Question 106 - In a wavefunction, let P(x) = A cos kx + B sin kx. By using the boundary conditions of x = 0 and x = l, where P(0) = P(l) = 0, prove by mathematical calculation that P(x) = B sin (npx / l) where p = 22 / 7 approximately, n is a rounded number. A, B and k are constants.
Answer 106 - When x = 0, P(0) = A cos (k x 0) + B sin (k x 0) = A = 0. When x = l, P(l) = A cos (k x l) + B sin (k x l) = 0 = B sin (k x l), then k x l = arc sin 0 = np, k = np / l. Finally P(x) = B sin kx = B sin (npx / l) as proven. The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; chuentat@hotmail.com; http://kangchuentat.wordpress.com.
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ELECTRICAL TECHNOLOGY - EXAMPLE 16.3 : In the design of a solar power system steps of calculations below are followed : (a) The power output of the inverter of the solar panel is 100 watts. What is the power input, Pin to the inverter when the efficiency of the inverter is 50 %? (b) If the rated power of the inverter is 300 watts, how many inverter is needed for the solar panel? (c) Charge controller of V = 12 volts is used to supply power to the inverter. What is the input current I to the inverter? (d) If the charge controller capacity is 10 A, how many charge controllers are needed? (e) If a biochemical mixer consumes 100 watts, running for 2 hours per day, what is the energy consumption in kilowatt hour per day? (f) What is the input energy needed when the efficiency of the inverter is 50 %? (g) If your area receives 2.88 hours of full sunlight per day, how much energy, in kilowatt hour can be produced per day when one solar panel can produce 20 watts of power? (h) If you know that you have to produce total energy as the answer for (f), how many solar panels are needed? (i) Each V = 12 V battery has 5 ampere hours. If the total energy needed is in answer (f), then how many batteries are needed to run the biochemical mixer if without sunlight for 3 days?
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Question 102 - (a) As an approximation, let v = Zc / 137 where v is the radial velocity for 1 s electron of an element, c is the speed of light, Z is the atomic number. For gold with Z = 79, find the radial velocity of its 1 s electron, in term of c and percentage of the speed of light. (b) As an approximation, let A x A = 1 - Z x Z / 18769 where A is the ratio of the relativistic and non-relativistic Bohr radius. Find the value of A.
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ELECTRICAL TECHNOLOGY - EXAMPLE 16.1 : According to Shockley equation, the I - V characteristic of a diode is approximated by I = IS [ exp (nVD / VT) - 1 ]. For silicon, let the reverse bias saturation current IS as 0.000000000001. If n is ideality factor with value of 1.5, VT as thermal voltage drop of 0.026 V at room temperature, what is the value of current I that passes through the silicon diode in the heater of evaporator when the forward voltage drop VD = 0.026 V? Please take note that exp is the exponential function with e(1) = 2.718, e(2) = 7.389.
PETROLEUM ENGINEERING - QUESTION 25.1 : Fact 1 : Dry air contains 20.95 % oxygen, 78.09 % nitrogen, 0.93 % argon, 0.039 % carbon dioxide, and small amounts of other gases by volume. Fact 2 : Volume occupied is directly proportional to the number of moles for ideal gases at constant temperature and pressure. Fact 3 : 12.5 moles of pure oxygen is required to completely burn 1 mole of pure octane. Fact 4 : Air-fuel ratio (AFR) is the mass ratio of dry air to fuel present in a combustion process such as in an internal combustion engine or industrial furnace. Fact 5 : Molecular weight of oxygen gas is 31.998 g / mole and molecular weight of nitrogen gas is 28.014 g / mole. (a) Find the molar ratio of nitrogen and oxygen, or (moles of nitrogen) / (moles of oxygen) in dry air, by assuming ideal features of nitrogen and oxygen gases. (b) How many moles of nitrogen are available if dry air is used to completely burn the 1 mole pure octane? (c) Find the mass of fuel of 1 mole of octane with molecular weight of 114.232 g / mole. (d) Find the mass of dry air with 12.5 moles of pure oxygen by assuming only oxygen and nitrogen gases exist in the air. (e) Find the air-fuel ratio (AFR) when octane is used as fuel. (f) Find the fuel-air ratio (FAR) when octane is used as fuel.
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