ELECTROSTATICS (II) PHYSICS SIMPLIFIED NOTES

Chapter Seven

ELECTROSTATICS (II)

Forces between Charged Bodies

• The magnitude of force in between charged bodies depend on two factors and can be illustrated by the use of gold-leaf electroscope.

Quantity of charges

• It was noted earlier that the divergence of the leaf of an, electroscope is proportional to the quantity of charge on the plate and the leaf.
• When a negatively charged rod near the cap of an electroscope the leave deflect
• Without moving the negatively charged rod (maintain a distance, x, between the cap and the rod), introduce another negatively charged rod but ensure that distance x is maintained, the divergence increases

Fig. 7. J: Leaf divergence is proportional to amount of charge

Distance of separation

When a positively charged pith ball is brought gradually close to another positively charged suspended pith ball the deflection of the suspended ball increases.

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If one brings a negatively charged pith ball near the suspended one, the two attract and the deflection increases as the suspended ball is approached.

Electric Field Patterns

• The space around a charged body where the force of attraction or repulsion is felt is called the
  electric field.
• Electric field is represented by lines along which the electrostatic forces act.
• These lines of force are called electric field lines. The direction of an electric field at a point is the direction in which a positively charged particle would move if placed at that point.

Electric fields have the following properties:

(i)    The electric lines of force are directed away from positive charges and towards negative charges

(ii) Unlike charges attract while like charges repel.

• shows the forces of attraction
• and (c) shows forces repulsion

Conclusion

Electric field lines:

(i) are directed away from a positive charge and towards the negative charge.
(ii) do not cross one another.

(iii) are parallel at a uniform field, widely spaced at weak fields, closely arranged at strong fields.

CHARGE DISTRIBUTION ON THE SURFACE OF A CONDUCTOR

To show charge distribution on surfaces of conductors

Different conductors are charged and the charges tested using proof plane at different point

Observations

• For spherical shape, divergence of the leaf is the same for all parts.
• For the pear-shaped conductor, the divergence varies from one part to another with the maximum at, sharp curve.

Explanation and Conclusion

Charge distribution for the sphere is even

For  pear-shaped conductors the charges are more concentrated at the sharp edge

Note:

The pear-shaped body, discharges faster than the spherical shape because of the high charge concentration at the sharp curvature which causes charge leakage.

The charge distribution for cuboids is shown in figure

Charge distribution on hollow conductors

A charged hollow conductor reveals that no charge is found on the inside surface.

The distribution of charge on a hollow conductor can be demonstrated using a cylindrical conductor. The cylindrical conductor is placed on an uncharged electroscope and a charged sphere on an insulating handle is lowered into it without touching

Using a negatively charged sphere equal and opposite charges are induced on inside and outside of the cylinder. The leaf of the electroscope diverges

If the sphere is made to touch the inner wall of the cylinder, the leaf remains diverged

When withdrawn and then tested for charges, is found to contain no charge (neutral).

Charges on Sharp Points

• When a highly charged metal rod is brought close to a Bunsen burner flame, it is observed that the flame is blown away
• If the charge on the wire is positive, the high concentration of positive charges at the sharp point of the wire causes ionization of the surrounding air to produce electrons and positive ions.
• Electrons in air are attracted towards the positive conductor while the heavy positive ions drift towards the flame, forming an electric wind which blow away the flame.
• If the wire is brought very close to the latter, the flame splits into two directions
• In this case, the negative ions in the flame are attracted to the rod, diverting part of the flame towards it.
• At the same time, positive ions are repelled away, diverting part of the flame away.

Lightning Arrestor

Example

A thundercloud which is positively charged on its base hangs over a tall building fitted with a lightning arrestor. Explain the action of the pointed edges of the lightning arrestor in such a situation.

Solution

• The positive charge on the base of the cloud induces a negative charge on the pointed edges of the lightning arrestor.
• Electrons concentrate on these pointed edges and the charge is very dense.
• The surrounding air molecules are ‘ionized’ with the production of both positive and negative ions.
• The negative ions are attracted by the positive charges on the cloud.
• Thus the charge on the base of the cloud is
• This prevents a large build-up of charges which otherwise might result in discharges to the earth in the form of lightning.
• The positive ‘ions’ produced are attracted by the pointed edges and the charge on the arrestor is
• Also even if lightning strikes, the huge tall building electrical charges are conducted through the metal rod of the arrestor to the earth and the building is saved from any damage.

CAPACITORS

• A capacitor is a device used for storing charge.
• It consists of two or more plates separated by either vacuum or a material medium

The material medium can be air, plastic or glass and is known as the ‘dielectric‘. A parallel- plate capacitor is represented by the symbol below.

Types of Capacitors

Capacitors are used in electric circuits for various purposes. Different types have different insulators (dielectric), depending on their uses. There are three main types of capacitors, namely, paper capacitors, electrolytic capacitors and variable capacitors.

Paper Capacitors

Paper capacitors consist of two long strips of metal foil between which are thin strips of paper, which act as the dielectric. The ‘sandwich’ is tightly rolled to form a small cylinder so that the arrangement is essentially parallel-plate capacitor of large surface area, occupying only a small volume, see figure 7.19 (a) and (b).

