# Subject Content

Study notes for the GCE ‘A’ level syllabus

## 19. Quantum Physics

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

• 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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## 20. Nuclear Physics

### 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

• $$\alpha$$ particle – helium nucleus
• $$\beta$$ particle – electron
• $$\gamma$$ particle – electromagnetic radiation

• $$A=-\frac{dN}{dt}=\lambda N$$
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.

## 15. Electromagnetism

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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:
• $$\rightarrow$$ : 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 $$F = BIl sin \theta$$.
• 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 $$F = Bqv sin\theta$$.

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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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## 04. Forces

### Types of Forces

• Static friction
• Frictional force between surfaces at rest with respect to each other.
• It increases with increasing applied force up to a maximum value (which is proportional to normal contact force).
• Kinetic friction
• Frictional force acting between surfaces in relative motion.
• Viscous forces
• Resistive force experienced by a solid moving in a fluid.
• Dependent on speed of object v, e.g. $$F_D\propto v$$ at low speeds and $$F_D\propto v^2$$ at high speeds.

## 03. Dynamics

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[accordion title=”1. Newton’s Laws of Motion”]

• Newton’s First Law:  a body will remain in its state of rest or uniform motion in a straight line unless acted upon by a resultant force.
• Newton’s Second Law the rate of change of momentum of a body is proportional to the resultant force acting on it and the change takes place in the direction of the resultant force.
• $$F =\frac{dp}{dt}$$ in general
• $$F =ma$$ when mass is constant.
• Newton’s Third Law:  if body A exerts a force on body B, then body B exerts an equal and opposite force on body A

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[accordion title=”2. Linear Momentum”]

• The linear momentum of a body is defined as the product of its mass and its velocity.
• Impulse is the product of the force acting on a body and the time interval during which the force is exerted. It is equal to the change in momentum of the body.
• For constant force, impulse = $$\Delta p =F \Delta t$$
• In general, impulse = $$\Delta p =\int {F .dt}$$

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[accordion title=”3. Collision Problems”]

• The principle of conservation of momentum states that the total momentum of a system of colliding objects remains constant provided no resultant external force acts on the system.
• Conservation of momentum applies to both elastic and inelastic collisions.
• $$m_1u_1+m_2u_2=m_1v_1+m_2v_2$$
• Conservation of kinetic energy applies only to elastic collisions.
• $$\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2$$
• Relative speed of approach = Relative speed of separation
• $$u_2-u_1=v_1-v_2$$

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## 02. Kinematics

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[accordion title=”1. Definitions”]

• Displacement is the distance travelled along a specified direction.
• Speed is the rate of change of distance travelled.
• Velocity is the rate of change of displacement.
• Acceleration is the rate of change of velocity.

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[accordion title=”2. One-Dimensional Motion with Constant Acceleration”]

• $$v=u+at$$
• $$s=(\frac{u+v}{2})t$$
• $$s=ut+\frac{1}{2}at^2$$
• $$v^2=u^2+2as$$

s: displacement
u: initial velocity
v: final velocity
a: acceleration
t: time

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[accordion title=”3. Two-Dimensional Motion”]

• Tip: Sometimes, you will require two equations to solve a kinematics problem. For a parabolic path in a projectile motion without resistive forces, you can draw a table such as the one below and fill in the blank with the information given in the question.
• In the case where a projectile is launched at an angle $$\theta$$ to the horizontal and undergoes the acceleration of free fall, the various horizontal and vertical components of displacement, velocity and acceleration can be expressed in the following way:
 Horizontal Vertical displacement, s $$(u \cos \theta)t$$ $$(u \sin \theta)t+\frac{1}{2}gt^2$$ initial velocity, u $$u \cos \theta$$ $$u \sin \theta$$ initial velocity, v $$u \cos \theta$$ $$u \sin \theta +gt$$ acceleration, a 0 $$g$$ time, t same for both dimensions

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