Transcript Electric potential energy
Electric Potential
We introduced the concept of potential energy in mechanics Let’s remind to this concept and apply it to introduce -by a force in general electric potential energy We start by revisit the work done on a particle of mass m -by a conservative force such as the gravitational force In general, work done by a force F moving a particle from point r a to r b
W a
b
b a
F d r
x 2
r t
0
b
x 1 In general we have to specify how we get from a to b, e.g., for friction force
However
, for a conservative force such as gravity we remember
F
where is the potential energy
With this we obtain
W a
b b
a
d r b a
d
b
Gravity as an example
Gravitational force derived from
G M r a
Independent of the path between
b
( ) and
a
( ) 0 h Pot. energy depends on h, not how to get there.
Electric potential energy
Can we find a function U =U (r) such that
U r
F
is the force exerted by a point charge q on a test charge q 0 ?
We expect the answer to be yes , due to the similarity between Coulomb force and gravitational force
F gravity
G m m
0
r
2
r
ˆ
G mm
0
r
, /
m
0 Potential Potential energy
F Coulomb
Let’s try In fact we see 1 4 0
F
Coulomb
0
r
2 1 4 0
r r
ˆ
0
dU dr r
ˆ
1 4
0
0
r
2
r
ˆ
simple because of radial symmetry where U(r)=U(r)
We conclude Electric potential energy of electrostatically interacting point charges q and q 0
1 4
0
0
r
U U repulsive potential qq 0 >0 attractive potential qq 0 <0 r r As always, potential defined only up to an arbitrary constant. Expression above uses U(r )=0 as reference point
...
We know already the superposition principle for electric fields and forces,
( )
1 2 3 can we find a net potential energy for q 0 interacting with several point charges? Force exerted on q 0 by charge q 3 at r 3 Force exerted on q 0 by charge q 2 at r 2 Force exerted on q 0 by charge q 1 at r 1 Net force q 0 experiences ( )
U r
1 ( ) 1 4 0
r q q
1 0
r
1 y ( )
U r
2 ( ) 1 4 0
r q q
2 0
r
2 r r 1 q 0 q 1 Note: textbook on p. 785 defines
r i
I prefer to keep r-dependence explicitly visible
r i
r
r 3 -r r 3 q 3
( )
( )
( )
( )
( )
r 2 ( ) ...
U r
x
( )
q
q
0 4 0
r q
1
r
1
r q
2
r
2
r q
3
r
3 ...
q
0 4 0
i
r q i
r i
The last expression answered the question about the potential energy of the charge q 0 due to interaction with the other point charges q1, q2, …, y q 1 r 1 Those point charges q1, q2, …, interact as well.
Each charge with all other charges r q 0 r 3 -r r 3 q 3 r 2 q 2 x If we ask for the total potential energy of the collection of charges we obtain
U
1 4 0
r i q q i j
r j
makes sure that we count each pair only once This is the energy it takes to bring the charges from infinite separation to their respective fixed positions r i
Clicker question
What is the speed of charge q after moving in the field E from the positive to the negative plate.
Neglect gravity.
1)
v
2 / 2)
v
2 / 3)
v
qEd
/ 2
m
4)
v
qEd
/ 4 0
m
5) None of the above + d
Electric potential
Goal: Making the potential energy a specific, test charge independent quantity We are familiar by now with the concept of creating specific quantities, e.g., Force on a test charge
F
qE
Electric field: test charge independent, specific quantity
E
Gravitational potential energy
G Mm
0
r
test mass independent, specific potential
m
0
G M r
Electric potential V
V
U q
0 Specific, test charge independent potential energy.
The SI unit of the potential is volt (V) .
Meaning of a potential difference Point a W a->b work done by electric force during displacement of charge q 0 from a to b.
W a
b
U b
U a
W a
b q
0
U q
0
U b q
0
U a q
0
V b
V a
V a
V b
Voltage of the battery Point b Alternatively we can ask : What is the work an external force, F, has to do to move charge q 0 This force is opposite to the electric force, F el , above.
Hence: from b to a
/
0
1
q
0
a b
Fd r
1
q
0
a b
1
q
0
b a
1
q
0
W
a
b
V
a
V
b
We know these two alternative interpretations already from mechanics
z a F g =-mg b
W a
b
b
a mg dz
mg z b
z a
)
a
z b
)
b
a b F=mg To slowly ( without adding kinetic energy ) move mass from b to a we need an external force acting against gravity
W
b a
mg dz
a
z b
)
b
Relation between electric potential & electric field
From
W a
b
b a
F d r
q
0
b a
E d r F
q E
0 and
W a
b q
0
V a
V b V a
V b
b a
E d r
We obtain the potential difference (voltage) from the path independent line integral taken between points a and b
Let’s calculate the potential of a charged conducting sphere by integrating the E-field
R 1.0
r 0.5
0.0
1.0
0 0.5
1 We start from
V a
point a becomes variable point in distance r
V b
b
E d r a
point b becomes reference point at r
Q
4 0
r
dr
r
2 :=0 1
r
2 r/R 3 4
For r
Q
4 0 for r>R
r
Q
4 0
r Q
4 0
R r
0
dr
R
dr
r
2
Q
4 0
R
0.0
An important application of our “potential of a conducting sphere”- problem
According to our considerations above we find at the surface of the conducting sphere: R
V surface
Q
4 0
R
E surface R
There is a dielectric breakdown field strength, E m , for all insulating materials including air For E>E m air becomes conducting due to discharge From
E m
V m R
max potential of a sphere before discharge in air sets in depends on radius Pinwheel: Electrons from the generator leave the pinwheel at point of small R. This charge collects on adjacent air molecules. Electrostatic repulsion propels the pinwheel
Demonstration: Surface Charge Density
How do we actually measure the charge on the proof plane ?