Scholarship Physics

Introduction

General Exam Advice

Students will be expected to demonstrate demonstrate:

• significant physics insight across a wide variety of situations
• ability to provide clear and concise explanations
• coherent and structured mathematical approaches to calculations
• depth and breadth of conceptual understanding

Spend time making sure that you understand the problem.

• Spend time thinking before you start writing.

Significant Figures

All answers should be rounded to the least number of significant figure of the given quantities. Also make sure not to "round as you go", as this can lead to rounding errors.

Table of Contents

• Waves
  • Interference
  • Beats
  • Standing waves in strings and pipes
  • Doppler Effect
• Mechanics
  • Center of Mass
  • Circular Motion
• Modern Physics
  • Bohr Model
  • Spectral Lines
  • The photoelectric effect
  • de Broglie wavelength
  • Nuclear Reactions
• Electricity and Magnetism
  • Capacitance
  • The Dielectric
  • DC Circuits
  • Flux and Induction
  • The Transformer
  • Self Inductance
  • Energy of an Inductor
  • AC circuits

Waves

Interference

Interference happens when two waves, each from individual point sources, pass through each other. Because of the way the troughs and crests from each of the waves add together when they meet, a pattern s formed which consists of lines along Huygens Principle which amplitude has been reduced to nothing.

Antinodal lines are when crests from one wave source meets crests from the others, forming a line of high-amplitude waves.

Along the nodal lines, crests from one wave source meet throughs from the other, forming a line of zero-amplitude waves.

Tbe pattern can be sharpened by using multiple sources. An additional condition is that the waves must be coherent meaning that they are monochromatic.

When we add more sources

• Brighter
• Fringes become sharper
• Separation between dark and light areas s is the same

We have nodes and anti-nodes when the two waves are either in phase or out of phase, this means that the difference in wave length is a multiple of the wavelength. Or $= n\lambda$ From this we get that for interference nodes and anti-nodes

$$\text{path difference} = n\lambda$$

which is where all our diffraction formulas come from

Requirements for diffraction interference

1. The two sources should have the same wavelength and frequency
2. The two sources should have a fixed phase difference
3. A separation between the two sources where the separation between the two sources should be greater than the wavelength of the sources
4. The same amplitude

If the numbers of fringes is too large, then the fringes become unobservable.

Beats

if two waves of slightly different frequencies $f_1$ and $f_2$ pass through each other, the combined wave will have an amplitude which varies between large and small. The variation in amplitude is known as beats. The frequency of variation in amplitude is the difference between the individual frequencies

$$f_b = |f_1-f_2|$$

If the waves are sound waves, the beats are heard as a wobble of loud and soft sounds.

Standing waves in strings and pipes

The $n$ th harmonic of the base frequency $f$ has frequency $nf$. Strings and open tubes have all the harmonics. Closed tubes have only odd harmonics.

When air is allowed to move, it is able to equalize the pressure

• Standing waves are formed by the interference of 2 coherent waves which are in opposite phase or travelling in opposite directions
• Caused in open tube by reflection due to change in impedance between inside and outside.

Doppler Effect

• The velocity is directly towards the source.

Mechanics

Center of Mass

the term system is used to describe two or more objects when they need to be considered a single object. The motion of a system can be treated in the same way as the motion of a single point object provided it is the center of mass of the system that is considered.

Circular Motion

Modern Physics

Bohr Model

According to the bohr model, the energy and the angular momentum of the single orbital electron in the hydrogen atom is restricted to certain values:

$$E_n = \frac{-hcR}{n^2}$$

where $R$ is a constant, called the Rydberg constant.

Each different value of $n$ relates to a particular energy level (energy state, quantum state). The first energy level, $E_1$ is also called the ground state -- this is the most stable state for the electron. The highest energy level is the level at which the electron is no longer bound to the nucleus, it is free; ionisation has occurred At this level, the electron has the least energy that a free electron can have.

The least energy a free electron can have is zero, An orbiting electron has less energy than a free electron, so orbiting electrons must have energies that have negative values.

