Work done by Gravity
z
z
z
Gravitational force near the
surface of the earth points down.
Apple thrown up loses kinetic
energy – we say that the work
done by the gravitational force on
the apple is negative.
The work transforms the kinetic
energy into a positive change of
the potential energy of the fieldapple system
Δ U = U f − U i = −W
Work done by Gravity
z
General equation for work is
z
F points down.
z
When the apple goes up d points
up and
r r
W = F •d
W = − Fd
z
In this case there is a positive
change in the potential energy.
Work done by Gravity
z
When the apple goes down
d points down
W = Fd
z
In this case there is a
negative change in the
potential energy.
Work done by Electric Field
z
z
Electric field near the surface of
the earth points down (ignore
gravity).
When the electron moves up the
work done on an electron by the
electric field is:
W = Fd
z
This results in a decrease in the
potential energy of the fieldelectron system.
Δ U = U f − U i = −W
Electric Potential Energy
z
When electrostatic force acts between charged
particles assign an electric potential energy, U
z
Difference in U of a charge at two different
points, initial i and final f is
ΔU = U f − U i
Electric Potential Energy
z
When the system changes from initial state i to
final state f electrostatic force does work
ΔU = U f − U i = −W
z
z
Electrostatic force is conservative
Work done by force is path independent
z
Work is same for all paths between points i and f
Electric Potential Energy
& Reference Point
Potential energy, U, is a scalar
z Need to choose a reference point where
U =0
z
z Choose
sea level to be zero altitude
z What if we define Denver to be zero altitude?
z Does the difference in altitude change?
z
Choose U =0 at i =∞ for electric potential
Electric Potential Energy
& Reference Point
z
Have several charges
initially at infinity so Ui
=0
z
Move charges close
together to state f
z
W∞ is work done by
force to move particles
together from infinity
ΔU =Uf −Ui =Uf
ΔU = −W
U f = −W∞
Electric Potential Energy
(Exercise)
z
A proton moves from point i to point f in a
uniform electric field.
z
Does the electric field do positive or negative
work on the proton?
Electric Potential Energy
(Exercise)
z
What is the work done by an electric field?
r r
W = F •d
r
r
F = qE
r r
W = qE • d = qEd cosθ
W = qEd cos(180) = − qEd
Negative work
Electric Potential Energy
(Exercise)
z
Does the electric potential energy of the proton
increase or decrease?
ΔU = U f − U i = −W
ΔU = −( − qEd ) = qEd
Increases
Electric Potential
z
Electric potential, V, is defined as the electric
potential energy, U, per unit charge
U
V =
q
z
Potential, V, is characteristic of the electric field
only
z
Unique value at any point in an electric field
Electric Potential
z
Electrostatic potential difference, ∆V, between
points i and f
Ui ΔU
W
ΔV = V f − Vi =
− =
=−
q
q
q
q
Uf
z
V is a scalar and can be +, -, or 0
z
Using reference point of Ui=0 at infinity
W∞
V =−
q
Electric Potential
Important difference between U and V
z Electric Potential Energy, U, is energy of a
charged object in an external E field
z
z Measured
z
in Joules (J)
Electric Potential, V, is property of E field
care if charged object is placed in E
field or not
z Measured in Joules per Coulomb (J/C)
z Doesn’t
Electric Potential Units
z
Define new SI unit for V (volt)
z
1 volt = 1 joule per coulomb, V=J/C
z
Define E field in volts per meter (V/m)
F
E=
q
N ⎛ N ⎞⎛ V ⋅ C ⎞⎛ J ⎞ V
= ⎜ ⎟⎜
⎟=
⎟⎜
C ⎝ C ⎠⎝ J ⎠⎝ N ⋅ m ⎠ m
Equipotential Surface
z
Equipotential surface - all points are at same
electric potential, V
W
ΔV = V f − Vi = −
q
z
W =0 if move between points i and f which
lie on same potential surface
z
True regardless of path taken between points
Equipotential Surfaces
z
For paths I and II
W = −q ΔV = − q (V1 − V1 ) = 0
W = −q ΔV = −q (V3 −V3 ) = 0
z
For paths III and IV
W = − qΔV = − q(V2 − V1 )