Fig. 7.19: Paper capacitor

Electrolytic Capacitors

These are made by passing a direct current between aluminium foils with a suitable electrolyte (aluminium borate) soaked in a paper.

When the current is passed for sometime, a very thin film of aluminium oxide is formed on the anode (marked positive). This film is an insulator and therefore acts as the dielectric. Electrolytic capacitors have much higher capacitance than the paper types.

Note:

The positive terminal of the capacitor should be connected to the positive side of the circuit; otherwise the thin film of aluminium oxide will break down.

The maximum working voltage should not exceed the recommended, lest the dielectric layer becomes a conductor.

Variable Air Capacitors

Variable air capacitors consists of fixed metal vanes connected to a metal frame and movable metal vanes joined to the central shaft and turned by a control knob

When the control knob is turned, overlap area of plates varies and so does the capacitance. Variable air capacitors are used in radio receivers for tuning.

Other types of capacitors include the plastic, ceramic and mica capacitors but their construction
and operation is similar to that of a paper capacitor.

Charging and Discharging Capacitors
To charge a capacitor

Procedure

• Set up the circuit as shown in figure 7.22.
• Close the switch and record the values of current at various time intervals. Tabulate your results as shown

• Plot a graph of current against time.
• Plot a graph of It against t.

Observation

The charging current is initially high but gradually reduces to zero, see figure 7.23.

Explanation

• When the capacitor is connected to the battery, negative charges flow from the negative terminal of a battery to plate B of the capacitor connected to it.
• At the same rate, negative charges flow from the other plate A of the capacitor towards the positive terminal of the battery.
• For this reason, equal positive and negative charges appear on the plates and oppose the flow of electrons which causes them. The charging current drops to zero when the capacitor is fully charged.

The graph of charge (it) against t is shown

The graph shows that charge increases with time and becomes a maximum when the capacitor is fully charged.

During charging, potential difference also develops across the plates of the capacitor, see figure 7.25.

(Practical kcse year 2013 and 2014)

As charge increases, the potential difference between the plates also increases.

When the charging current reduces to zero, the potential difference between the plates of the capacitor will be seen to be the same as the battery voltage.

A resistor in charging a capacitor increases time for charging the capacitor

Discharging a charged capacitor

The capacitor is connected through a resistor and the current noted with time

Plot a graph of current I (mA) against time t (s).

Explanation

• The milliameter reading is seen to reduce from a maximum value to a minimum,
• The ammeter deflection occurs in a direction opposite to that during charging.
• During discharging, the charges flow in the opposite direction, from the plate B to A until the positive charges on A are neutralized.
• This goes on for some time until the charge on the plates is zero. When this happens, the capacitor is said to be discharged.
• During discharging, potential difference across the capacitor practically diminishes to zero. Below is the curve showing how potential difference falls.

Capacitance

• The capacitance of a capacitor is a measure of the amount of charge the capacitor can store when connected to a given voltage, and is defined as the charge stored per unit voltage.
• Capacitance C =^Q/_V where Q is the charge in coulombs and V the voltage.
• The SI unit of capacitance is the farad (F).
• One farad (1 F) is the capacitance of a body if a charge of one coulomb raises its potential by one volt.

Note:

One farad is a very large unit of capacitance and in practice, smaller units such as microfarads
(µF), nanofarads (nF) and picofarads (pF) are used.

1 µF= 10^-6F

1 nF = 10^-9F

1 pF = 10^-12F

Factors affecting capacitance of a parallel-plate capacitor

(i)   Area of the plates that are overlapping: An increase in the area of an overlap of the plates decreases the potential difference between the plates, hence capacitance increases.

(ii) Distance between the plates and area overlapping: When the plates are moved closer to each other but not touching, it results in decrease in electric potential between the plates and hence capacitance of the plates is increased.

(iii) Dielectric used between the plates:

When an insulating material medium is used, the potential difference between the plates decreases. A decrease in potential difference shows an increase

It follows that capacitance is directly proportional to the area of overlap and inversely proportional to the distance of separation. It also-depends on the nature of the dielectric.

So, C = ^ƹA/_d where ƹ is a constant dependent on the medium between the plates and is called permittivity of the insulating material. If the plates are in vacuum, the constant is denoted by ƹ_o (epsilon nought) and its value is 8.85 x 10-12 Fm^-1.

Example 1

Two plates of a parallel-plate capacitor are 0.6 mm apart and each has an area of 4 cm². Given that the potential difference between the plates is 100 V, calculate the charge stored in the capacitor. (Take ƹ_o = 8.85 X 10-¹² Fm-^I)

Solution

Combinations of Capacitors

Just like resistors, capacitors can be combined in series or parallel to provide an effective value.

Capacitors in Series

Consider the series arrangement of capacitors in figure 7.30.

