• Define fundamental laws and concepts in electricity and magnetism

• Understand laws of electromagnetism

• Discuss practical applications of electromagnetism

• Solve problems using knowledge based on electricity and magnetism

4.1.1 Coulomb's Law

The electric force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

Electric field is the region where electric forces act.

The electric field intensity at a point in an electric field is the electric force acting upon a unit positive charge placed at that point.

It is a vector quantity.

The  electric  field at a particular point in space due to a charge q is determined by placing (or imagining) a positive test charge q0  at the point. A positive charge is used by convention. Then, the magnitude of the electric field is given by

Electric Potential Energy

Work  must  be  done  to separate two bodies having opposite charges since they attract one another. This work is called the electric potential energy.

( OR )

Work must be done to bring closer two bodies having the same kind of charge since they repel one another. This work is called the electric potential energy.

The SI unit of electric potential energy is joule (J).

The electrical potential energy U is given by the relationship

U = (k q1q2 ) / r

The electric potential at a point in an electric field is the work done in bringing a unit positive charge from infinity to that point against electric force.

The electric potential energy at any point due to a charge q is determined by using a positive test charge q0  (as in the case of the E field), i.e.,

U = (k q0 q)/r

Then, the electric potential (V, or voltage) is defined

V = U/q0 = (k q)/r

Electric potential = potential energy / charge

and as energy / charge, has units of J C-1. However, the unit of joule per coulomb is given the name volt (V) in SI system. Like electric potential energy, the sign of the electric potential is determined by the signs of the charges.

Since the electric potential energy and the work done in bringing charges together are equal in magnitude, an alternative form of the electric potential equation is

V = W/q0

Of particular interest is the potential or voltage difference between two points. This is equal to the work done in moving a test charge between two points, i.e.,

ΔV = Vb – Va = Wab/q0

The potential difference can be related to the electric field. Consider the case of a uniform electric field. In moving a positive test charge from point 'a' to point 'b' against the field, the work done is

Wab = F d = q0 E d 

Since E = F/q0. Then,

ΔV = Wab/q0 = E d

which is commonly written as V = Ed

with  being understood as in the case of x for Δx . Notice that the electric field  E = V/d  has units of V m-1, which is equivalent to N C-1 (E = F/q0)

1. Compute  (a)  the electric field intensity E in air at a distance from a charge q1 = 5 x 10-9 C   (b)  the  force  F on a charge  q2= 4 x 10-9 C placed 30 cm from q1.

2. A  point  charge –30μC  is  placed  at  the origin of coordinates. Find the electric field at the point x = 5 cm on the x-axis.

3. Two charged bodies exert a force on each other of 24 mN. What will be the force between the same two bodies if the distance between them is doubled?  [6 mN]

4. What is the potential at the point midway between charges of +2 μC and +5 μC, which are 6 m apart?

5. How  far  apart would an electron and a proton have to be to have a mutual electrical attraction of 1.0 N? (r = 1.5 x 10-14 m) What is the electric field 60 cm away from a charge of +3.0μC? (E = 7.5 x 104 N C-1 away from charge)

6. What  is  the magnitude and direction of a vertical electric field that would support the weight of an electron? (E =  (mg)/q = 5.6 x 10-11 N C-1)

7. Two charges of +5μC and -4.0 μC are located at positions (x, y) of (-20cm, 0) and (10 cm, 0) respectively. (a) What is the electric potential energy of the charges? (b) How much work is required to move the -4.0μC charge to the position (20 cm, 0)?

8. The electric potential at a particular location is 9.0 x 104 J C-1 as determined by using a test charge of +3.0μC. What is the electric potential energy of the charge at this point? (Ans ; U = q V = 0.27 J)

Electric current 

An electric current is a flow of electrons from a place of lower potential to a place of higher potential.

1. The SI unit of charge is the coulomb (C).

2. (a) Materials which allow current to pass through them are called conductors. Most metals such as copper, silver, mercury and carbon (graphite) are conductors.

(b)  Materials which do not allow current to pass through them are called insulators. Some examples are most non-metals such as rubber and plastic.

3.(a) A conventional current flows from the positive terminal of a cell through the circuit to the negative terminal.

