Subject Content

• The root-mean-square value of an alternating current is equivalent to the steady direct current that would dissipate heat at the same rate as the alternating current in a given resistor.
• For a sinusoidal source,
  (a) the root mean square value of the current is given by $$I_{rms}=\frac{I_o}{\sqrt{2}}$$.
  (b) the mean or average power < P > absorbed by a resistive load is half the maximum power.
  $$<P>=\frac{1}{2}{P}_o=\frac{1}{2}{I}_o{V}_o=\frac{1}{2}{I_o}^{2}R=\frac{V_o^2}{2R}$$.
  
• An a.c. transformer is a device for increasing or decreasing an a.c. voltage. It consists of a primary coil of N_p turns and voltage V_p and secondary coil of N_s turns and voltage V_s wrapped around an iron core.
• For an ideal transformer (assuming no energy is lost), the following equation is obeyed
  $$\frac{N_s}{N_p}=\frac{V_s}{V_p}=\frac{I_p}{I_s}$$.
• Power loss in the transmission lines is minimized if the power is transmitted at high voltages (i.e. low currents) since $$P_{loss}=I^{2}R$$ where I is the current through the cables and R is the resistance of the cables.
• The equation $$P=\frac{V^2}{R}$$ is often mistakenly used to suggest that power lost is high when voltage of transmission is high. In fact, V refers to the potential difference across the cables, which often have but a fraction of the overall resistance through which the current passes.

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[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 $$\mathrm{\Phi }$$ of the metal. Einstein’s photoelectric equation states that

$$hf=\mathrm{\Phi }+K.E{.}_{max}=h{f}_o+K.E{.}_{max}$$

• 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 $$\mathrm{\Phi }$$ 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

$$hf=\mathrm{\Phi }+e{V}_s$$.

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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 = E_higher – E_lower.
• 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 $$\lambda ={\displaystyle \frac{h}{p}}$$ 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 $$\mathrm{\Psi }$$ where the square of the amplitude of the wave function $$|\mathrm{\Psi }{|}^2$$ 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 $$|{\mathrm{\Psi }}^2|$$ 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 $$k=\sqrt{{\displaystyle \frac{8{\pi }^{2}m(U-E)}{h^2}}}$$

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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: $${}_7^{14}N+{}_2^{4}He\to {}_8^{17}O+{}_1^{1}H$$
• for a reaction that releases energy, mass-energy of reactants = mass-energy of products + E,
  where $$E=m{c}^2$$ 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. $${}_{92}^{235}U+{}_0^{1}n\to {}_{92}^{236}U\to {}_{56}^{144}Ba+{}_{36}^{90}Kr+2_0^{1}n+Energy$$
• nuclear fusion occurs when two light nuclei combine to form a single more massive nucleus, leading to the release of energy.
  e.g. $${}_1^{2}H+{}_1^{3}H\to {}_2^{4}He+{}_0^{1}n+Energy$$

• 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 $$x=x_0{e}^{-\lambda t}$$,
  $$\frac{x}{x_0}=\frac{1}{2}=e^{-\lambda t_{\frac{1}{2}}}$$
  $$\Rightarrow -ln2=-\lambda t_{\frac{1}{2}}$$
  $$\Rightarrow t_{\frac{1}{2}}=\frac{ln2}{\lambda }$$
• You may also use $$\frac{x}{x_0}={\frac{1}{2}}^{\frac{t}{t_{1/2}}}$$, as shown in the following video.

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[accordion title=”1. Definitions”]

• The magnetic flux density at a point is defined as the force acting per unit current per unit length of the conductor when the conductor is placed at right angles to the field.
• One tesla is the uniform magnetic flux density which, acting normally to a long straight wire carrying a current of 1 ampere, causes a force per unit length of 1 N m^–1 on the conductor.

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[accordion title=”2. Magnetic Fields”]

• The following are the vector symbols used in diagrams to represent the direction of vectors in 3 dimensional space:
  • $$\to$$ : on the plane of the page
  • $$\otimes$$ : into of the page
  • $$\odot$$ : out of the page
• The following are some important points to take note when representing a magnetic field by magnetic field lines:
  • Magnetic field lines appear to originate from the north pole and end on the south pole.
  • Magnetic field lines are smooth curves.
  • Magnetic field lines never touch or cross.
  • The strength of the magnetic field is indicated by the distance between the lines – closer lines mean a stronger field.

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[accordion title=”3. Force on a Current-Carrying Conductor in a Magnetic Field”]

• When a wire of length $$l$$ carrying a current $$I$$ lies in a magnetic field of flux density $$B$$ and the angle between the current $$I$$ and the field lines $$B$$ is $$\theta$$, the magnitude of the force $$F$$ on the conductor is given by $$F=BIlsin\theta$$.
  
• The directions of the vectors can be recalled by using the Fleming’s Left-Hand Rule.
  

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[accordion title=”4. Force on a Moving Charge in a Magnetic Field”]

• A charge $$q$$ travelling at constant speed $$v$$ at an angle $$theta$$ to a magnetic field of flux density $$B$$ experiences a force $$F=Bqvsin\theta$$.

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[accordion title=”5. Magnetic fields of current-carrying conductors”]

• Long straight wire
  
• Flat circular coil
• Solenoid

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[accordion title=”6. Ferromagnetic Materials”]

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[accordion title=”7. Force between Two Parallel Current-Carrying Conductors”]

•  Like currents attract and unlike currents repel.

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