24 July

19. Quantum Physics

Seng Kwang Tan Subject Content

[accordions autoHeight=’true’]

[accordion title=”1. Particle Nature of Light”]

  • A photon is a quantum of electromagnetic radiation.
  • The energy of a photon is given by E=hf, where h is Planck’s constant (6.63 $$\times$$ 10-34 J s) and f is its frequency.

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[accordion title=”1.1 Photoelectric Effect”]

  • The photoelectric effect is the emission of electrons from a metal surface when electromagnetic radiation of sufficiently high frequency is shone on it.
  • The energy of an incident photon is the sum of the maximum kinetic energy $$K.E._{max}$$ of the emitted electrons from the metal surface and the work function $$\Phi$$ of the metal. Einstein’s photoelectric equation states that

$$hf=\Phi +K.E._{max}=hf_o +K.E._{max}$$

  • where $$f_o$$ is the threshold frequency or minimum frequency of the electromagnetic radiation below which no electrons are emitted from the metal surface regardless of the intensity of the radiation.
  • The work function $$\Phi$$ of a metal is the minimum energy needed to remove an electron from the metal surface.
  • $$K.E._{max}$$ can be measured by applying a voltage to prevent the emitted electrons from reaching the electrode that collects them. This voltage is known as the stopping voltage $$V_s$$ and since the charge of an electron is e, the equation can be rewritten as

$$hf=\Phi + eV_s$$.

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[accordion title=”1.2 Line Spectra”]

  • An atom is in the ground state when its electron occupies the lowest energy level. When the atom gains energy, its ground state electron makes a transition to a higher energy level. The atom is said to be in an excited state.
  • At this excited state, the electron is unstable. It will jump to a lower energy level by emitting a photon whose energy is equal to the energy difference between the two levels. The photon energy is given hf = Ehigher – Elower.
  • The emission line spectra are the spectra of light radiated by individual atoms in a hot gas when the electrons in the atoms jump from higher energy levels to lower energy levels. Each spectrum consists of coloured lines on a dark background.
  • The absorption line spectra consists of dark lines on a coloured background. When a beam of white light is passed through a cool gas, photons whose energies are equal to the excitation energies of the gas atoms, are absorbed. These photons are re-emitted in all directions, so the intensity of these wavelengths in the transmitted white light beam is reduced.

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[accordion title=”2. Wave Nature of Particles”]

  • Louis de Broglie postulated that, because photons have wave and particle characteristics, perhaps all forms of matter have both properties. Electron diffraction provides evidence for the wave nature of particles.
  • The de Broglie wavelength of a particle is given by $$\lambda = \dfrac{h}{p}$$ where p is the momentum (mv) of the particle and h is Planck’s constant.

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[accordion title=”3. X-ray Spectrum”]

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[accordion title=”4. Heisenberg Uncertainty Principle”]

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[accordion title=”5. Wave Function and Probability”]

  • An electron can be described by a wave function $$\Psi$$ where the square of the amplitude of the wave function $$|{\Psi}|^2$$ gives the probability of finding the electron at a point.

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[accordion title=”6. Quantum Tunneling”]

  • Classically, an electron of energy E approaching a potential barrier, whose height U is greater than E, cannot penetrate the barrier but would simply be reflected and return in the opposite direction.
  • However, quantum mechanics predicts that since $$|{\Psi}^2|$$ is non-zero beyond the barrier, there is a finite chance of this electron tunnelling through the barrier and reaching the other side of the barrier.
  • The transmission coefficient T represents the probability with which an approaching electron will penetrate to the other side of the barrier. The transmission coefficient T is given by $$T=e^{-2kd}$$ where $$k=\sqrt{\dfrac{8\pi^2m(U-E)}{h^2}}$$

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18 July

20. Nuclear Physics

Seng Kwang Tan Subject Content

The Nucleus

  • existence and size demonstrated using the Rutherford $$\alpha$$-scattering experiment.
  • consists of nucleons (protons and neutrons)
  • isotopes of an element share the same number of protons but different number of neutrons.