1. A current of 2 A is flowing through a conductor. How long does it take for 10°C of charge to pass any point?

A) 0.2 s B) 5 s C) 12 s D) 20 s

2. 50 J of energy is dissipated for every 10 C of charge passing through a resistor. Which instrument would allow this to be measured?

A) An ammeter in series with the resistor

B) An ammeter in parallel with the resistor

C) A voltmeter in series with the resistor

D) A voltmeter in parallel with the resistor

3. An electrical quantity is defined by "the energy converted by a source in driving unit charge round a complete circuit". What is this quantity called?             

A)  Current C) Potential difference B) Electromotive force D) Power

A  battery  or  cell  is  a  device for producing electrical energy. We may therefore define  the  e.m.f.,  E,  as  the  total  energy  per coulomb it produces. E can also be defined as the total power per ampere it produces. So if the battery current is I when it  is  connected  to  a  resistance  R,  the  total  power  produced by the battery is EI.

Part  of  the  power  is  delivered  to  the  resistance  R  and  the rest to the internal resistance r of the battery. So

EI = power in R + power in r

=  IV + Iv

where V  is the p.d. across R and v is the p.d. across r.  Dividing by I

E   = V + v

or E   =  p.d. across external resistance + p.d. across internal resistance

  E    =    V     +     Ir

Current Law (KCL)

The algebraic sum of the currents flowing into a junction is zero.

∑  I  =  0

Voltage Law (KVL)

The algebraic sum of the voltages around a closed loop is zero.

∑ V  =  0

Current law is a statement of conservation of charge. Whatever current enters a given point in a circuit   must  leave  that  point because charge cannot build up or disappear at a point.

Voltage law is equivalent to the law of conservation of energy. Any charge that moves around any closed loop in a circuit must gain as much energy as it loses.

There are limitations on the  number  of times that the current law and the voltage law can be used.

The  number of times the current law can be used is one fewer than the number of junction points in the circuit.

The voltage law can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation.

In general, the number of independent equations needed must be at least equal the number of unknowns in order to solve a particular circuit problem.

Rules to be followed while applying Kirchhoff’s laws in any circuit:

(i) Label all quantities, known and unknown, including an assumed direction for each unknown current. If the assumed direction of a particular quantity is opposite to the  actual direction,  the numerical value of the quantity will come out negative.

(ii) If one traverses through a source of emf from the negative terminal to the positive terminal, the change in potential is +E.

(iii) If one  traverses  through a source of emf from the positive terminal to the negative terminal, the change in potential is -E.

(iv) If one  traverses  through a resistor R in the same direction as the assumed current direction, the change in potential is –IR.

(v) If one traverses through a resistor R in the direction opposite to the assumed current direction, the change in potential is +IR.

Electrical  energy  can  be  transformed  to  many  other useful forms of energy. In electric motors, for example, most of the electrical energy supplied is changed to the mechanical energy of rotation. 

In a loudspeaker most of the electrical energy is transformed into sound energy.

When an electric current  passes  through a resistance wire there is a transfer of energy from electrical to heat energy. Basically, this is due to the moving electrons 'colliding'  with  the vibrating atoms of the metal, and giving up their energy to the atoms. The heating  effect of an electric current is widely used in the home and in industry; light bulbs use the heating effect of a current, for example.

Electrical energy is obtained when a quantity of charge moves between two points that have a potential difference. Thus, from the definition of the volt, 1 J (joule) of energy is obtained when 1 coulomb moves between two points at a p.d. of 1 V. So if 20 coulombs move between two points having a p.d. of 4 V, 80 J of energy are obtained. Generally, if Q coulombs move through two points having a p.d. of V volts, the energy W in joules is given by

W = QV (1)

A more useful formula for an energy is one which contains current, I. Now Q = It, where t is the time for which the current flows. So, from (1) , the energy W in joules is given by

W = VIt (2)

The practical unit of electrical energy is kilowatt hour (kWh).

Power  is  defined  generally  as  the  rate  of  energy  transfer.  So power  may  be calculated from

Power             P = energy transferred/ time taken

In the case of electrical machines, the energy transferred in a time t second is VIt. So the power P used is given by

P = (IVt)/t

 P  = VI

Here  P  is  in  watts,  W,  when  I  is in amperes and V is in volts. 1 W is the rate of working at 1 J / s .

