elevation I have a
scalar.
constant Voltage (equipotentials).
Voltage, it has an electrical
potential energy [Joules] = qV
•Equi-potential surfaces and field lines are always mutually
perpendicular
and larger values of |E|
E. Field and E. potential
What is Electric Field?
analogous to a steepness
depends which direction you
as a function of position, we can
compute Voltage.
find the change in elevation.
This does not depend on our path taken.
E. Field and E. potential
V i =V(∞)= 0 by our convention.
dz
of Voltage in a given direction. z
z
VE ∇−= rv The Electric field vector “potential gradient”
http://www.falstad.com/vector2de/
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CPS question
Two identical charge, +Q and +Q, are fixed in space. The electric potential (V) at the point X midway between the charges is: A) Zero B) Non-Zero
+Q +Q Point X
CPS Question
Drawn are a set of equipotential lines. Consider the electric field at points A and B. Which of the following statements is true?
0V 10V 20V 30VPoint A
Point B
A) |EA| > |EB| B) |EA| < |EB| C)|EA| = |EB| D)Not enough information given. E) None of the above
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CPS Question
Two charges, +Q and -Q, are fixed in space. The electric field at the point X midway between the charges is: A) Zero B) Non-Zero
+Q -Q Point X
Point P
The magnitude of the electric field at point P is: A) Zero B) Non-Zero
The magnitude of the voltage at point P is: A) Zero B) Non-Zero
All points on conductor must be at the same
electrical potential.
Imagine point a at Voltage V a .
Since E=0 everywhere inside the conductor (no steepness), integral to
point b is always 0. V ab =0.
E. Potential inside conductors