Full As level Physics Note according to Camridge Syllabius

Measurement

Base quantities and their units; mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol):

Base Quantities	SI Units
	Name	Symbol
Length	metre	M
Mass	kilogram	kg
Time	second	S
Amount of substance	mole	mol
Temperature	Kelvin	K
Current	ampere	A
Luminous intensity	candela	cd

Derived units as products or quotients of the base units:

Derived	Quantities Equation	Derived Units
Area (A)	A = L2	m2
Volume (V)	V = L3	m3
Density (ρ)	ρ = m / V	kg m-3
Velocity (v)	v = L / t	ms-1
Acceleration (a)	a = Δv / t	ms-1 / s = ms-2
Momentum (p)	p = m x v	(kg)(ms-1) = kg m s-1

Derived Quantities	Equation	Derived Unit		Derived Units
		Special Name	Symbol
Force (F)	F = Δp / t	Newton	N	[(kg m s-1) / s = kg m s-2
Pressure (p)	p = F / A	Pascal	Pa	(kg m s-2) / m2 = kg m-1 s-2
Energy (E)	E = F x d	joule	J	(kg m s-2)(m) = kg m2 s-2
Power (P)	P = E / t	watt	W	(kg m2 s-2) / s = kg m2 s-3
Frequency (f)	f = 1 / t	hertz	Hz	1 / s = s-1
Charge (Q)	Q = I x t	coulomb	C	A s
Potential Difference (V)	V = E / Q	volt	V	(kg m2 s-2) / A s = kg m2 s-3 A-1
Resistance (R)	R = V / I	ohm	Ω	(kg m2 s-3 A-1) / A = kg m2 s-3 A-2

Prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units:

Multiplying Factor	Prefix	Symbol
10-12	pico	P
10-9	nano	N
10-6	micro	Μ
10-3	milli	M
10-2	centi	C
10-1	decid	D
103	kilo	K
106	mega	M
109	giga	G
1012	tera	T

Estimates of physical quantities:

When making an estimate, it is only reasonable to give the figure to 1 or at most 2 significant figures since an estimate is not very precise.

Physical Quantity	Reasonable Estimate
Mass of 3 cans (330 ml) of Coke	1 kg
Mass of a medium-sized car	1000 kg
Length of a football field	100 m
Reaction time of a young man	0.2 s

• Occasionally, students are asked to estimate the area under a graph. The usual method of counting squares within the enclosed area is used. (eg. Topic 3 (Dynamics), N94P2Q1c)
• Often, when making an estimate, a formula and a simple calculation may be involved.

EXAMPLE

Estimate the average running speed of a typical 17-year-old‟s 2.4-km run.

velocity = distance / time = 2400 / (12.5 x 60) = 3.2 ≈3 ms^-1

Which estimate is realistic?

	Option	Explanation
A	The kinetic energy of a bus travelling on an expressway is 30000J	A bus of mass m travelling on an expressway will travel between 50 to 80 kmh-1, which is 13.8 to 22.2 ms-1. Thus, its KE will be approximately ½ m(182) = 162m. Thus, for its KE to be 30000J: 162m = 30000. Thus, m = 185kg, which is an absurd weight for a bus; ie. This is not a realistic estimate.
B	The power of a domestic light is 300W.	A single light bulb in the house usually runs at about 20W to 60W. Thus, a domestic light is unlikely to run at more than 200W; this estimate is rather high.
C	The temperature of a hot oven is 300 K.	300K = 27 0C. Not very hot.
D	The volume of air in a car tyre is 0.03 m3.		Estimating the width of a tyre, t, is 15 cm or 0.15 m, and estimating R to be 40 cm and r to be 30 cm, volume of air in a car tyre is = π(R2 – r2)t = π(0.42 – 0.32)(0.15) = 0.033 m3 ≈ 0.03 m3 (to one sig. fig.)

Distinction between systematic errors (including zero errors) and random errors and between precision and accuracy:

Random error: is the type of error which causes readings to scatter about the true value.

Systematic error: is the type of error which causes readings to deviate in one direction from the true value.

Precision: refers to the degree of agreement (scatter, spread) of repeated measurements of the same quantity. {NB: regardless of whether or not they are correct.}

Accuracy: refers to the degree of agreement between the result of a measurement and the true value of the quantity.

	→ → R Error Higher → → → → → → Less Precise → → →
↓ ↓ ↓ S Error Higher Less Accurate ↓ ↓ ↓

Assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties (a rigorous statistical treatment is not required).

For a quantity x = (2.0 ± 0.1) mm,

Actual/ Absolute uncertainty, Δ x = ± 0.1 mm

Fractional uncertainty, Δxx = 0.05

Percentage uncertainty, Δxx ´ 100% = 5 %

If p = (2x + y) / 3 or p = (2x - y) / 3 , Δp = (2Δx + Δy) / 3

If r = 2xy³ or r = 2x / y³ , Δr / r = Δx / x + 3Δy / y

Actual error must be recorded to only 1 significant figure, &
The number of decimal places a calculated quantity should have is determined by its actual error.