An electrical appliance, which takes a current I when the potential difference across its  terminals is V, uses power IV. An electric motor, which is connected to a 50 V supply and takes a current of 5 A, uses power of 50 x 5 or 250 W. An electric stove carrying a current of 4 A when the mains potential difference is 240 V uses a power of 960 W. An electric lamp filament carrying a current of ¼ A when the mains p.d. is 240 V uses a power of 60 W.

Joule was the first person to investigate the heating effect of an electric current. His conclusions, known as Joule's law of electrical heating, were:

By Joule's law of electrical heating the heat energy W developed in a wire is proportional to:

(a) the time t (for a given resistance and current);

(b) the square of the current or I2 (for a given resistance and time);

(c) The resistance R of the wire (for a given current and time).

As  we  have  seen  previously,  the  power  or energy per second in a resistance R carrying a current I is P = I2R. So if t is the time, the energy or heat produced, W = I2Rt. For resistances in series, I is the same in each resistance. So a large resistance produces more heat than a small resistance.

For resistances in parallel, however, the p.d. across each is the same. Now we have shown  previously  that P = V2 / R , so the energy or heat produced, W = V2t / R. It follows that, for a fixed p.d., the smaller resistance produces more heat than a larger resistance. Lamps in parallel in lighting circuits have the same p.d. Since the power P in the lamp is inversely proportional to R , a 120 W lamp filament has a smaller resistance than a 60 W lamp one.

A  current  of  3 A  flows for 2 min through a wire of resistance 20. If the wire is totally  immersed inside 0.1 kg of water in a can of heat capacity 40 J/K, calculate the temperature rise of the water (c = 4200 J/kg K for water).

Heat supplied = I2Rt = 32 x 20 x 120 = 9 x 2400 = 21600 J

1. An  electric lamp is marked 12 V, 36 W. Calculate (a) the resistance of the  lamp when in use; (b) the energy in joules expended each minute.

2. An  electric lamp is marked 100 W, 250 V. If the lamp is connected to a 250 V  mains,  calculate (a) the current taken; (b) the cost of using the lamp for  100hr at 1 p per kilowatt hour.

3. Define watt, kilowatt-hour.

A washing machine for use on 240 V mains has a ⅓ h.p. motor and a heating  element  rated  at 2 kW. What current does it take when in use, and what is the cost of using it for 40 min each week for a period of 12 weeks if the cost  of electricity is 2 p per unit? (Assume 1 h.p. = 0.75 kW)

4. An electric lamp is labelled 12 volt, 60 watt. Given this information state;

(i)    one necessary precaution when using the lamp.

(ii)   the energy the lamp will dissipate each second when correctly used.

(iii)  the current then passing through the lamp.

5. A  battery  sends  a  current  through two wires of resistance 10 and 20.  Compare the rate of production of heat  in  the  10 wire with that in the 20 wire  when  they  are  connected  (a) in series,  (b) in parallel,  across the battery.

6. The heating element of an electric kettle, containing 1 litre of water, initially at a  temperature of 20°C,  is  connected  to  a  250 V d.c. supply and the  water  commences  to  boil  in  9  min. If  the  current  through  the  heating  element  is  4 A,  calculate  (a)  the  thermal  capacity  of the kettle; (b) the   cost involved when the price of electrical energy is 6p per kilowatt hour. Neglect heat losses. (O and C)

7. A small house with a mains supply at 250 V has two 2 kW electric heaters and six 100 W lamps. The  power  and  the  lighting  circuits are entirely separate, and each has its own main fuse. What current passes through each of the fuses when both heaters and all the lamps are in use? (O)   

From earliest times it was observed that certain ferrous ores possessed magnetic properties. That is, they  are  able  to  attract  other small pieces of iron such as iron filings, and if freely suspended, a piece of this ore would always take up a particular direction  relative to the earth's surface. For this reason the early navigators called it 'lodestone'  and  the  simplest  forms  of  compass  consisted  of a piece of lodestone mounted on wood, floating in a bowl of water.

Magnetic substances (for all practical purposes) are confined to ferrous metals such as iron or steel. Such substances are therefore called ferromagnetic.

The  magnetism in  a  magnet is  concentrated near the ends of the magnet. These are called the poles of the magnet. The 'quantity of magnetism' at the poles of a magnet is called the pole strength and they are always equal but opposite in nature.

