Download Lecture 10 ppt version

EXERCISES
Try roughly plotting the
potential along the axis for
some of the pairs
Exercises on sheet similar to this
True
True or False?
False
Electric Potential
1.
Electric potential is a scalar quantity.
2.
The potential at some point due to several point
charges is the sum of the potentials due to each charge
True
separately.
3.
Electric field lines always point towards regions of
lower potential.
4.
The value of the electric potential can be chosen to be
zero at any convenient point.
True
True
True
Capacitance
Chapter 23
23.1
Electrostatic potential energy
23.2
Capacitors and Capacitance
23.3
Capacitors in parallel and series
23.4
The storage of electrical energy
The uses of capacitors
Energy can be stored as
potential energy in an electric
field, using a capacitor.
The energy for the electronic flash of a camera is
stored in a capacitor in the flash unit
Capacitors are used in large numbers in
common electronic devices such as TV sets.
NB In this chapter the potential energy is defined in terms
of the work done by an external force (ie W = qV) in bringing
charges in from infinity.
See question A2 on Sheet 5 - a version of this one -
More on Potential Energy
When we bring a point charge q from far away to a region
where there are other charges, we must do work (qV), which
is stored as electrostatic potential energy. For a system of
charges this is the total work needed to assemble the
charges.
When positive charge is placed on an isolated conductor, the
potential, V, of the conductor increases.
The ratio of the charge to the potential is called the
capacitance of the conductor.
Capacitance
A capacitor is made up of
two isolated conductors of
any shape…..called plates
(whatever their shape).
Conductors are
equipotential surfaces
with potential
difference between
plates, known as V
(rather than V). The
charge on the plates,
q, and V are
proportional to one
another
q = CV
Capacitance C is the proportionality constant. It is dependent only on the
geometry of the plates, and not on their charge or potential
Charging a capacitor
The capacitance is a measure of how much charge
must be put on the plates to produce a certain
potential difference between them. The greater the
capacitance the more charge is required.
In the circuit the battery maintains a potential
difference between its terminals, with positive
terminal at higher potential.
When switch is closed, electrons move through
electric field along wire, towards plate l, and away
from plate h to the positive terminal.
Plate h and positive terminal are at same potential
• No electric field in wire
Plate l and negative terminal are at same potential
• Capacitor is fully charged
CHECKPOINT: Does the capacitance C of a capacitor
A. increase,
B. decrease, or
C. remain the same
(a) when the charge q on it is doubled?
(b) when the potential difference V across it is
doubled?
Calculating the capacitance
We always need to know the geometry of a
capacitor, but once we do, then we can
follow this plan:
(1) Assume a charge q on the plates
(2) Calculate the electric field E between the
plates in terms of q, using Gauss’ Law
(3) Knowing E, calculate the potential
difference V between the plates
(4) Calculate C from C = q/V
First we can make
some simplifying
assumptions
Calculating the electric field
We use Gauss’ Law
qenc
 E  d A  0
We can choose a Gaussian surface such
that E will have uniform magnitude on it,
and the vectors E and dA will be parallel.
Gauss’ Law then reduces to
E  q / A 0   /  0
where A is the area of the positive plate,
completely enclosed by the Gaussian surface
Gauss’ Law and two parallel plates oppositely charged
E   / 0
Calculating the potential difference
The potential difference between the plates of
a capacitor is related to the field E by
f
V   E  d s
i
In general we can choose a path that follows
an electric field line, from the negative to the
positive plate. For this path E and ds will have
opposite directions, so we can rewrite V as
Now we can find the
capacitance of some
particular cases

V   Eds

start on negative plate
1. A parallel-plate capacitor
(1) q
(2) Gauss’ Law  E
(3) Potential difference V
(4) C = q/V
0 A
q
q
q
q
C 



q
V Ed  d
d
d
0
0 A
C
0 A
The capacitance depends only on the geometry, ie the
area and the separation of the plates.
d
C increases with increase in area and decreases with
increase in separation.
2. A cylindrical capacitor
Ch 23 Problem 41
Find the capacitance of a long cylindrical conductor
with radius a, and linear charge density λ, surrounded
by a coaxial cylindrical conducting shell with inner
radius b, and linear charge density –λ.
(1) q
This will be done fully in
Wednesday Workshop
(2) Gauss’ Law  E
(3) Potential difference V
(4) C = q/V
2. A cylindrical capacitor
Problem 23-41
Choose the Gaussian
surface to be a cylinder of
length l, and radius r,
between the plates.
Find E, then integrate along
path (green) to find V.
From C = q/V we find:
L
C  20
ln( b / a)
(1) q
(2) Gauss’ Law  E
(3) E  V
(4) C = q/V
Again, the capacitance depends only on the geometry, in this case L, b and a.
3. A spherical capacitor
Example 23-42
ab
C  40
ba
(1) q
(2) Gauss’ Law  E
(3) E  V
(4) C = q/V
4. An isolated sphere C  40 R
NB all formulae derived have
factor 0 x length dimensions
Energy stored in an electric field
When a small amount of positive charge dq is
moved from the negative conductor to the
positive conductor, its potential energy is
increased by
dU = V dq
where V is the potential difference between
the conductors.
variables
The work needed to charge a
capacitor is the integral of V dq
from the initial charge of q = 0 to
the final charge of q = Q.
f
Q
Q
0
0
W   dW   Vdq  
i
Energy stored
in a capacitor:
2
q
Q
dq 
C
2C
1 Q2 1
1
U
 QV  CV 2
2 C
2
2
Energy density
The potential energy of a charged
capacitor can be thought of as being
stored by the electric field…..
..or the work required to charge the
capacitor can be thought of as the
work required to create the electric
field.
eg For a parallel-plate capacitor,
V = E d, (d is the plate separation)
C = 0A/d
1
U  CV 2
2
1
U   0 E 2 ( Ad )
2
(23.7)
True A
True or False?
False F
1.
Near an isolated, uniformly charged plane, the magnitude of the
electric field decreases with the first power of the distance
False
from the plane.
2.
Outside an isolated, uniformly charged spherical shell, the
magnitude of the electric field decreases with the square of
the distance from the centre of the shell
3.
Inside the cavity of an isolated, uniformly charged spherical
shell, the magnitude of the electric field is everywhere zero.
True
4.
Inside an isolated, infinitely long, uniformly charged cylindrical
shell, the magnitude of the electric field is everywhere zero.
True
True
MID-SEMESTER TEST
PHYS1022 will be on Wednesday 10 November at 11 am
It will consist of two questions taken from Wolfson chapters 20-23.
Please prepare one A4 sheet of revision notes which you may bring with you.
This will be handed in with your answers but will not be marked.
The test is worth 10% of your final mark for this course.
The main purpose of the test is to assess your progress and provide feedback.