For eg, suppose g has been initially calculated to be 9.80645 ms^-2 & Δg has been initially calculated to be 0.04848 ms^-2. The final value of Δg must be recorded as 0.05 ms^-2 {1 sf }, and the appropriate recording of g is (9.81 ± 0.05) ms^-2.

Distinction between scalar and vector quantities:

	Scalar	Vector
Definition	A scalar quantity has a magnitude only. It is completely described by a certain number and a unit.	A vector quantity has both magnitude and direction. It can be described by an arrow whose length represents the magnitude of the vector and the arrow-head represents the direction of the vector.
Examples	Distance, speed, mass, time, temperature, work done, kinetic energy, pressure, power, electric charge etc. Common Error: Students tend to associate kinetic energy and pressure with vectors because of the vector components involved. However, such considerations have no bearings on whether the quantity is a vector or scalar.	Displacement, velocity, moments (or torque), momentum, force, electric field etc.

Representation of vector as two perpendicular components:

In the diagram below, XY represents a flat kite of weight 4.0 N. At a certain instant, XY is inclined at 30° to the horizontal and the wind exerts a steady force of 6.0 N at right angles to XY so that the kite flies freely.

By accurate scale drawing	By calculations using sine and cosine rules, or Pythagoras‟ theorem
Draw a scale diagram to find the magnitude and direction of the resultant force acting on the kite. R = 3.2 N (≡ 3.2 cm) at θ = 112° to the 4 N vector.	Using cosine rule, a2 = b2 + c2 – 2bc cos A R2 = 42 + 62 -2(4)(6)(cos 30°) R = 3.23 N Using sine rule: a / sin A = b / sin B 6 / sin α = 3.23 / sin 30° α = 68° or 112° = 112° to the 4 N vector

Kinematics

Displacement, speed, velocity and acceleration:

Distance: Total length covered irrespective of the direction of motion.
Displacement: Distance moved in a certain direction.
Speed: Distance travelled per unit time.
Velocity: is defined as the rate of change of displacement, or, displacement per unit time
{NOT: displacement over time, nor, displacement per second, nor, rate of change of displacement per unit time}
Acceleration: is defined as the rate of change of velocity.

Using graphs to find displacement, velocity and acceleration:

• The area under a velocity-time graph is the change in displacement.
• The gradient of a displacement-time graph is the {instantaneous} velocity.
• The gradient of a velocity-time graph is the acceleration.

The 'SUVAT' Equations of Motion

The most important word for this chapter is SUVAT, which stands for:

• S (displacement),
• U (initial velocity),
• V (final velocity),
• A (acceleration) and
• T (time)

of a particle that is in motion.
Below is a list of the equations you MUST memorise, even if they are in the formula book, memorise them anyway, to ensure you can implement them quickly.

1.	v = u +at	derived from definition of acceleration: a = (v – u) / t
2.	s = ½ (u + v) t	derived from the area under the v-t graph
3.	v2 = u2 + 2as	derived from equations (1) and (2)
4.	s = ut + ½at2	derived from equations (1) and (2)

These equations apply only if the motion takes place along a straight line and the acceleration is constant; {hence, for eg., air resistance must be negligible.}
 

Motion of bodies falling in a uniform gravitational field with air resistance:

Consider a body moving in a uniform gravitational field under 2 different conditions:

Without Air Resistance:

Assuming negligible air resistance, whether the body is moving up, or at the highest point or moving down, the weight of the body, W, is the only force acting on it, causing it to experience a constant acceleration. Thus, the gradient of the v-t graph is constant throughout its rise and fall. The body is said to undergo free fall.

With Air Resistance:

If air resistance is NOT negligible and if it is projected upwards with the same initial velocity, as the body moves upwards, both air resistance and weight act downwards. Thus its speed will decrease at a rate greater than 9.81 ms^-2 . This causes the time taken to reach its maximum height reached to be lower than in the case with no air resistance. The max height reached is also reduced.
At the highest point, the body is momentarily at rest; air resistance becomes zero and hence the only force acting on it is the weight. The acceleration is thus 9.81 ms^-2 at this point.
As a body falls, air resistance opposes its weight. The downward acceleration is thus less than 9.81 ms^-2. As air resistance increases with speed, it eventually equals its weight (but in opposite direction). From then there will be no resultant force acting on the body and it will fall with a constant speed, called the terminal velocity.

Equations for the horizontal and vertical motion:

	x direction (horizontal – axis)	y direction (vertical – axis)
s (displacement)	sx = ux t sx = ux t + ½ax t2	sy = uy t + ½ ay t2 (Note: If projectile ends at same level as the start, then sy = 0)
u (initial velocity)	ux	uy
v (final velocity)	vx = ux + axt (Note: At max height, vx = 0)	vy = uy + at vy2 = uy2 + 2asy
a (acceleration)	ax (Note: Exists when a force in x direction present)	ay (Note: If object is falling, then ay = -g)
t (time)	t	T

Parabolic Motion: tan θ = v_y / v_x
θ: direction of tangential velocity {NOT: tan θ = s_y / s_x }

Dynamics

Newton's laws of motion:

Newton's First Law
Every body continues in a state of rest or uniform motion in a straight line unless a net (external) force acts on it.