For most practical purposes, the poles are assumed to be at the ends of the magnet. A more  exact  result  is obtained by assuming each pole to be 1/12 of the length of the magnet in  from each  end.  The distance between the poles is called the effective length.

If  a magnet is mounted so that it is free to turn in a horizontal plane, it will come to rest with one end pointing approximately towards the North Magnetic Pole. This end of the magnet is called the north seeking or red pole. The other end of the magnet is called the south seeking or blue pole. 

Like poles repel each other and unlike poles attract each other.

Flux  density  B  is  the  number  of magnetic flux lines per square metre cross-section, taken at right angles to the direction of the flux.

i.e.  B (tesla) = Φ (webers)/ A (m2)

It  is  measured  in  weber per square metre (Wb m-2). This unit is also known as the tesla (T).

 The  concept  of flux density is similar to that of pressure (force/area) in mechanics.  The denser the flux lines (higher B), the stronger the magnetic field.

The direction of the field is the direction in which a north pole would be urged in that field, that is, away from the red pole and towards a blue pole. Hence flux lines can be regarded as running out at the red, and running in at the blue.

Any line of force leaving a red pole must end up on a blue, often the blue pole of the magnet itself. Others, like those leaving the ends, go wandering off and ultimately end on the earth's blue.

Two flux lines never run together nor cross one another, nor does one ever divide. They are all separate  and continuous,  entering  the bar at the blue end, running continuously through the bar, fanning out again as they leave the red end.

The  attractive  and  repulsive  forces  between  magnetic  poles  vary  directly as the product  of  the  pole  strengths  and  inversely as the square of the distance between them. This law is known as Coulomb's 

law

Hence Coulomb's law states that:

F = (m1m2)/ (4πμ0d2)

where:  F = force in newton

 m1, m2 = pole strengths in weber

d = distance between the pole in metre

μ0= 4 π x 10 –7henry per metre (H m-1)

μ0 is a constant known as the permeability of free space (vacuum). If the poles are similar, the force is repulsive otherwise it is attractive.

Whenever a  direct current flows in a wire, the wire is surrounded by a magnetic field (Fig 2).  The  direction  of  this  field  can be obtained using the right-hand-grip rule (Fig 3). 

Retentivity is the ability of a substance to retain its magnetism, once it is magnetised. The greater the retentivity, the longer will the magnet last. A permanent bar magnet must have high retentivity, whilst an electro‑magnet must have low retentivity.Smith   Pg.232

Various qualities of steel are used in shipbuilding, and some of these are more susceptible to magnetic influences than others. In magnetism, the terms hard and soft iron are used to express different magnetic properties.

Hard iron is difficult to magnetise, but once magnetised, it retains its poles more or less permanently. Magnetism in hard iron is termed permanent magnetism.

In ships, most magnetism of this type is located in the shell and deck plating. The magnetic poles become permanently fixed by the amount of hammering the ship receives during construction. Their position thus bears some relation to the direction in which the ship is lying whilst it is being built.

Soft iron is easily magnetised. In fact, a bar of soft magnet has only to be held in the earth's magnetic field for it to become a magnet. The poles are not permanent. They will disappear the moment the bar is removed from the magnetising influence. Magnetism of this kind is called transient induced magnetism.

In ships, magnetism of this type is located chiefly in beams and girders, such as frames, deck beams, masts, derrick and derrick posts, funnel and so on.

When considering the nature of the earth's magnetic field it is sometimes convenient to regard the earth as if it had a large magnet inside, as shown in Fig 5. The magnetic lines of force emanate from the earth's Magnetic South Pole and end up on the earth's Magnetic North Pole.

A freely suspended magnet will line itself up with the earth's line of force wherever it happens to be. For instance, such a magnet would be vertical at MN, red end pointing into the earth. At A and B, it would be inclined at the angles shown. At C it would be horizontal, whereas at D and E it would be inclined again, but this time with the red end upwards.

Finally at MS, it would again be vertical, but with red end pointing upwards.

The area illustrated  in  Fig 5  where a freely suspended compass needle would be vertical are called the magnetic poles.

