Electromagnetism, the Atom and Radioactivity

Physics · WAEC and JAMB · SS2 and SS3

This topic closes the syllabus and supplies the transformer and half-life calculations that appear almost every year. The examiner wants the rules named correctly, Faraday and Lenz for induction, and clean nuclear equations that balance on both mass number and atomic number.

What you need to know

A magnetic field surrounds every magnet and every current-carrying conductor. Field lines run from north to south outside a magnet, never cross, and are closest together where the field is strongest.
A straight current-carrying conductor has circular field lines around it; the right-hand grip rule gives their direction. A solenoid carrying current behaves like a bar magnet, and the end from which current flows anticlockwise is the north pole.
An electromagnet is a solenoid wound on a soft iron core. Soft iron is used because it magnetises strongly and loses its magnetism as soon as the current stops, which is what makes the electric bell, the relay, the telephone earpiece and the scrapyard crane work. Steel is used for permanent magnets because it retains magnetism.
A current-carrying conductor placed in a magnetic field experiences a force F = BIL sin(theta), maximum when the conductor is perpendicular to the field. Fleming's left-hand rule gives the direction and is the motor rule: first finger field, second finger current, thumb motion.
Faraday's law of electromagnetic induction states that the magnitude of the induced emf is directly proportional to the rate of change of magnetic flux linkage. Lenz's law states that the induced current always flows in a direction that opposes the change producing it, which is simply conservation of energy in another dress.
Fleming's right-hand rule gives the direction of induced current in a generator. An induced emf appears only while the flux is changing: a magnet held still inside a coil induces nothing at all, no matter how strong it is.
A simple a.c. generator converts mechanical energy to electrical energy using slip rings; replacing the slip rings with a split-ring commutator gives a d.c. generator. A motor is the reverse device, converting electrical energy to mechanical energy.
A transformer works only on alternating current, because it needs a changing flux in the soft iron core. For an ideal transformer, Vs/Vp = Ns/Np, and if it is 100 per cent efficient then VpIp = VsIs, so a step-up in voltage is always a step-down in current.
Transformer losses are reduced in named ways: a laminated core cuts eddy currents, soft iron of low hysteresis loss cuts energy lost in repeated magnetisation, thick low-resistance copper windings cut I^2R heating, and winding the coils on the same limb cuts flux leakage. Transmission is done at high voltage and low current precisely to cut I^2R losses in the cables.
An atom has a small, dense, positively charged nucleus containing protons and neutrons, with electrons in orbits around it. The atomic number Z is the number of protons, the mass number A is protons plus neutrons, and isotopes are atoms of the same element with the same Z but different A, such as carbon-12 and carbon-14.
The photoelectric effect is the emission of electrons from a metal surface when light of sufficiently high frequency falls on it. Einstein's equation is hf = W + (1/2)mv^2(max), where W is the work function and the threshold frequency f0 = W/h. Below the threshold frequency no electrons are emitted no matter how bright the light, which is the evidence that light arrives as quanta.
Radioactivity is the spontaneous disintegration of unstable nuclei with emission of radiation. Alpha particles are helium nuclei, positively charged, heavily ionising and stopped by paper. Beta particles are fast electrons, negatively charged, more penetrating and stopped by a few millimetres of aluminium. Gamma rays are electromagnetic waves, uncharged, weakly ionising and only reduced by thick lead or concrete.
In alpha emission the mass number falls by 4 and the atomic number by 2; in beta emission the mass number is unchanged and the atomic number rises by 1; gamma emission changes neither. Every nuclear equation must balance separately in mass number and in atomic number.
Half-life is the time for half the nuclei in a sample to decay, and it is fixed for a given isotope and unaffected by temperature, pressure or chemical combination. After n half-lives the remaining fraction is (1/2)^n. Nuclear fission splits a heavy nucleus into lighter ones and drives nuclear reactors; nuclear fusion joins light nuclei into heavier ones and powers the sun.

Key terms

Magnetic flux: The product of the magnetic flux density and the area perpendicular to the field, measured in webers.
Electromagnetic induction: The production of an emf in a conductor whenever the magnetic flux linking it changes.
Lenz's law: The induced current always flows in such a direction as to oppose the change that is producing it.
Transformer: A device that uses mutual induction to change the voltage of an alternating supply with little loss of power.
Isotopes: Atoms of the same element having the same atomic number but different mass numbers.
Half-life: The time taken for half the atoms in a given sample of a radioactive element to decay.
Work function: The minimum energy required to liberate an electron from the surface of a metal.

Formulae

F = B*I*L*sin(theta)
induced emf: E = -N*(change in flux)/(time)
transformer: Vs/Vp = Ns/Np
ideal transformer: Vp*Ip = Vs*Is
transformer efficiency = (Vs*Is)/(Vp*Ip) * 100
Einstein photoelectric equation: h*f = W + (1/2)*m*v^2
threshold frequency f0 = W/h
energy of a photon E = h*f = h*c/lambda
fraction remaining after n half-lives = (1/2)^n
number of half-lives n = total time / half-life
E = m*c^2 for mass-energy conversion

Worked examples

A step-down transformer has 1000 turns in the primary and 50 turns in the secondary coil. It is connected to a 240 V a.c. mains supply and delivers a current of 2.0 A to a load. If the transformer is 80 per cent efficient, calculate the secondary voltage, the power delivered to the load and the current in the primary coil.

Secondary voltage: Vs/Vp = Ns/Np, so Vs = Vp x Ns/Np = 240 x 50/1000.
Vs = 240 x 0.05 = 12 V.
Power delivered to the load (output power) = Vs x Is = 12 x 2.0 = 24 W.
Efficiency = (output power / input power) x 100, so 80 = (24 / input) x 100.
Input power = 24 x 100/80 = 30 W.
Primary current Ip = input power / Vp = 30 / 240 = 0.125 A.

Answer: Secondary voltage = 12 V; power output = 24 W; primary current = 0.125 A

A radioactive substance has a half-life of 6 hours. If the initial mass of the sample is 80 g, calculate the mass remaining after 24 hours and the mass that has decayed.

Number of half-lives n = total time / half-life = 24 / 6 = 4.
Fraction remaining = (1/2)^4 = 1/16.
Mass remaining = 80 x 1/16 = 5.0 g.
Step through to confirm: 80 g, then 40 g after 6 h, 20 g after 12 h, 10 g after 18 h, 5 g after 24 h.
Mass decayed = 80 - 5.0 = 75 g.

Answer: 5.0 g remains undecayed and 75 g has decayed

The mistake to avoid

Candidates invert the turns ratio and turn a step-down transformer into a step-up one, reporting 4800 V from a 240 V supply. Sanity-check it: more turns on the secondary means higher secondary voltage, fewer turns means lower. In half-life work, the other standard error is subtracting the half-life repeatedly from the time instead of dividing; count how many half-lives fit into the total time, then halve the mass that many times.

In the exam

Nuclear equation questions are free marks if you balance the top numbers and the bottom numbers separately, so write the sums out rather than guessing the daughter nucleus. State Faraday's law and Lenz's law in full when asked, and name the rule you are using, left hand for motors and right hand for generators. For transformer questions, always check whether the examiner wants the ideal case or an efficiency figure, because the primary current changes completely between the two.