Newton's Second Law
The rate of change of momentum of a body is directly proportional to the net force acting on the body, and the momentum change takes place in the direction of the net force.

Newton's Third Law
When object X exerts a force on object Y, object Y exerts a force of the same type that is equal in magnitude and opposite in direction on object X.

The two forces ALWAYS act on different objects and they form an action-reaction pair.

Linear momentum and its conservation:

Mass: is a measure of the amount of matter in a body, & is the property of a body which resists change in motion.

Weight: is the force of gravitational attraction (exerted by the Earth) on a body.

Linear momentum: of a body is defined as the product of its mass and velocity ie p = m v

Impulse of a force (I): is defined as the product of the force and the time Δt during which it acts

ie I = F x Δt {for force which is const over the duration Δt}

For a variable force, the impulse I = Area under the F-t graph { ∫Fdt; may need to “count squares”}

Impulse is equal in magnitude to the change in momentum of the body acted on by the force.
Hence the change in momentum of the body is equal in mag to the area under a (net) force-time graph.
{Incorrect to define impulse as change in momentum}

Force: is defined as the rate of change of momentum, ie F = [ m (v - u) ] / t = ma or F = v dm / dt
The {one} Newton: is defined as the force needed to accelerate a mass of 1 kg by 1 m s^-2.

Principle of Conservation of Linear Momentum: When objects of a system interact, their total momentum before and after interaction are equal if no net (external) force acts on the system.

• The total momentum of an isolated system is constant
• m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂ if net F = 0 {for all collisions }

NB: Total momentum DURING the interaction/collision is also conserved.

(Perfectly) elastic collision: Both momentum & kinetic energy of the system are conserved.

Inelastic collision: Only momentum is conserved, total kinetic energy is not conserved.

Perfectly inelastic collision: Only momentum is conserved, and the particles stick together after collision. (i.e. move with the same velocity.)

For all elastic collisions, u₁ – u₂ = v₂ – v₁

ie. relative speed of approach = relative speed of separation

or, ½ m₁u₁² + ½ m₂u₂² = ½ m₁v₁² + ½ m₂v₂²

In inelastic collisions, total energy is conserved but Kinetic Energy may be converted into other forms of energy such as sound and heat energy.

Forces

Hooke's Law:

Within the limit of proportionality, the extension produced in a material is directly proportional to the force/load applied

F = kx
Force constant k = force per unit extension (F/x)

Elastic potential energy/strain energy = Area under the F-x graph {May need to “count the squares”}

For a material that obeys Hooke‟s law,

Elastic Potential Energy, E = ½ F x = ½ k x²

Forces on Masses in Gravitational Fields:
A region of space in which a mass experiences an (attractive) force due to the presence of another mass.

Forces on Charge in Electric Fields:
A region of space where a charge experiences an (attractive or repulsive) force due to the presence of another charge.

Hydrostatic Pressure p = ρgh

{or, pressure difference between 2 points separated by a vertical distance of h }

Upthrust: An upward force exerted by a fluid on a submerged or floating object; arises because of the difference in pressure between the upper and lower surfaces of the object.

Archimedes' Principle: Upthrust = weight of the fluid displaced by submerged object.

ie Upthrust = Vol_submerged x ρ_fluid x g

Frictional Forces:

• The contact force between two surfaces = (friction² + normal reaction²)^½
• The component along the surface of the contact force is called friction
• Friction between 2 surfaces always opposes relative motion {or attempted motion}, and
• Its value varies up to a maximum value {called the static friction}

Viscous Forces:

• A force that opposes the motion of an object in a fluid
• Only exists when there is (relative) motion
• Magnitude of viscous force increases with the speed of the object

Centre of Gravity of an object is defined as that pt through which the entire weight of the object may be considered to act.

A couple is a pair of forces which tends to produce rotation only.

Moment of a Force: The product of the force and the perpendicular distance of its line of action to the pivot

Torque of a Couple: The produce of one of the forces of the couple and the perpendicular distance between the lines of action of the forces. (WARNING: NOT an action-reaction pair as they act on the same body.)

Conditions for Equilibrium (of an extended object):

1. The resultant force acting on it in any direction equals zero
2. The resultant moment about any point is zero

If a mass is acted upon by 3 forces only and remains in equilibrium, then

1. The lines of action of the 3 forces must pass through a common point
2. When a vector diagram of the three forces is drawn, the forces will form a closed triangle (vector triangle), with the 3 vectors pointing in the same orientation around the triangle.

Principle of Moments: For a body to be in equilibrium, the sum of all the anticlockwise moments about any point must be equal to the sum of all the clockwise moments about that same point.

Work, Energy and Power

Work Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force

W = F s cos θ

Negative work is said to be done by F if x or its compo. is anti-parallel to F
If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. {May need to find area by “counting the squares”. }

By Principle of Conservation of Energy,

Work Done on a system = KE gain + GPE gain + Work done against friction}

Consider a rigid object of mass m that is initially at rest. To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s.