The north magnetic pole is situated in a region north of Hudson Bay, in the general vicinity of  70°N, 97°W. The south magnetic pole is in the Antarctic, in the general vicinity of 73°S, 155°E.

It should be noted that these are very far from being at opposite ends of a diameter. In fact, the earth's  magnetic field itself is far from being symmetrical - on no account   should  it be regarded as a nice tidy arrangement of great circles from north to south magnetic poles.

Dip

Smith  Pg.237   Fig.75   1.5"x2"We  have  already seen that a freely suspended magnet will in general point at some angle down into the ground. The angle of dip is the angle a freely suspended magnet makes with the horizontal. Fig  6  shows the dip circle, used for measuring the dip angle.

In north magnetic  latitudes,  i.e. north of the magnetic equator, the dip is given a + sign, and in south magnetic latitudes a - sign, to remind us that although the angle of dip may be the same at two places, at the north end, it is tilted down, and at the other, up.

The  narrow belt surrounding the earth roughly midway between the magnetic poles where the earth's magnetic field is horizontal is called the magnetic equator.

The  approximately  position  of  the  magnetic  equator  is  shown  in Fig 7. From a maximum  of  about  14°S  latitude  in  Brazil, it crosses the Equator at about 17°W, reaches  its  maximum   of about 11°N latitude near Cape Guardafui, and crosses the Equator again in about 170°W.

Since  the  compass  needles are so mounted that they can only rotate in a horizontal plane, they can only be moved by a horizontal force, or at least, a force that has a horizontal component.

For this reason the earth's total field intensity I is split up into two components, 

i.e. H, the horizontal component of the earth's field I,

Z, the vertical component of the earth's field I.

From the  right-angled  triangle  in  Fig 8,  the  following formulae can be deduced, giving the relation between H, Z, I and dip, i.e.

H is the component that affects the compass, that is, gives the needles their directive force, their ability to point north. Its importance can therefore hardly be over-emphasised.

At the magnetic equator, there is no dip, so H is equal to I, the earth's total field, and Z, is zero.

As  we  move  towards  either pole, the dip increases, H diminishes and Z increases. With dip 45°, H and Z are equal. With dip 60°, H is only equal to half the earth's total field (since H = I cos 60°), until finally, at the pole, H is zero, and the whole of the earth's field is vertical.

It is easy to see from this that the compass is most efficient where H is greatest, i.e., at the magnetic equator, and is virtually useless as we get near the magnetic poles. This is usually summed up in the words:

The directive force of the compass needle varies directly as H.

The importance of the vertical component Z

From our point of view the vertical force Z has only a nuisance value. It plays no part in giving directive force to the compass. On the contrary, by inducing poles in vertical soft iron (V.S.I.) such as masts, derrick posts, frames and funnel, it causes undesirable deviation of the compass.

However, for this reason it is obviously important to know the value of Z, so that we may know, not only the strength of the poles in V.S.I., but also which is the blue end and which the red end.

The value of Z in turn depends on the value of the dip, i.e. the greater the dip, the greater the force Z will be.

Thus, at the magnetic equator dip is zero, Z is zero, and there are no poles in V.S.I. As we move into north orsouth magnetic latitudes, dip increases, Z increases, and the strength of the poles in V.S.I. increases, until finally, at the magnetic poles. Z has its maximum value, being then equal to the earth's total field I.

Furthermore, remember that polarity induced in V.S.I. will always be red next to the earth's blue, and vice-versa. Thus, in the northern hemisphere, a bar of V.S.I. will have a red pole at its lower end. In the southern hemisphere, a blue pole at its lower end (Fig 9).

We  have  seen that a compass needle does not point to true north, but lines itself up with  the  direction  of  the  earth's  field  at  that  place.  This  direction  is called the  magnetic meridian at that place (Fig 10).

The  magnetic  meridian  at  any  place  is  the direction assumed by a compass needle, free to move only in a horizontal plane, and under the influence of the  earth's field alone.

Once the above definition is learnt, it is easy to define magnetic variation as follows:

The magnetic variation at any place is the angle between the true north and the  magnetic meridian at that place.

 It is named westly if magnetic north is westward of true north, and eastly if magnetic  north is eastward of true north. 

The value of the variation at any one point is not constant. It is subject to a slight daily swing of a few minutes of arc to the west, and back again, during the 24 hours. Also, magnetic storms cause a slight swing. These are both negligible, as far as navigation is concerned.

