# ELECTROSTATICS II - Form 3 Physics Notes

## Charge Distribution on the Surface of a Conductor

• The quantity of charge per unit area of the surface of a conductor is called charge density.
• The charge distribution on a conductor depends on the shape of the conductor.
• Generally, the charge concentration on a spherical conductor is uniform while that on a sharp point is high.
• The high charge concentration at sharp points makes it easier to gain or lose charges.
• The effects of high charge concentration at sharp points can be seen in the following cases:

### Electric Wind

• When a highly charged sharp point is brought close to a candle flame, the flame is observed to drift away as if there was wind.
• The high charge concentration at the sharp point ionizes the surrounding air producing both positive and negative charges.
• Opposite charges are attracted to the point while similar charges are repelled away from the point blowing away the flame.
• If the point is brought very close, the flame splits into two; one part moves towards the point and the other part away from the point.
• This is because a flame has both positive and negative ions.
• The negative ions are attracted towards the point while the positive ions are repelled away from the point.

### Lightning Arrestors

• When clouds move in the atmosphere, they rub against the air particles and produce a large amount of static charges by friction.
• These charges induce large amounts of the opposite charge on the earth.
• Hence a high potential difference is created between the earth and cloud.
• This makes air to be a charge conductor.
• The opposite charges attract each other and neutralize, causing thunder and lightning.
• Lightning can be very destructive to buildings and other structures.
• Lightning arrestors are used to safeguard such structures.
• It consists of a thick copper plate buried deep under the ground.
• The plate is connected by a thick copper wire to the spikes at the top of the building.
• The arrestor assumes the same charge as the earth. At the spikes, a high charge density builds up and a strong electric field develops between the cloud and the spikes.
• The air around the spikes is ionized. The opposite charges attract each other and neutralize. Excess electrons flow to the ground through the thick copper wire.
• It is for this reason that people are advised not to take shelter under trees when it is raining.

## Applications of Static Charges

### Electrostatic Precipitator

• One of the causes of air pollution globally is increased industrialization.
• Some industries have indeed responded to this challenge by installing electrostatic precipitators which are found within the chimneys.
• An electrostatic precipitator consists of a cylindrical metal plate fixed along the walls of the chimney and a wire mesh suspended through the middle.
• The plate is charged positively by connecting it to a high voltage, approximately 50,000V and the wire mesh charged negatively.
• As a result, a strong electric field exists between the plate and the wire mesh.
• The ionized pollutant particles get attracted; some to the plate and others to the wire mesh.
• The deposits are removed occasionally. The same principle is used in fingerprinting and photocopying.

### Spray Painting

• The nozzle of the spraying can is charged. When spraying, the paint droplets acquire similar charge and spread out finely due to repulsion.
• As the droplets approach a metallic body, they induce opposite charge which then attracts them to the metal surface. This ensures that little paint is used.

## Dangers of Static Charges

• When a liquid flows through a pipe, its molecules rub against each other and against the walls of the pipe and become charged.
• If the liquid is flammable like petrol, it is likely to cause sparks or even explosion.
• This can also happen to fuels when they are packed in plastic containers.
• It is therefore advisable to store fuels and other flammable liquids in metallic containers so that any charges generated can continually leak out.
• This also explains why long chains hang underneath fuel tankers as they move.

## Electric Field

• This is the region around a charged body where its influence (attraction and repulsion) can be felt.
• It is represented lines of force called electric field lines.
• The direction of an electric field is the direction in which a positive charge would move if placed at that point.
• Electric field lines have the following properties:
• Originate from a positive charge and terminate at a negative charge
• Do not cross each other i.e. do not intersect
•  Are parallel at uniform field, close together at strong fields and widely spaced at weaker fields.

### Electric Field Patterns

• The electric field pattern between two charged bodies obeys the law of electrostatics.
• Below are some patterns between charged bodies:

NB: At the neutral point, the resultant effect is zero.

## Capacitors

• A capacitor is a device used for storing charge.
• It consists of two or more metal plates separated by a vacuum or a material medium (insulator). This material is known as a ‘dielectric’.
• Other materials that can be used as a dielectric include air, plastic, glass e.t.c. the symbol of a capacitor is shown below:
• There are three main types of capacitors namely
• Paper capacitors,
• Electrolytic capacitors and
• Variable capacitors.
• Others include plastic, ceramic and mica capacitors.

### Charging a Capacitor

#### Experiment: To Charge a Capacitor

Apparatus : Uncharged capacitor of 500μF, 5.0V power supply, rheostat, voltmeter, milliammeter, switch, connecting wires and a stop watch.