Since F is constant, acceleration is constant,

Therefore, using the equation:

v² = u² +2as,
as = 12 (v² - u²)

Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v,

The kinetic energy, EK

= Work done by the force F = Fs = mas = ½ m (v2 - u2)

Gravitational potential energy: this arises in a system of masses where there are attractive gravitational forces between them. The gravitational potential energy of an object is the energy it possesses by virtue of its position in a gravitational field.

Elastic potential energy: this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them.

Electric potential energy: this arises in a system of charges where there are either attractive or repulsive electric forces between them.

The potential energy, U, of a body in a force field {whether gravitational or electric field} is related to the force F it experiences by:
F = - dU / dx.

Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h₁ to a height h₂. Since the object moves up at a constant speed, F is equal to mg.

The change in potential energy of the mass

= Work done by the force F = F s = F h = m g h

Efficiency: The ratio of (useful) output energy of a machine to the input energy.

ie =	Useful Output Energy	x100% =	Useful Output Power	x100%
	Input Energy		Input Power

Power {instantaneous} is defined as the work done per unit time.

P =	Total Work Done	=	W
	Total Time		t

Since work done W = F x s,

P =	F x s	=	Fv
	t

• for object moving at const speed: F = Total resistive force {equilibrium condition}
• for object beginning to accelerate: F = Total resistive force + ma

Wave Motion

Displacement (y): Position of an oscillating particle from its equilibrium position.

Amplitude (y₀ or A): The maximum magnitude of the displacement of an oscillating particle from its equilibrium position.

Period (T): Time taken for a particle to undergo one complete cycle of oscillation.

Frequency (f): Number of oscillations performed by a particle per unit time.

Wavelength (λ): For a progressive wave, it is the distance between any two successive particles that are in phase, e.g. it is the distance between 2 consecutive crests or 2 troughs.

Wave speed (v): The speed at which the waveform travels in the direction of the propagation of the wave.

Wave front: A line or surface joining points which are at the same state of oscillation, i.e. in phase, e.g. a line joining crest to crest in a wave.

Ray: The path taken by the wave. This is used to indicate the direction of wave propagation. Rays are always at right angles to the wave fronts (i.e. wave fronts are always perpendicular to the direction of propagation).

From the definition of speed, Speed = Distance / Time
A wave travels a distance of one wavelength, λ, in a time interval of one period, T.
The frequency, f, of a wave is equal to 1 / T
Therefore, speed, v = λ / T = (1 / T)λ = fλ

v = fλ

Example 1: A wave travelling in the positive x direction is showed in the figure. Find the amplitude, wavelength, period, and speed of the wave if it has a frequency of 8.0 Hz. Amplitude (A) = 0.15 m Wavelength (λ) = 0.40 m Period (T) = 1f = 18.0 ≈ 0.125 s Speed (v) =fλ = 8.0 x 0.40 = 3.20 m s-1 A wave which results in a net transfer of energy from one place to another is known as a progressive wave.

Intensity {of a wave}: is defined as the rate of energy flow per unit time {power} per unit cross-sectional area perpendicular to the direction of wave propagation.

Intensity = Power / Area = Energy / (Time x Area)

For a point source (which would emit spherical wavefronts),

Intensity = (½mω²x_o²) / (t x 4πr²) where x₀: amplitude & r: distance from the point source.
Therefore, I ∝ x_o² / r² (Pt Source)

For all wave sources, I ∝ (Amplitude)²

Transverse wave:

A wave in which the oscillations of the wave particles {NOT: movement} are perpendicular to the direction of the propagation of the wave.

Longitudinal wave:

A wave in which the oscillations of the wave particles are parallel to the direction of the propagation of the wave.

Polarisation is said to occur when oscillations are in one direction in a plane, {NOT just “in one direction”} normal to the direction of propagation.

{Only transverse waves can be polarized; longitudinal waves can’t.} Example 2: The following stationary wave pattern is obtained using a C.R.O. whose screen is graduated in centimetre squares. Given that the time-base is adjusted such that 1 unit on the horizontal axis of the screen corresponds to a time of 1.0 ms, find the period and frequency of the wave. Period, T = (4 units) x 1.0 = 4.0 ms = 4.0 x 10-3 s f = 1 / T = 14 x 10-3 = 250 Hz

Superposition

Principle of Superposition: When two or more waves of the same type meet at a point, the resultant displacement of the waves is equal to the vector sum of their individual displacements at that point.

Stretched String

A horizontal rope with one end fixed and another attached to a vertical oscillator. Stationary waves will be produced by the direct and reflected waves in the string.

Or we can have the string stopped at one end with a pulley as shown below.

Microwaves

A microwave emitter placed a distance away from a metal plate that reflects the emitted wave. By moving a detector along the path of the wave, the nodes and antinodes could be detected.

Air column

A tuning fork held at the mouth of a open tube projects a sound wave into the column of air in the tube. The length of the tube can be changed by varying the water level. At certain lengths of the tube, the air column resonates with the tuning fork. This is due to the formation of stationary waves by the incident and reflected sound waves at the water surface.

Stationary (Standing) Wave) is one

• whose waveform/wave profile does not advance {move},
• where there is no net transport of energy, and
• where the positions of antinodes and nodes do not change (with time).