The only change of importance to the navigator is the secular change, that is, a slow but fairly regular change which goes on persistently over a long period - centuries - and accumulates to a considerable amount.

The annual amount of the secular change is always quoted on the chart, in a statement such as "variation 14°W. 1936, decreasing 11' annually". This change must always be allowed  for, i.e.  always  work, whether  on  the  chart  or  in  calculations,  with  the variation for the current year.

The deviation of the compass is caused by the iron in the ship. Without any such iron, the  compass  needle  would  lie  in  the  magnetic  meridian. Deviation may be very simply  defined  as the amount in degrees the compass needle is deflected to, east or west of the magnetic meridian.

If  the north end of the needle is deflected to the eastward of the magnetic meridian, the deviation is named east, if westward, it is named west.

Fig. 11  shows  the  effect  of permanent magnetism of the ship on the deviation of thcompass. Since the ship’s bow is magnetised red, it repels the red end of the compass needle to the left and produces a westward deviation from the magnetic meridian. 

Magnetism

1. Two  magnetic poles of pole strength 0.52 Wb and 0.28 Wb are 20 cm apart in air.  Assuming  that the permeability of air is the same as that of vacuum, calculate the force between the poles.  [2.31 x 105 N]

2.  (a)   Explain what is meant by-     

(i) the flux of a magnet,

(ii)  the flux density of magnetic field.

(b)  Find the  flux near the pole of a magnet whose flux density is 2.5 x 10-2 tesla  and whose cross-sectional area is 0.5cm2 . [ 1.25 x 10-6Wb ]

3. The north pole of magnet A is separated from the south pole of magnet B by 5 cm  as shown in the figure below. Given that the pole strengths of magnets A and B are respectively 0.80 Wb and 0.30 Wb and the permeability of free space,  is 4 x 10-  7 H m-1, calculate the force of attraction between the two magnets. [ 6.08x 103 kN ]

Suppose a loop of wire is connected to a galvanometer (an instrument for measuring very small current in both directions), as shown in Fig 1, and a permanent magnet is positioned so that the magnetic flux from the magnet passes through the loop. 

When a coil is placed in a magnetic field, magnetic flux lines are imagined to be passing through the coil. This is known as flux linkage. 

Changing the magnetic field through the coil by moving either the coil or magnet, or both will result in a change of flux linkage.

The electromotive force, emf, produced in the coil is given by Faraday's law, which states that

The induced emf in a wire loop is directly proportional to the rate of change of magnetic flux through that loop.

E  =  rate of change of flux linking the coils

If the rate of change is in weber per second, then the induced emf is in volts.

When a permanent magnet approaches a conducting loop, a current is established in the loop. This current in turn sets up its own magnetic field around the loop. 

The induced magnetic field interacts with the field of the permanent magnet and will either attract or repel the permanent magnet.

Suppose a rectangular loop of length L and width W is rotated in a uniform magnetic field B as shown in Fig 4. An emf will be produced in the coil given by the formula:

E = Eo sin ωt 

where, E is the instantaneous emf

Eo is the maximum emf or amplitude, and

Ω is the angular velocity, given by 2 πf

The amplitude, Eo is given by     Eo=  2πfN B A

where, N is the number of turns,

B is the magnetic flux density,   and

A is the area of the coil.

Note that B through  the  coil  can  be increased by either using a stronger magnet or  wrapping the coil around a soft iron core.

An inductor is a coil of wire which may be wound on a core of iron or other material.

In Fig 6a, the current rises virtually instantaneously to its final value when switch S is closed and the current falls to zero almost instantaneously when the switch is opened.

When  two coils are positioned so that a portion of the magnetic flux produced by one coil  passes  through  the  other, the  coils  are said to be magnetically coupled. The mutual inductance  of  a pair of coils is a measure of the effectiveness of one coil in inducing a voltage in the other. It is also measured in henries.

It is often desirable either to step up or step down the voltage of a particular alternating current source. A transformer provides a convenient means of doing this efficiently. 

Fig. 7 shows the construction of a transformer. It comprises a core, a primary winding and a  secondary winding. The  purpose of  the  soft-iron core is to link  the flux produced  in  the  primary  winding with the secondary winding. Soft-iron is used to make the core because it has low  retentivity and can be easily magnetised in one direction, demagnetised and magnetised the other way continuously.   