Nuclear Reactions

  • nuclear reactions involve two or more reactants.
  • represented using the form: $${^{14}_7N}+{^4_2He}\rightarrow{^{17}_8O}+{^1_1H}$$
  • for a reaction that releases energy, mass-energy of reactants = mass-energy of products + E,
    where $$E = mc^2$$ and m is the mass defect (difference in mass between the products and reactants).
  • binding energy is the energy released when the nucleus is formed from its separate protons and neutrons. The same amount of energy is required to break up a nucleus into its constituent nucleons.

  • binding energy per nucleon ($$\frac{B.E.}{A}$$) is an indication of the stability of a nucleus, where B.E .is binding energy and A is the nucleon number. You need to know how to sketch its variation with nucleon number. (The following video explains the shape of the $$\frac{B.E.}{A}$$ versus A graph and why it peaks at $$^{56}Fe$$.

  • nuclear fission is the disintegration of a heavy nucleus into two lighter nuclei of comparable mass with the emission of neutrons and release of energy.
    e.g. $${^{235}_{92}U}+{^1_0n}\rightarrow{^{236}_{92}U}\rightarrow{^{144}_{56}Ba}+{^{90}_{36}Kr}+2^1_0n+Energy$$
  • nuclear fusion occurs when two light nuclei combine to form a single more massive nucleus, leading to the release of energy.
    e.g. $${^2_1H}+{^3_1H}\rightarrow{^4_2He}+{^1_0n}+Energy$$

  • The following quantities are always conserved:
    • proton number & neutron number
    • momentum
    • mass-energy

Radioactive Decay

  • spontaneous and random emission of radiation from a radioactive nucleus.
    • $$\alpha$$ particle – helium nucleus
    • $$\beta$$ particle – electron
    • $$\gamma$$ particle – electromagnetic radiation

http://youtu.be/Qlb5Z8QBpcI

  • $$A=-\frac{dN}{dt}=\lambda N$$
    where A is the rate of disintegration or activity, N is the number of radioactive nuclei and $$\lambda$$ is the decay constant.
  • $$x=x_0{e^{-\lambda t}}$$
    where x could represent the activity, number of undecayed particles or received count rate.
  • half-life ($$t_{\frac{1}{2}}$$) is the average time taken for half the original number of radioactive nuclei to decay.
  • From $$x=x_0{e^{-\lambda t}}$$,
    $$\frac{x}{x_0}=\frac{1}{2}=e^{-\lambda t_{\frac{1}{2}}}$$
    $$\Rightarrow{-ln2}=-\lambda t_{\frac{1}{2}}$$
    $$\Rightarrow{t_{\frac{1}{2}}}=\frac{ln 2}{\lambda}$$
  • You may also use $${\frac{x}{x_0}}={\frac{1}{2}}^{\frac{t}{t_{1/2}}}$$, as shown in the following video.

13 June

P-N Junction

Seng Kwang Tan Teaching Resources

The following is the transcript for a video that I will be making to explain how a P-N junction works.

A p-n junction, as the name suggests, is the boundary between two types of semiconductors: P-type and N-type.

For an intrinsic or pure semiconductor such as silicon which has 4 valence electrons, each atom is bond to 4 other neighbouring atoms.
The p-type semiconductor is one with excess holes due to the addition of dopants to intrinsic semiconductors. Elements such as boron or phosphorus from Group III of the periodic table all contain three valence electrons, causing them to function as acceptors when used to dope silicon. When an acceptor atom replaces a silicon atom in the crystal, a vacant state ( an electron “hole”) is created, which can move around the lattice and functions as a charge carrier.

The n-type semiconductor is one doped with Group V elements which have five valence electrons, allowing them to act as a donor; substitution of these atoms for silicon creates an extra free electron. Therefore, a silicon crystal doped with boron creates a p-type semiconductor whereas one doped with phosphorus results in an n-type material.

When the two types of semiconductors are put together, electrons diffuse across the boundary to combine with holes, creating a depletion region where there are no charge carriers. An electric field is also set up in the depletion region because the group III atoms are now negatively charged, having gained one more electron and the group V atoms are now positively charged, having each lost an electron. This electric field prevents further charges from diffusing across the boundary.