A stationary wave is formed when two progressive waves of the same frequency, amplitude and speed, travelling in opposite directions are superposed. {Assume boundary conditions are met}

	Stationary Waves	Stationary Waves Progressive Waves
Amplitude	Varies from maximum at the anti-nodes to zero at the nodes.	Same for all particles in the wave (provided no energy is lost).
Wavelength	Twice the distance between a pair of adjacent nodes or anti-nodes.	The distance between two consecutive points on a wave, that are in phase.
Phase	Particles in the same segment/ between 2 adjacent nodes, are in phase. Particles in adjacent segments are in anti-phase.	All particles within one wavelength have different phases.
Wave Profile	The wave profile does not advance.	The wave profile advances.
Energy	No energy is transported by the wave.	Energy is transported in the direction of the wave.

Node is a region of destructive superposition where the waves always meet out of phase by π radians. Hence displacement here is permanently zero {or minimum}.

Antinode is a region of constructive superposition where the waves always meet in phase. Hence a particle here vibrates with maximum amplitude {but it is NOT a pt with a permanent large displacement!}

Dist between 2 successive nodes / antinodes = λ / 2

Max pressure change occurs at the nodes {NOT the antinodes} because every node changes fr being a pt of compression to become a pt of rarefaction {half a period later}

Diffraction: refers to the spreading {or bending} of waves when they pass through an opening {gap}, or round an obstacle (into the “shadow” region). {Illustrate with diag}
For significant diffraction to occur, the size of the gap ≈ λ of the wave

For a diffraction grating, d sin θ = n λ ,
d = dist between successive slits {grating spacing} = reciprocal of number of lines per metre

When a “white light” passes through a diffraction grating, for each order of diffraction, a longer wavelength {red} diffracts more than a shorter wavelength {violet} {as sin θ ∝ λ}.

Diffraction refers to the spreading of waves as they pass through a narrow slit or near an obstacle.

For diffraction to occur, the size of the gap should approximately be equal to the wavelength of the wave.

Coherent waves: Waves having a constant phase difference {not: zero phase difference / in phase}

Interference may be described as the superposition of waves from 2 coherent sources.

For an observable / well-defined interference pattern, the waves must be coherent, have about the same amplitude, be unpolarised or polarised in the same direction, & be of the same type.

Two-source interference using:

1. Water Waves

Interference patterns could be observed when two dippers are attached to the vibrator of the ripple tank. The ripples produce constructive and destructive interference. The dippers are coherent sources because they are fixed to the same vibrator.

2. Microwaves

Microwave emitted from a transmitter through 2 slits on a metal plate would also produce interference patterns. By moving a detector on the opposite side of the metal plate, a series of rise and fall in amplitude of the wave would be registered.

3. Light Waves (Young‟s double slit experiment)

Since light is emitted from a bulb randomly, the way to obtain two coherent light sources is by splitting light from a single slit.

The 2 beams from the double slit would then interfere with each other, creating a pattern of alternate bright and dark fringes (or high and low intensities) at regular intervals, which is also known as our interference pattern.

Condition for Constructive Interference at a pt P:

Phase difference of the 2 waves at P = 0 {or 2π, 4π, etc}

Thus, with 2 in-phase sources, * implies path difference = nλ; with 2 antiphase sources: path difference = (n + ½)λ

Condition for Destructive Interference at a pt P:

Phase difference of the 2 waves at P = π { or 3π, 5π, etc }
With 2 in-phase sources, + implies path difference = (n+ ½ λ), with 2 antiphase sources: path difference = n λ

Fringe separation x = λD / a,
if a<<D {applies only to Young's Double Slit interference of light, ie, NOT for microwaves, sound waves, water waves}

Phase difference Δφ betw the 2 waves at any pt X {betw the central & 1st maxima) is (approx) proportional to the dist of X from the central maxima.

Using 2 sources of equal amplitude x₀, the resultant amplitude of a bright fringe would be doubled {2x₀}, & the resultant intensity increases by 4 times {not 2 times}. { I_Resultant ∝ (2 x₀)² }

Electric Fields

Electric field strength / intensity at a point is defined as the force per unit positive charge acting at that point {a vector; Unit: N C^-1 or V m^-1}

E = F / q → F = qE

• The electric force on a positive charge in an electric field is in the direction of E, while
• The electric force on a negative charge is opposite to the direction of E.
• Hence a +ve charge placed in an electric field will accelerate in the direction of E and gain KE {& simultaneously lose EPE}, while a negative charge caused to move (projected) in the direction of E will decelerate, ie lose KE, { & gain EPE}.

Representation of electric fields by field lines

Coulomb's law:

The (mutual) electric force F acting between 2 point charges Q₁ and Q₂ separated by a distance r is given by:

F = Q₁Q₂ / 4πε_or² where ε₀: permittivity of free space

or, the (mutual) electric force between two point charges is proportional to the product of their charges & inversely proportional to the square of their separation.

Example 1:

Two positive charges, each 4.18 μC, and a negative charge, -6.36 μC, are fixed at the vertices of an equilateral triangle of side 13.0 cm. Find the electrostatic force on the negative charge.