The electron can jump from one energy level to another. To jump to a higher energy level it must take in energy. When it jumps to a lower level it gives out energy in the form of electromagnetic radiation or a photon.

The frequency of the photon emitted when a electron jumps between two energy levels is given by

$$E = hf$$

The wavelength of the electromagnetic radiation involved in an energy ump between $S$ and $L$ is given by

$$\frac{1}{\lambda} = R\left( \frac{1}{S^2} + \frac{1}{L^2} \right)$$

Spectral Lines

If the atoms of a gas / vapour are given extra energy, their electrons will jump to higher energy level. These excited electrons will continuously return to a more stable lower energy level, emitting photons of energy as they do so. As for the hydrogen atom, these photons have certain energy values only, and so the frequencies of the emitted electromagnetic radiation also have certain values only. If the emitted radiation is in the visible part of the electromagnetic spectrum, the emitted photons can be seen as a series of discrete coloured lines called an emission line spectrum.

If white light is shone through a gas/vapour, the electrons absorb photons whose energies allow a jump from one energy level to another, (electrons only absorb photons whose energy level exactly matches the energy difference between two different levels). If this light is spread into a spectrum the missing frequencies correspond to the absorbed photons. These missing frequencies can be seen as the dark lines within the spread of colours. This is called an absorption line spectrum.

📝 Note: the light spreads are made using a prism or diffraction grating.

The photoelectric effect

When light is incident on a metal surface, electrons will be emitted from the metal. For an electron to be emitted it must be given energy; firstly, enough to free it from the metal surface and enough for it to move away from the surface. If a photon does not have enough energy to do this then no electrons will be emitted. The energy minimum amount of energy needed for this for a specific metal is called the work function $\phi$.

$$E_f = \phi + E_k$$

• The photoelectric effect will not happen if the energy of the individual photons do not exceed the work function.
• Each photon can only free one electron, so the rate which electrons are emitted is proportional to the intensity.

Comparison between the wave and the particle model of photons

de Broglie wavelength

The de Broglie wavelength is the wavelength associated with a particle due to its momentum which comes from the de Broglie relationship of light

$$p = \frac{h}{\lambda}$$

Nuclear Reactions

The mass of a particle is derived from both its matter and its energy. The mass of a particle changes if its energy changes. The relationship between mass and energy is:

$$E = mc^2$$

💡 Tip: bring this up

The difference between the mass of a nucleus and the total mass of its constituent particles when separated infinitely far apart is called the mass deficit. The corresponding energy is called the binding energy which is the energy required to separate the protons and neutrons so that they are infinitely far apart.

BEnP or binding energy per nucleon is a more useful quantity for the comparison of different nuclei. The greater the binding energy per nucleon, the more stable the nucleus. This is because more energy is required in order to break all the nucleons apart.

BEnP increases sharply as nucleon number increases up until $56$ nucleons or Iron-56 where it peaks and then gradually decreases.

Nuclear reactions will release energy if the binding energy per nucleon is greater than the binding energy per nucleon of the reactants. The nuclear force or the residual strong force is the force between two or more nucleons. It is responsible for the binding of protons and neutrons in the nucleus. This force:

• is powerfully attractive between nucleons at a small distance of about 1 fm
• rapidly decreases to virtually nothing at distances beyond 2.5 fm
• at distances less than 0.7 fm is becomes repulsive At short distances it is stronger than electrostatic repulsion between protons. But beyond 2.5 fm the electrostatic force dominates and becomes the major force between protons.

Electricity and Magnetism

Capacitance

A parallel plate capacitor is constructed from two parallel metal plates; its purpose is to 'store' charge. Negative charge on one plate is separated from positive charge on the other plate by insulating material.

The ability of a capacitor to 'store' charge depends on its capacitance, C, measured in farads, F. Capacitance depends on the construction of the capacitor, sad so is a constant for a particular capacitor.

$$C = \frac{\epsilon_0 \epsilon_rA}{d}$$

where $\epsilon_r$ is the dielectric constant for the insulating material between the plates and is dimensionless. When connected in a circuit, the amount of charge stored is given by

$$Q = VC$$

The electric field between toe oppositely charged plates is related to the voltage across them and the distance between them by

$$V = Ed$$

The rate of discharge of a capacitor is proportional to the amount of charge 'stored' in the capacitor, hence it decays exponentially with time constant

$$\tau = RC$$

The energy stored across two plates is

$$\frac{1}{2}QV = \frac{1}{2}CV^2$$

Capacitance adds in parallel and adds reciprocally in series.