Procedure

1. Set up the apparatus as shown above.
2. Close the switch and record the values of current, I at various time intervals. Tabulate your values in the table below:
 Time, t(s) 0 10 20 30 40 50 60 70 Current, I(mA) It (mAs)
3. Plot a graph of current, I against time, t
4. Plot a graph of It against time.

Observations

• The charging current is initially high but gradually reduces to zero. A graph of current, I against time appears as shown below:
• The charging current drops to zero when the capacitor is fully charged. As the p.d. across the capacitor increases the charge in the capacitor also increases up to a certain value.
• When the capacitor is fully charged, the p.d across the capacitor will be equals the p.d of the source.
• A graph of p.d across the capacitor against time is exponential. A graph of It against time is also exponential.

NB The product It represents the amount of charge in the capacitor.

### Discharging a Capacitor

#### Experiment: To Discharge a Capacitor

Apparatus : A charged capacitor, resistor, galvanometer, switch and connecting wires.

Procedure

1. Set up the apparatus as shown above.
2. Close the switch and record the values of current at various time intervals in the table below.
 Time, t(s) 0 10 20 30 40 50 60 70 Current, I(mA)
3. Plot a graph of current, I against time, t.

Observations

• The value of current is seen to reduce from maximum value to zero when the capacitor is fully discharged.
• The galvanometer deflects but in the opposite direction to that during charging.
• During discharging, the p.d. across the capacitor reduces to zero when the capacitor is fully discharged.
• The graphs below show the variation between current, I and time, t and between the p.d across the capacitor and time, t.
• A graph of charge in the capacitor, Q against time, t during discharging also appears like that of p.d against time i.e. p.d across the capacitor is directly proportional to the charge stored.

### Capacitance

• Capacitance of a capacitor is defined as the measure of the charge stored by the capacitor per unit voltage;
C = Q/V
Hence Q = CV
Recall: Q = It
Therefore Q= CV = It
• The SI Unit of capacitance is the farad, F.
• A farad is the capacitance of a body if a charge of one coulomb raises its potential by one volt.
i.e. 1 μF = 10-6 F
1 nF = 10-9 F
1 pF = 10-12 F

#### Factors Affecting Capacitance of a Capacitor

• The capacitance of a parallel plate capacitor depends on three factors, namely:
• Area of overlap of the plates, A
• Distance of separation, d between the plates
• Nature of the dielectric material

#### Experiment: To Investigate the Factors Affecting Capacitance

Apparatus: 2 aluminium plates, K and L of dimensions 25cm * 25cm , Insulating polythene support , uncharged electroscope , Glass plate , earthing wireand a free wire.

Procedure

• Fix the plates on the insulating support so that they stand parallel and close to each other as shown above.
• Charge plate K to a high voltage and then connect it to the uncharged electroscope. Earth the second plate, L.
• While keeping the area of overlap, A the same vary the distance of separation, d and observe the leaf divergence.
• While keeping the distance of separation, d constant vary the area of overlap, A and observe the leaf divergence.
• While keeping both the area of overlap and the distance of separation, d constant introduce the glass plate between the plates of the capacitor and observe what happens to the leaf.

Observations

• When the distance of separation is increased the leaf divergence also increased.
• When the area of overlap is increased the leaf divergence decreased.
• When the glass plate is introduced between the plates, the leaf divergence increased.
Note that the leaf divergence here is a measure of the potential, V of plate K.
Hence the larger the divergence the greater the potential and thus the lower the capacitance ( since C = Q/V, but Q is constant).

Conclusion

• From the above observations, it follows that the capacitance is directly proportional to the area of overlap between the plates and inversely proportional to the distance of separation.
• It also depends on the nature of the dielectric material.
C ∝ A/d
C = εA/d where ε is a constant called permittivity of the dielectric material (epsilon).
• If between the plates is a vacuum, then ε = εo , known as epsilon nought and is given by 6.85 × 10-12 Fm-1.
Hence C = εoA/d

Example 4.1

1. How much charge is stored by a 300μF capacitor charged up to 12V? give your answer in (a) μC (b) C
(ans 3600μC/0.0036C)
Solution
1. Q= CV = 300×12 =3600μC
2. 3600 × 10-6 =0.0036C
2. What is the average current that flows when a 720μF capacitor is charged to 2V in 0.03s?
(ans 0.24A)
Solution
Q = CV =It
I= 720 × 10-6 × 2/0.03 =0.24A.
3. Find the separation distance between two plates if the capacitance between them is 1.0 × 10-12 C and the enclosed area is 5.0 cm2. Take εo = 6.85× 10-12 Fm-1 .
(ans d = 1.425 × 10-4 m)
Solution
C = εA/d
d = 6.85 × 10-12 × 5.0 × 10-4/1.0 × 10-12
= 1.425 × 10-4 m