F = Q1Q2 / 4πεor2 = (8.99 x 109) [(4.18 x 10-6)(6.36 x 10-6) / (13.0 x 10-2)2] = 14.1 N (Note: negative sign for -6.36 μC has been ignored in the calculation) FR = 2 x Fcos300 = 24.4 N, vertically upwards

Electric field strength due to a Point Charge Q : E = Q / 4πε_or²
{NB: Do NOT substitute a negative Q with its negative sign in calculations!}

Example 2:

In the figure below, determine the point (other than at infinity) at which the total electric field strength is zero.

From the diagram, it can be observed that the point where E is zero lies on a straight line where the charges lie, to the left of
the -2.5 μC charge.

Let this point be a distance r from the left charge.

Since the total electric field strength is zero,

E_6μ = E_-2μ

[6μ / (1 + r)²] / 4πε_or² = [2.5μ / r²] / 4πε_or² (Note: negative sign for -2.5 μC has been ignored here)

6 / (1 + r)² = 2.5 / r²

√(6r) = 2.5 (1 + r)

r = 1.82 m

The point lies on a straight line where the charges lie, 1.82 m to the left of the -2.5 μC charge.

Uniform electric field between 2 Charged Parallel Plates: E = Vd,
d: perpendicular dist between the plates,
V: potential difference between plates

Path of charge moving at 90° to electric field: parabolic.
Beyond the pt where it exits the field, the path is a straight line, at a tangent to the parabola at exit.

Example 3:

An electron (m = 9.11 x 10^-31 kg; q = -1.6 x 10^-19 C) moving with a speed of 1.5 x 10⁷ ms^-1, enters a region between 2 parallel plates, which are 20 mm apart and 60 mm long. The top plate is at a potential of 80 V relative to the lower plate. Determine the angle through which the electron has been deflected as a result of passing through the plates.

Time taken for the electron to travel 60 mm horizontally = Distance / Speed = 60 x 10^-3 / 1.5 x 10⁷ = 4 x 10^-9 s
E = V / d = 80 / 20 x 10^-3 = 4000 V m^-1
a = F / m = eE / m = (1.6 x 10^-19)(4000) / (9.1 x 10-³¹) = 7.0 x 10¹⁴ ms^-2
v_y = u_y + at = 0 + (7.0 x 10¹⁴)( 4 x 10^-9) = 2.8 x 10⁶ ms^-1
tan θ = v_y / v_x = 2.8 x 10⁶ / 1.5 x 10⁷ = 0.187
Therefore θ = 10.6°

Effect of a uniform electric field on the motion of charged particles

• Equipotential surface: a surface where the electric potential is constant
• Potential gradient = 0, ie E along surface = 0 }
• Hence no work is done when a charge is moved along this surface.{ W=QV, V=0 }
• Electric field lines must meet this surface at right angles.
• {If the field lines are not at 90° to it, it would imply that there is a non-zero component of E along the surface. This would contradict the fact that E along an equipotential = 0. }

Electric potential at a point: is defined as the work done in moving a unit positive charge from infinity to that point, { a scalar; unit: V } ie V = W / Q

The electric potential at infinity is defined as zero. At any other point, it may be positive or negative depending on the sign of Q that sets up the field. {Contrast gravitational potential.}

Relation between E and V: E = - dV / dr

i.e. The electric field strength at a pt is numerically equal to the potential gradient at that pt.

NB: Electric field lines point in direction of decreasing potential {ie from high to low pot}.

Electric potential energy U of a charge Q at a pt where the potential is V: U = QV
Work done W on a charge Q in moving it across a pd ΔV: W = Q ΔV

Electric Potential due to a point charge Q : V = Q / 4πε_o

Current of Electricity

Electric current is the rate of flow of charge. {NOT: charged particles}

Electric charge Q passing a point is defined as the product of the (steady) current at that point and the time for which the current flows,

Q = I t

One coulomb is defined as the charge flowing per second pass a point at which the current is one ampere.

Example 1:
An ion beam of singly-charged Na⁺ and K⁺ ions is passing through vacuum. If the beam current is 20 μ A, calculate the total number of ions passing any fixed point in the beam per second. (The charge on each ion is 1.6 x 10^-19 C.)

Current, I = Q / t = Ne / t where N is the no. of ions and e is the charge on one ion.

No. of ions per second = N / t = I / e = (20 x 10^-6) / (1.6 x 10^-19) = 1.25 x 10^-14

Potential difference is defined as the energy transferred from electrical energy to other forms of energy when unit charge passes through an electrical device,

V = W / Q

P. D. = Energy Transferred / Charge = Power / Current or, is the ratio of the power supplied to the device to the current flowing,

V = P / I

The volt: is defined as the potential difference between 2 pts in a circuit in which one joule of energy is converted from electrical to non-electrical energy when one coulomb passes from 1 pt to the other, ie 1 volt = One joule per coulomb

Difference between Potential and Potential Difference (PD):
The potential at a point of the circuit is due to the amount of charge present along with the energy of the charges. Thus, the potential along circuit drops from the positive terminal to negative terminal, and potential differs from points to points.

