Transcript Potential and Field

Ch – 30 Potential and Field
Learning Objectives – Ch 30
• To establish the relationship between and V.
• To learn more about the properties of a
conductor in electrostatic equilibrium.
• To introduce batteries as a practical source of
potential difference.
• To find the connection between current and
potential difference for a conductor.
• To find the connection between charge and
potential difference for a capacitor.
• To analyze simple capacitor circuits.
Finding potential (V) from field (E)
-ΔU =
ΔV = ΔU/q, E =F/q , therefore
If you want to find a value for Vf , instead of ∆V, you
must specify a position, si , where Vi =0. This position is
often at infinity.
Finding E field from potential (V)
Es = - dV/ds
Given this graph of V, make a graph of Ex
Assume the equation of the parabola is of
the form y = Kx2 where K is a constant
Answer
E = -dV/dx
1500
1000
500
0
0
E (V/m)
for parabola K =
50,000 V/m2
E0-2cm =(-)100,000x
E2-4cm = 0
E4-6cm = (-)-20V/.02m
=1000V/m
1
2
3
4
-500
5
6
7
E (V/m)
-1000
-1500
-2000
-2500
x (cm)
Given this graph of Ex , make a
graph of V (V = 0 at x = 0)
Answer
∆V = area under the
curve (V0 = 0 at x=0)
∆ V1-3 = (-)-200V x .02m
= +4 V
∆ V3-4 = ∆ V1-3 = +4V
5
4
4
3
3
V
V
∆ V4-6 = (-) 200V x .02m
= -4 V
2
..
2
1
1
0
0
1
2
3
4
x (cm)
5
6
7
A point charge of + 5/9nC is
located at the origin
• Determine the values of x at which the
potential is 100, 200, 300, 400, 500V.
• Graph V vs x along an x axis with the
charge at the origin.
• Describe E, the electric field on both sides
of the point charge (e.g.positive, negative,
constant, increasing, decreasing).
Graph of V vs x
600
500
400
V
V
x(cm)
100 ± 5.0
200 ± 2.5
300 ± 1.67
400 ± 1.25
500 ± 1.0
300
200
100
0
-6
-4
-2
0
2
4
6
x (cm)
For values of x > 0, E is positive and decreasing with increasing value of
x
For values of x < 0, E is negative and decreasing with increasing values
of |x|
Geometry of Potential and Field
• The direction of the electric field is
perpendicular to the equipotential surfaces
• E always points in the direction of
decreasing potential
• Field strength (magnitude) of E is inversely
proportional to the spacing ∆s between
equipotential surfaces
Which set of equipotential surfaces
is valid for the electric field shown?
Answer
• Answer is c
• Field strength
(magnitude) of E is
inversely proportional
to the spacing ∆s
between equipotential
surfaces
Kirchoff’s Loop Law
• The sum of all
potential differences
encountered while
moving around closed
path is zero
• This is a result of the
conservation of
energy for a
conservative force
Conductor in electrostatic
equilibrium
• For a sphere of
charge Q, outside the
conductor (r >R), E =
kQ/r2 with the
maximum field
strength at the
surface of the sphere
E = kQ/R2
• Inside the sphere,
E=0.
Conductor in electrostatic
equilibrium
• To find ∆V between 2
points outside the
sphere, integrate E
along a line between
the two points.
• At the surface of the
sphere, there is a
nonzero potential.
Conductor in electrostatic
equilibrium
• Inside, since E = 0,
∆V =0, which means
the potential inside is
constant
• When a conductor is
in electrostatic
equilibrium, the entire
conductor has the
same potential, not
necessarily the same
charge
Conductor in electrostatic
equilibrium
• At the surface of the
conductor
• Same charge and same
potential here .
• Same potential, but
different charge and
different electric field
here
Connecting Potential and Current
• I = AJ = A σ E
• In the battery shown:
Ewire = |∆Vwire |
L
We can derive:
I = ∆Vwire
R
Where R = L / A σ
R = ρL / A
R is a property of a specific
wire, depending on the
material, length and area.
Current Ranking Task
• Rank the currents
I1–I5 at the five
labeled points in this
figure, from greatest
to least. Explain.
Fat and Skinny
• Which current is
greater? Both wires
are made of the same
material.
• Explain.
2 conductive rods
• Two conductive rods
have been connected
to a 6 V battery for “a
long time.” What are
the values of:
• ∆V12 _______
• ∆ V34 _______
• ∆ V23_______
Equipotential Map
• Compare the field
strengths Ea and Eb.
Are they equal, or is one
larger than the other?
Explain.
• Compare the field
strengths Ec and Ed.
Are they equal, or is one
larger than the other?
Explain.
• Draw the electric field
vectors at points a
through e.
Answers
• Electric field strength
is the gradient of the
potential
• Ea > Eb
• Ec >Ed
E field vectors and potential
• Is the potential at point 1
greater than, less than or
equal to the potential at
point 2? Explain.
• Determine a value of
∆V12 (1 is initial position,
2 is final position).
• Draw a series of
equipotential surfaces
spaced every 5 V.
Answer
• E points in direction of
decreasing potential
so V1 > V2
• |∆V12 |= 200 V/m x 10
cm = 20 V and since it
is decreasing:
∆V12 = -20V
• Arbitrary assumption
that V2 = 0 for
equipotentials
0V
10V
20V
What is the Value for E?
• Assume V is a linear
function as in a
capacitor.
• Draw an arrow on the
figure showing
direction.
Answer
• E = 1000 V/m
Answer