That is, until a potential difference is applied. The p-n junction serves now as a diode. We shall illustrate this with a single cell attached to the device. In the reverse-biased mode, the positive terminal is connected to the n-type semiconductor while the negative terminal is connected to the p-type end. This causes more electrons to move away from the depletion region in the n-type semiconductor and for more holes to be filled in the p-type semiconductor. The result is a widened depletion region and a larger opposing electric field.

In the forward-biased mode, the positive terminal of the cell is connected to the p-type semiconductor while the negative terminal is connected to the n-type end. The potential difference provided offers the electrons in the n-type semiconductor a push to overcome the small electric field formed across the depletion region and flow across to the p-type semiconductor which it then passes from hole to hole into the positive terminal.

12 June

Planning Question

Seng Kwang Tan 21 Practical / Planning

The following is the script that I used for recording the video above for our e-learning day. I’m posting it here as I can edit it easily via wordpress’s mobile app, and because I have LaTeX enabled here.

Aim:

The aim of this planning question is to investigate how the resonant frequency of a wire vibrating in its fundamental mode depends on the tension in the wire.

The independent variable is the tension in the string which can be varied by hanging masses at one end of the wire and dangling that end over the edge of a table on a pulley. The tension is represented by the symbol.

The dependent variable is the resonant frequency of the fundamental mode in the wire. The fundamental mode consists of two nodes at both ends of the wire and one antinode in the middle.

In other words, in our experiment, we shall vary tension and measure the resonant frequency of the fundamental mode.

The length of wire between the supports is kept constant throughout the experiment and we shall use the same wire throughout so that the mass per unit length is kept constant.

Procedure:

The experiment will be set up according to this diagram. One end of the wire is first tied to a fixed object. The other end is hanging over a pulley clamped on the edge of the laboratory table and tied to a mass hanger. To control the length of the wire, place a bridge at each end. Only the length of the wire between the two bridges will be vibrating. We shall keep this length constant for the whole experiment.

Record the mass of the mass hanger m and determine its weight. The weight mg is taken to be equal to the tension T on the string.

A uniform horizontal magnetic field is generated by a pair of large electromagnetic coils on both sides of the wire. The wire is connected on both ends to a function generator. An alternating current is produced by the function generator and its frequency can be varied using the same apparatus. The function generator should have a display that enables us to read the frequency. 

The resulting magnetic force acting on the wire will be driving the oscillation of the wire at the frequency shown on the function generator. By adjusting the frequency of the alternating current until the fundamental mode of a standing wave is formed on the wire, we can record the resonant frequency fo for the corresponding tension T.

We shall repeat the experiment for different values of m by adding known amounts of mass (e.g. 50 g increments each time) onto the mass hanger. All the values for the mass m and resonant frequency for the fundamental modes fo should be recorded and tabulated.

The tension in the wire is then calculated using the equation T = mg.

Assume that the resonant frequency for the fundamental mode fo and the tension T follows the equation  fo = kTn where k and n are constants. Then lg fo = lg k + n lg T. Plotting a graph of lg fo versus lg T, we can conclude that the assumption is correct if a linear relationship is observed and we can obtain the values of n and k from the gradient and the vertical intercept of the graph respectively.

As a safety consideration, the person conducting the experiment should wear goggles as the wire at high tension might suddenly snap or come loose. Always handle power supply with care.

As a precaution to improve reliabiliy, we can place a white card behind the vibrating wire so it can be seen easily. To make sure that the weight of each of the slotted masses is as indicated, measure them on a weighing balance. FInally, ensure that the pulley is smooth by measuring the tension in the wire using a force meter to check that it is indeed equal to the weight of the slotted mass.

01 June

Boyle’s Law

Seng Kwang Tan 08 Temperature and Ideal Gases, Demonstrations

Using a hand-operated vacuum pump, we can demonstrate the relationship between pressure and volume of a gas. According to Boyle’s law, the pressure of a gas of constant mass and temperature will be inversely proportional to its volume.

In our demonstration, we will reduce the ambient pressure within the sealed container, hence allowing the higher internal pressure of a balloon to cause it to expand. When the volume within the balloon increases, the internal pressure can be observed to decrease until it is in equilibrium with the surrounding pressure.

While the relationship between pressure and volume is not exactly obeying Boyle’s law due to additional factors such as the tension due to the elastic property of the balloon, it does demonstrate an inverse relationship.