Potential Difference refers to the difference in potential between any given two points.
For example, if the potential of point A is 1 V and the potential at point B is 5 V, the PD across AB, or V_AB , is 4 V. In addition, when there is no energy loss between two points of the circuit, the potential of these points is same and thus the PD across is 0 V.

Example 2:
A current of 5 mA passes through a bulb for 1 minute. The potential difference across the bulb is 4 V. Calculate:

(a) The amount of charge passing through the bulb in 1 minute.

Charge Q = I t = 5 x 10^-3 x 60 = 0.3 C

(b) The work done to operate the bulb for 1 minute.

Potential difference across the bulb = W / Q
4 = W / 0.3
Work done to operate the bulb for 1 minute = 0.3 x 4 = 1.2 J

Electrical Power, P = V I = I² / R = V² / R

{Brightness of a lamp is determined by the power dissipated, NOT: by V, or I or R alone}

Example 3:
A high-voltage transmission line with a resistance of 0.4 Ω km^-1 carries a current of 500 A. The line is at a potential of 1200 kV at the power station and carries the current to a city located 160 km from the power station. Calculate

(a) the power loss in the line.

The power loss in the line P = I² R = 500² x 0.4 x 160 = 16 MW

(b) the fraction of the transmitted power that is lost.

The total power transmitted = I V = 500 x 1200 x 10³ = 600 MW
The fraction of power loss = 16 / 600 = 0.267

Resistance is defined as the ratio of the potential difference across a component to the current flowing through it ,

R = VI

{It is NOT defined as the gradient of a V-I graph; however for an ohmic conductor, its resistance equals the gradient of its V-I graph as this graph is a straight line which passes through the origin}

The Ohm: is the resistance of a resistor if there is a current of 1 A flowing through it when the pd across it is 1 V, ie,
1 Ω = One volt per ampere

Example 4:
In the circuit below, the voltmeter reading is 8.00 V and the ammeter reading is 2.00 A. Calculate the resistance of R.

Resistance of R = V / I = 8 / 2 = 4.0 Ω

Temperature characteristics of thermistors:

The resistance (i.e. the ratio V / I) is constant because metallic conductors at constant temperature obey Ohm's Law.	As V increases, the temperature increases, resulting in an increase in the amplitude of vibration of ions and the collision frequency of electrons with the lattice ions. Hence the resistance of the filament increases with V.
A thermistor is made from semi-conductors. As V increases, temperature increases. This releases more charge carriers (electrons and holes) from the lattice, thus reducing the resistance of the thermistor. Hence, resistance decreases as temperature increases.	In forward bias, a diode has low resistance. In reverse bias, the diode has high resistance until the breakdown voltage is reached.

Ohm's law: The current in a component is proportional to the potential difference across it provided physical conditions (eg temp) stay constant.

R = ρL / A {for a conductor of length l, uniform x-sect area A and resistivity ρ}

Resistivity is defined as the resistance of a material of unit cross-sectional area and unit length. {From R = ρl / A , ρ = RA / L}

Example 5:
Calculate the resistance of a nichrome wire of length 500 mm and diameter 1.0 mm, given that the resistivity of nichrome is
1.1 x 10^-6 Ω m.

Resistance, R = ρl / A = [(1.1 x 10^-6)(500 x 10^-3)] / π(1 x 10^-3 / 2)² = 0.70 Ω

Electromotive force (Emf) is defined as the energy transferred / converted from non-electrical forms of energy into electrical energy when unit charge is moved round a complete circuit. ie EMF = Energy Transferred per unit charge

E = WQ

EMF refers to the electrical energy generated from non-electrical energy forms, whereas PD refers to electrical energy being changed into non-electrical energy. For example,

EMF Sources	Energy Change	PD across	Energy Change
Chemical Cell	Chem → Elec	Bulb	Elec → Light
Generator	Mech → Elec	Fan	Elec → Mech
Thermocouple	Thermal → Elec	Door Bell	Elec → Sound
Solar Cell	Solar → Elec	Heating element	Elec → Thermal

Effects of the internal resistance of a source of EMF:

Internal resistance is the resistance to current flow within the power source. It reduces the potential difference (not EMF) across the terminal of the power supply when it is delivering a current.

Consider the circuit below:

The voltage across the resistor, V = IR,
The voltage lost to internal resistance = Ir
Thus, the EMF of the cell, E = IR + Ir = V + Ir
Therefore If I = 0A or if r = 0Ω, V = E

Nuclear Physics

Experimental evidence for a small charged nucleus in an atom:

Results of an experiment where a beam of alpha particles is fired at a thin gold foil, where n= no of alpha particles incident per unit time.

Most of the α-particles passed through the metal foil were deflected by very small angles.
A very small proportion was deflected by more than 90°, some of these approaching 180°
From these 2 observations it can be deduced that: the nucleus occupies only a small proportion of the available space (ie the atom is mostly empty space) & that it is positively charged since the positively-charged alpha particles are repelled/deflected.
Nucleon: A particle within the nucleus; can be either a proton or a neutron
Nuclide: An atom with a particular number of protons and a particular number of neutrons
Proton number Z {old name: atomic number}: No. of protons in an atom
Nucleon number N {mass number}: Sum of number of protons and neutrons in an atom
Isotopes: are atoms with the same proton number, but different nucleon number {or different no of neutrons}

Mass defect and Nuclear Binding Energy::

Energy & Mass are Equivalent: E = mc² → ΔE = (Δm)c²

Nuclear Binding Energy:

• Energy that must be supplied to completely separate the nucleus into its individual nucleons/particles.