### Arrangement of Capacitors

#### Series Arrangement

• Consider three capacitors; C1, C2 and C3 arranged as shown below:

Recall V = V1 + V2 + V3 and Q = CV
• When capacitors are connected in series, the charged stored in them is the same and equals the charge in the circuit.
i.e. Q = Q1 = Q2 =Q3
Therefore V1 = Q/C1 , V2 = Q/C2 , and V3 = Q/C3
V = Q/C1 + Q/C2 + Q/C3
Dividing through by Q, we obtain V/Q = 1/C1 + 1/C2 + 1/C3
Since V/Q = 1/C
1/C = 1/C1 + 1/C2 + 1/C3
Where C is the combined capacitance.
• In a special case of two capacitors in series, the effective/combined capacitance ,
C = C1C2/(C 1 + C 2 ).

#### Capacitors in Parallel

• When capacitors are arranged in parallel, the potential drop across each of them is the same.

Q1 = C1V, Q2 = C2V, Q3 = C3V
• The total charge, Q = Q1 + Q2 + Q3
Q = C1V + C2V + C3V = V(C1 + C2 + C3 )
Dividing through by V, we obtain Q/V = C1 + C2 + C3
Since C = Q/V,
C = C1 + C2 + C3
• Hence the combined capacitance for capacitors in parallel is the sum of their capacitance.

Example 4.2

1. In the circuit below, calculate:
1. The effective capacitance of the capacitors
2. The charge on each capacitor
3. The p.d across the plates of each capacitor

Solution
a. C = (12× 24)/(12 + 24) =8μF
b. Q1 = Q2 = CV = 8 × 6 = 48μC
c. V 1 = 48/12 = 4V, V2 = 48/24 = 2V
2. The figure below shows an arrangement of capacitors connected to a 2V d.c supply.
Determine:
1. The combined capacitance of the arrangement
2. The total charge in the circuit
( ans 0.7778μF,3.778μC)
solution
a. CBD =(3×3)/(3+3) = 1.5μF
CAE = 2 + 1.5 = 3.5μF
C = (3.5×1)/(3.5+1) = 0.7778μF
b. Q = CV = 0.7778 × 2 = 3.778μC.

Assignment 4.1

• The figure below shows part of a circuit connecting 3 capacitors. Determine the effective capacitance across AC.

### Energy Stored by a Capacitor

• During charging, the addition of electrons to the negatively charged plate involves doing work against the repulsive force.
• Also the removal of electrons from the positively charged plate involves doing some work against the attractive force.
• This work done is stored in the capacitor in the form of electrical potential energy.
• This energy may be converted to heat, light or other forms.
• A graph of p.d, V against charge, Q is a straight line through the origin whose gradient gives the capacitance of the capacitor.
• The area under this graph is equal to the work done or energy stored in the capacitor.
i.e. E = ½QV but Q = CV
Hence E = ½CV2 =Q2/2C

Example 4.3

1. The figure below shows two capacitors connected to a 12V supply

Determine:
1. The effective capacitance of the circuit
2. Charge on each capacitor
3. Energy stored in the combination
(ans. 18μF, 72μC, 5.46 × 10-3 J)
a.12+6 = 18μF
b. Q1= 12 × 12 = 144μC
c. E= ½ CV2 = ½ × 18×10-6 × 122 = 5.46 × 10-3 J
2. In the figure below, calculate the energy stored in the combined capacitor.

{ans. 5.4×10-6 )
C = 2×3 /2+3 =1.2μF
E = ½ ×1.2 ×10-6 ×22 = 5.4 × 10-6 J

### Application of Capacitors

1. Rectification (smoothing circuits)
• In the conversion of alternating current to direct current using diodes, a capacitor is used to maintain a high d.c. voltage.
• This is called smoothing or rectification.
2.  Reduction of sparking in the induction coil
• A capacitor is included in the primary circuit of the induction coil to reduce sparking.
3. In tuning circuits
• A variable capacitor is connected in parallel to an inductor in the tuning circuit of a radio receiver.
• When the capacitance of the variable capacitor is varied, the electrical oscillations between the capacitor and the inductor changes.
• If the frequency of oscillations is equal to the frequency of the radio signal at the aerial of the radio, that signal is received.
4. In delay circuits
• Capacitors are used in delay circuits designed to give intermittent flow of current in car indicators.
5. In camera flash
• A capacitor in the flash circuit of a camera is charged by the cell in the circuit. When in use, the capacitor discharges instantly to flash.

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