EXERCISES

Try roughly plotting the potential along the axis for some of the pairs

Exercises on sheet similar to this

True or False?

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 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 False

True

True

Electric Potential

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

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.

Energy can be stored as potential energy in an electric field, using a capacitor.

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 EnergyWhen 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

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.

Charging a capacitor

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.

• No electric field in wire

• Capacitor is fully charged

Plate h and positive terminal are at same potential

Plate l and negative terminal are at same potential

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

0

encqE d A

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

where A is the area of the positive plate, completely enclosed by the Gaussian surface

00 // AqE

Gauss’ Law and two parallel plates oppositely charged

0/E

Now we can find the capacitance of some particular cases

Calculating the potential difference

The potential difference between the plates of a capacitor is related to the field E by

sdEVf

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

V Eds

start on negative plate

1. A parallel-plate capacitor

The capacitance depends only on the geometry, ie the area and the separation of the plates.

C increases with increase in area and decreases with increase in separation.

d

AC 0

(1) q

(2) Gauss’ Law E

(3) Potential difference V

(4) C = q/V

0

0 0

Aq q q qC

qV Ed dd dA

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

(2) Gauss’ Law E

(3) Potential difference V

(4) C = q/V

2. A cylindrical capacitor

This will be done fully in Wednesday Workshop

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:

Again, the capacitance depends only on the geometry, in this case L, b and a.

)/ln(2 0 ab

LC

(1) q

(2) Gauss’ Law E

(3) E V

(4) C = q/V

2. A cylindrical capacitor Problem 23-41

3. A spherical capacitorExample 23-42

4. An isolated sphere

ab

abC

04

RC 04

(1) q

(2) Gauss’ Law E

(3) E V

(4) C = q/V

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.

2

0 0 2

f Q Q

i

q QW dW Vdq dq

C C

22

2

1

2

1

2

1CVQV

C

QU Energy

stored in a capacitor:

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.

)(2

1 20 AdEU

eg For a parallel-plate capacitor,

V = E d, (d is the plate separation)

C = 0A/d2

2

1CVU

(23.7)

True or False?

1. Near an isolated, uniformly charged plane, the magnitude of the electric field decreases with the first power of the distance 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.

4. Inside an isolated, infinitely long, uniformly charged cylindrical shell, the magnitude of the electric field is everywhere zero.

True

False

True

True

True A False F

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.