Month: July 2013

The Potential Divider Rule

Determining the relative brightness of light bulbs using the potential divider rule

How Internal Resistance affects brightness of light bulbs

How emf is induced

Electromagnetic Induction and its Effects

To explain a phenomenon that happens due to Electromagnetic Induction, we can use an acronym CFILE to structure our answer.

C stands for cutting of flux or changing of flux linkage. This is necessary for electromagnetic induction to happen. When a wire is pulled through a magnetic field perpendicular to it, it is said to be cutting the field lines. When a magnet enters a coil of wires, we can say that the magnetic flux linkage is increasing.

F stands for Faraday’s law, which states that the induced e.m.f. in a circuit is directly proportional to the rate of change of flux-linkage or to the rate of cutting of magnetic flux.

I stands for an induced current. However, do note that this is only possible if there is a closed circuit or a path for the current to flow.

L stands for Lenz’s law, which states that the direction of the induced e.m.f. is such that it tends to oppose the flux change causing it, and does oppose it if induced current flows. 

E stands for effect. This is really just stating how an induced current or Lenz’s law causes the phenomenon in question.

• 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.
  $$<P>=\frac{1}{2}P_o=\frac{1}{2}{I_o}{V_o}=\frac{1}{2}{I_o}^2R =\frac{V_o^2}{2R}$$.
  
• 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 $$P_{loss}=I^2R$$ 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 $$\Phi$$ 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 = 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 $$\lambda = \dfrac{h}{p}$$ 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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