OR

• The energy released {not lost} when a nucleus is formed from its constituent nucleons.

B.E. per nucleon is a measure of the stability of the nucleus.
Mass Defect: The difference in mass between a nucleus and the total mass of its individual nucleons

Zm_p + (A - Z)m_n - Mass of Nucleus

Thus, Binding Energy. = Mass Defect x c²
In both nuclear fusion and fission, products have higher B.E. per nucleon {due to shape of BE per nucleon-nucleon graph}, energy is released {not lost} and hence products are more stable.

Energy released = Total B.E. after reaction (products) - Total B.E. before reaction (reactants)

Nuclear fission: The disintegration of a heavy nucleus into 2 lighter nuclei. Typically, the fission fragments have approximately the same mass and neutrons are emitted.
Fig below shows the variation of BE per nucleon plotted against the nucleon no.

Warning!!! Graph is NOT symmetrical.

Principle of Conservation of Energy-Mass:

Total energy-mass before reaction = Total energy - mass after reaction
Σ (mc² + ½ mv²)_reactants = Σ (mc² + ½ mv²)_products + hf {if γ - photon emitted}

Energy released in nuclear reaction= Δmc² = (Total rest mass before reaction – Total rest mass after reaction) x c²

Radioactive decay:

Radioactivity is the spontaneous and random decay of an unstable nucleus, with the emission of an alpha or beta particle, and is usually accompanied by the emission of a gamma ray photon.
Spontaneous: The emission is not affected by factors outside the nucleus
Random: It cannot be predicted when the next emission will occur {Evidence in fluctuation in count-rate}
Decay law: dN/dt = -λN, where N= No. of undecayed {active} nuclei at that instant;

N = N₀e_-λt ; A = A₀e^-λt ; C = C₀e^-λt

Background radiation refers to radiation from sources other than the source of interest.

True count rate = Measured count rate – Background count rate

Nature of α, β & γ:

	Alpha Particles	Beta particles	Gamma Particles
Notation	α	β	Γ
Charge	+ 2e	- e	No charge
Mass	4u	1/1840 u	Massless
Nature	Particle {He nucleus}	Particle {electron emitted from nucleus}	Electromagnetic Radiation
Speed	Monoenergetic (i.e. one speed only)	Continuous range (up to approximately 98% of light)	100% speed of light

Decay constant λ is defined as the probability of decay of a nucleus per unit time {or,the fraction of the total no. of undecayed nuclei which will decay per unit time}
Activity is defined as the rate at which the nuclei are disintegrating; A = dN/dt = λN

A₀ = λ N₀

Number of undecayed nuclei ∝ Mass of sample

Number of nuclei in sample = (Sample Mass / Mass of 1 mol) x N_A
where, Mass of 1 mol of nuclide= Nucleon No {or relative atomic mass} expressed in grams {NOT: in kg!!}
{Thus for eg, mass of 1 mole of U-235 = 235 g = 235 x 10^-3kg, NOT: 235 kg}
Application of PCM to radioactive decay:
It is useful to remember that when a stationary nucleus emits a single particle, by PCM, after the decay,the
ratio of their KE = ratio of their speeds = reciprocal of the ratio of their masses
Half-life is defined as the average time taken for half the number {not: mass or amount} of undecayed nuclei in the sample to disintegrate,
or, the average time taken for the activity to be halved.

t_½ = (ln2) / λ

Example:
Antimony-124 has a half-life of 60 days. If a sample of antimony-124 has an initial activity of 6.5 x 10⁶Bq, what will its activity be after 1 year (365 days)?
Using A = A₀e^-λt & t_½ = (ln2) / λ
A = 9.6 x 10⁴Bq

Biological effect of radiation:

Radiation damage to biological organisms is often categorized as: somatic and genetic.
Somatic damage refers to any part of the body except the reproductive organs.
Somatic damage harms that particular organism directly. Some somatic effects include radiation sickness (nausea, fatigue, and loss of body hair) and burns, reddening of the skin, ulceration, cataracts in the eye, skin cancer, leukaemia, reduction of white blood cells, death, etc.
Genetic damage refers to damage to reproductive organs.
Genetic effects cause mutations in the reproductive cells and so affect future generations – hence, their effects are indirect. (Such mutations may contribute to the formation of a cancer.)
Alternatively,

• Ionising radiation may damage living tissues and cells directly.
• It may also occur indirectly through chemical changes in the surrounding medium, which is mainly water. For example, the ionization of water molecules produces OH free radicals which may react to produce H₂O₂, the powerful oxidizing agent hydrogen peroxide, which can then attack the molecules which form the chromosomes in the nucleus of each cell.

                                                      Good luck