• When capacitors are arranged this way, there is an equal distribution of charge on the plates.
• Hence, Q₁ = Q₂ = Q₃‘ This is so because once the battery draws electrons from one plate of capacitor C₁‘ the negative charge on the negative plate of C₁ induces a positive charge on one of the plates of C₂ and this process is repeated until charges appear on all other capacitor plates.
• Let the charge on each capacitor be Q. The potential difference across the individual capacitors will be given by;

CAPACITORS IN PARALELL

Parallel arrangement

In the parallel arrangement, all capacitors have the same potential difference across them.

Let the potential difference across them be equal to V and Q₁ Q₂ and Q₃ be the charge on each of the capacitors. The total charge;

Q _T=Q₁ +Q₂+Q₃  Q₁ = C₁V, Q₂ = C₂V and Q₃= C₃V
Therefore, Q = C₁ V + C₂ V + C₃ V

Thus, Q/V = C₁ + C₂ + C₃

But Q =C

V

So, C = C₁ + C₂ + C₃, where C is the combined capacitance.  In case of n capacitors of equal capacitance C₁‘ the combined capacitance C = nC₁

Example 2

Three capacitors of capacitance 1.5µF, 2 µand 3 µF are connected to a potential difference
of 12.0 V as shown in figure 7.32. Find:

(a) the combined capacitance.

(b) the total charge.

(c) the charge on each capacitor.

(d) the voltage across the 2 F capacitor.

(More examples in KCSE mirror page 211-213)

(a) Identify the factors that affect the capacitance of a capacitor.

(b) (i) On the set of axis show how the charge of a capacitor varies with time as a capacitor discharges.

Time, it

(ii) Draw a simple circuit to show how a capacitor can be discharged.
(c) Show that for two capacitors C₁ and C₂ arranged in series the effective capacitance C_T is given by the relation:

Energy Stored in a Charged Capacitor

Energy stored in a capacitor is in form of electrical potential energy. The energy may be converted to heat, light or other forms.

A plot of potential difference V against charge Q for a charging capacitor gives a straight line through the origin, as

The area of ΔOAB =¹_/2 QV

But QV = work done in moving a charge Q through a potential difference of V volts. This is the energy stored in a charged capacitor.

Work done (W) = average charge x potential difference

= ½QV

= ½CV² (since Q = CV)

= ^Q²/_2C(since V =^Q/_V)

Note that slope of graph yields =capacitance

ExampleS

A 2 µF capacitor is charged to a potential difference of 120 V. Find the energy stored in it.

Solution

W = ½CV2
= ½ x 2 x lQ^-6 X 120²

= 1.44 X 10-² J

Example 6

A 20 µF capacitor is charged to 60 V and isolated. It is later connected across an uncharged 100 µF capacitor. Calculate the final potential difference across the combination.

Solution

Let C₁ = 20 µF. C₂ = 100 µF.
V=60V

Let Q be the initial charge on C₁

Q = V₁C₁

Q= 60x20x 10^-6C
= 1.2 X 10-³C

When the two capacitors are connected in parallel. the potential difference across them is the same, say V E. Also, Q = Q_I + Q₂‘ where Q_I and Q₂ are the charges on the first and second

capacitors respectively.

But Q = C₁V_I, Q₂ = C₂V₂

VI =V₂=V

Q = C₁V_I + C₂V₂
= V(C_I + C₂)

1.2 x 10-³ = V x 120 x 1O^-6F

V = 1.2 X 10^-3

120 X 10-6

=l0V

Example 7

A 5 µF capacitor is charged to a potential difference of 200 V and isolated. It is then connected in parallel to a 10 µF capacitor. Find:

(a) the resultant potential difference.

(b) the energy stored before connection.

(c) energy in the two capacitors after connection. Is the energy conserved? Explain your answer.

Solution

(a)   When the 5µF capacitor is charged to 200 V, it will acquire a charge;
Q = CV

=5x 10^-6 x200
= 1.0 x 10^-3 C

Let VI be the resultant potential. IfC_I = 5µF and C₂ = 10 µF, then;
C\V\ + C₂V\ = 1.0 X 10-³

V – 1.0 X 10-³

C₁ +C₂

_ 1.0 x 10-³
– 15 X 10-6

= 66.7V

(b) Energy stored before connection = ½ CV²

= ½ x 5 x 10^-6 X 200²
= 0.1 J

(c) Energy in the two capacitors = ½ x 5 X 10^-6 X 66.7² + ½ x 10 X 10^-6 X 66.7²
= 1 X 66.7² (15 x 10^-6)

= 0.03336 J

The energy is not conserved. Some of it is converted into heat in the connecting wires.

Applications of Capacitors

Capacitors have extensive uses. Some of these are described below.

1. Rectification (Smoothing Circuits)

When converting a.c. to d.c. using a diode, d.c. voltages appear varying from minimum to maximum. To maintain a high d.c. voltage, capacitors are included in the circuit.

2. Reduction of Sparking in Induction Coil Contact

A capacitor is included in the primary circuit of induction coil to eliminate sparking at the contacts.

3. In Tuning Circuits

In the tuning circuit of a radio receiver, a variable capacitor is connected in parallel to an inductor. 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 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 is included in a flash circuit of a camera. It is easily charged by a cell in the circuit. When in use, the capacitor discharges instantly to flash.