A voltage or current that changes in magnitude and direction periodically is called an alternating voltage or current.

i(t)   =   Im sin wt     where  Im= maximum current

Root Mean Square Values

 The value of an alternating current or voltage changes from one instant to the next. The average value over a complete cycle is zero.

The root mean square (rms) value of an alternating current or voltage (also called the  effective value) is defined as the steady direct current or voltage which would give the same heating effect.

For sine waves, rms or effective value =  peak value/√2

In an ac circuit there are three basic elements that determine the current flow for a given alternating voltage source.  These  are  resistance  R,  inductance  L  and capacitance C.

An inductor or choke is a coil of wire with a core of either air or a magnetic material such as iron.

By Ohm’s law   V = IXL or    I = V/XL  where V and I are effective voltage and current.

                                        XL = V/I  is the inductive reactance measured in ohm (W)

The opposition of an inductor to a.c. is called its inductive reactance.

 XL = wL  =  2pfL

where  f = frequency (Hz) and L = inductance (henry, H)

An inductor allows a.c. and d.c. to pass but it opposes a.c. more than steady d.c.

Its reactance increases as the frequency of the a.c. increases.

A capacitor is a device that stores electrical energy in the form of an electric field,

A capacitor consists of two parallel metal plates separated by an insulator called the dielectric. It does not allow d.c. to flow through it and it behaves as if a.c. does flow through.

1. An air core solenoid has 900 turns of wires. When a maximum current of 100 mA flows through the coil, it produces a flux of 1.33 x 10-7Wb inside the solenoid. If the flux takes 75 ms to grow from zero to its maximum level, find the emf induced in the coil. [1.6 mV]

2. State the factors that will affect the  induced emf of  a coil in a magnetic field.

A flat search coil of  500 turns,  each  of  area  2.5  x  10-4 m2, is connected to a galvanometer. If the coil rotates at 15.7 rad s-1 in a magnetic field of 0.3 T, find the maximum emf induced. [ 0.59 V ]

3.  A  transformer has 200 turns in a primary coil. The input voltage is  240 V  a.c. How many turns are on the secondary coil if the output voltage is 3600 V a.c? [ 3000 ]

4.  The current in the primary coil of a transformer is 100% efficient and the current in the secondary coil is 2 A. Find the output voltage if the input power is 6 W. [ 3 V ]

Alternating voltage and current

5. A  voltage  source  of  120 V , 60  Hz is connected in series with an inductor  which  has  inductance  125 mH.  Find  the current in the circuit. [2.6 A]

6. A series RC circuit has a resistance of 100 Ω and a capacitance of 30 mF. If  the  voltage  source  operates  at  120 V, 60 Hz,  find the current in the circuit.  [0.9 A]

7. A Series  RLC  circuit with a voltage source of 120 V, 60 Hz has a resistance of 100 Ω  and capacitive and inductive reactances of 88 Ω  and 17 Ω  respectively. Compute the current in the circuit.  [ 1.1 A ]

8. A  series RLC circuit with a 125 Ω   resistor and inductive and capacitive reactance of 113 Ω  and 53 Ω  respectively is driven by 220 V-60 Hz source. What is (a) power factor and (b) true power lost of the circuit?   [ 0.90, 314 W ]

9.  What is the resonance frequency of a series RLC circuit that resistance of 100 Ω, inductance of 50 mH and capacitance of 0.1 mF. [2.2 kHz] 

1. Calculate the values of two equal charges if they repel one another with a force of 0.1 N when situated 50 cm apart in vacuum.

2.  If  the  force  acting on a charge Q, 6 cm from a charge of +50 x 10-8 C is 0.24 N,  find the magnitude of Q.

3.  Find the force between two charges of  +1 μC  and  +2 μC   when they are 0.03 m apart.  (20 N)

4.  Calculate the  total energy  provided  by a battery of e.m.f. 2.50 V when it causes a steady current of 0.40 A to flow for 15 min through an electric bulb. If the battery had an internal resistance of 2.00 Ω, calculate the heat dissipated in the electric bulb in that time. (L)

5. Find the loop currents of the following circuit.