The Dielectric

Applying voltage causes charge to accumulate on the two plates, separated by an insulating dielectric.

To be a good dielectric a material should be

1. Insulating
2. Easily Polarized / Able to hold electric charge

Dielectric strength is the maximum electric field that a material can withstand without breaking down or becoming conductive

Dielectric Loss is the amount of energy that is dissipated as heat when an alternating electric field is applied to a material.

Work is done to remove the dielectric as energetically it decreases $C$ but also because there is an attractive force between the dielectric and the plates.

💡 Tip: Always think about what is kept constant.

📝 Note: The electric field extends past the plates, this applies to both magnetic and electric with the overlapping of fields in the magnetic case.

Fields polarise the dielectric creating opposite charges on the dielectric material. the polarisation of the dielectric creates an induced electric field which interacts with the existing electric field leading to an attractive force.

The dielectric must completely fill the space between the two plates. Dielectrics increase capacitance because the dielectric becomes polarised by the charge stored across the two plates, this opposes the electric field of the two plates which decreases voltage and hence increases capacitance.

DC Circuits

• Kirchkov's Voltage Law
• Kirchkov's Current Law

📝 Note: The power dissipated in parallel is the same, even when we add more parallel paths.

$$P = \frac{V^2}{R}$$

Flux and Induction

The symbol for magnetic flux is $\phi$. It has the unit weber Wb. It is the product of the area of a loop of wire and the magnetic field strength perpendicular to the loop of wire.

Faraday's Law says that the induced voltage through a loop of wire is given by

$$\text{EMF} = -\frac{d\phi}{dt}$$

Lenz's Law states that the direction of the induced voltage is such that it will opposes the change that is causing it.

The Transformer

When current flows in a coil there is a magnetic flux along its core. In a transformer, the magnetic flux due to a current in one coil is guided by an iron core through another coil. If the current in the primary is changing, the flux changes and changing flux creates a changing current in the second one.

$$\frac{V_1}{V_2} = \frac{N_1}{N_2}$$

Self Inductance

When current flow in a coil, there is a magnetic flux along its core. When this current is changing, the magnetic flux is changing, this changing magnetic flux induces a back EMF which opposes the direction of the current that created it (Lenz's Law).

$$\text{EMF} = - \frac{dI}{dt}$$

When a switch is connected and disconnected, there is a changing current and thus a large back EMF is produced which entirely opposes the voltage.

$$\tau = \frac{L}{R}$$

Energy of an Inductor

When a current is passed through an inductor, part of the energy supplied by the voltage source is changed to heat by the resistor, but the rest is stored in the inductor. In the magnetic field produced around the wire / inductor.

$$E = \frac 1 2 L I^2$$

📝 Note: Discuss the interaction of the induced and original magnetic field

Eddy currents are currents through a block a metallic material that results from a changing magnetic flux through the material. This leads to wasted energy in the form of heat. The eddy current produces a magnetic field which opposes the original magnetic field. large area = lager eddy currents = more energy wasted.

Inductors store magnetic field as current flows through it, when the current is cut off. The energy storied in the magnet field drives a current which decreases exponentially

📝 Note: Capacitors maintain voltage, Inductors maintain current

📝 Note: Wires are uncharged!

💡 Tip: Always consider the loss of energy due to heat, RESISTANCE = HEAT LOSS

AC circuits

Although capacitors and pure inductors do not have resistance, they do affect the flow of current. The measure of the effect on the current of a capacitor or an inductor is called the reactance $X$

The combined effect of the resistance and reactance in a circuit is called the impedance $Z$

The current across all 3 components is the same, the voltage across an inductive leads the supply voltage by 90 degree, and the voltage across a capacitor lags by 90 degrees. The sum of the voltages across all is the supply voltage.