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# Comparision

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### Comparision

1. 1. Analogies (Synoptic)
2. 2. Comparisons of Spring & Capacitor <ul><li>Summary of capacitors </li></ul><ul><li>A capacitor is a device that stores charge. </li></ul><ul><li>The charge stored by a given capacitor is proportional to the voltage applied across it. </li></ul><ul><li>Different capacitors store different quantities of charge for the same voltage applied. </li></ul><ul><li>The constant of proportionality is designated C , the capacitance of a capacitor. </li></ul><ul><li>• Q = CV </li></ul><ul><li>Summary of springs </li></ul><ul><li>A spring stores Potential energy </li></ul><ul><li>When a force is applied to a spring, the extension is proportional to the force applied. </li></ul><ul><li>Different springs extend by different amounts for the same force applied. </li></ul><ul><li>The constant of proportionality is designated k , the spring constant or spring stiffness. </li></ul><ul><li>F = kx </li></ul>The independent variable in the case of capacitors is V and in springs F. So: Q = CV x = 1/k F Factors that affect: C include the distance between the plates, the area of the plates and the dielectric material. k include theYoung’s modulus of the material from which it is made.
3. 3. Cont. Energy Stored Although these appear to be very dissimilar phenomena, the expression for the energy stored is of the same general form in each case because the quantities concerned ( Q and C ; F and x ) are proportional to each other.
4. 4. Questions <ul><li>Name two devices which store energy: one mechanical device and one electrical. Explain how they each store energy. (4) </li></ul><ul><li>Describe how the energy is stored and give the equation for the energy stored. </li></ul><ul><li>Explain why the mathematical models for springs and capacitors give rise to similar expressions for stored energy. </li></ul>A capacitor stores charge. When a p.d. is applied across the plates of a capacitor, charge is transferred between the plates. The charge stored is proportional to the p.d applied. The electrical energy stored is related to the p.d. and the charge stored. A stretched spring stores elastic strain energy. When a force is applied to the spring it stretches. The extension is proportional to the force applied and energy stored in the stretched spring is related to the force and the extesion. If the p.d were constant, the energy stored would be Q V, but since V varies we must consider a small change in Q for which V can be considered constant, then the total energy stored is the sum of the small changes i.e. the area under the V/Q graph. E = ½ CV 2 If the force were constant, the energy stored would be Fx but since F varies we must consider a small change in x over which F can be considered constant, then the total energy stored is the sum of the small changes i.e. the area under the F/x graph. E = ½ Fx 2
5. 5. Decay general decay equation X = X 0 e − kt radioactivity k = λ (decay constant) capacitor discharge k = 1 /RC (time constant) count rate X = R activity X = A number of nuclei X = N current X = I charge X = Q voltage X = V
6. 6. Cont. <ul><li>A capacitor stores electric charge. If its plates are connected by a resistor a current flows and the capacitor is charges. </li></ul><ul><li>The atoms in a radioactive substance have unstable nuclei. Their decay is a random process. </li></ul>For both graphs λ is known as the decay constant and it can be shown that λ is related to the half life by the equation: λ = ln2 = 0.693 ≈ 0.7
7. 7. Cont <ul><li>Capacitor </li></ul><ul><li>Q=Q 0 e (-t/RC) </li></ul><ul><li>Current I = Q/RC </li></ul><ul><li>Radiation </li></ul><ul><li>N=N 0 e (-  t) </li></ul><ul><li>Activity A =  N </li></ul>
8. 8. Electric & Gravitational Fields <ul><li>Gravitational Field </li></ul><ul><li>Objects with mass Objects </li></ul><ul><li>Constant G </li></ul><ul><ul><li>Applies to all objects </li></ul></ul><ul><li>inverse square relationship. </li></ul><ul><li>All masses attract. </li></ul><ul><li>Weak except for massive bodies </li></ul><ul><li>F = G MM/r 2 </li></ul><ul><li>Gravitational Field Strength </li></ul><ul><ul><li>g = F/m = GM/r 2 </li></ul></ul><ul><li>Electric Field </li></ul><ul><li>Objects with charge </li></ul><ul><li>Constant  </li></ul><ul><ul><li>depends on the medium in which the field exists </li></ul></ul><ul><li>inverse square relationship. </li></ul><ul><li>Like charges repel </li></ul><ul><ul><li>Unlike charges attract </li></ul></ul><ul><li>Strong at close range. </li></ul><ul><li>F =  QQ/r 2 </li></ul><ul><li>Electric Field Strength </li></ul><ul><ul><li>E = F/Q =  Q/r 2 </li></ul></ul>
9. 9. Question 1 <ul><li>(i) In the study of the decay of a radioactive source the activity A appears as the subject of the equation A = λ N . Explain in words what is meant by activity and state its unit. (3) </li></ul><ul><li>(ii) The discharge of a capacitor through a resistor is analogous to radioactive decay. Write down the equation for capacitor discharge which is analogous to A = λ N. Explain the analogy between A and the subject of your equation. (4) </li></ul><ul><li>(iii) Outline another analogy in physics and state the similar mathematical relationships which describe the two different phenomena. (3) </li></ul><ul><li>Activity: (the number of) nuclei/atoms/particles (1) decaying per unit time (1) Unit: bequerel/Bq/s–1 [Not s–1 if above] (1) </li></ul><ul><li>(ii) I  Q (1) 1/ RC [dependent on 1st mark] (1) A and I both involve rate of change/change with time (1) A of nuclei atoms/particles/decays and I of charge (1) 7 </li></ul><ul><li>EITHER E- and g - fields/forces (1) Both involve inverse square laws (1) E = kq/r 2 and g = Gm/r 2 (1) OR Springs and capacitors (1) F = kx and V = Q/C OR E = ½ Fx = ½ VQ (1) An action produces a change OR energy stored (1) OR Other analogy spelled out in a similar manner, but not radioactive decay and the discharge of a capacitor. 3 </li></ul>
10. 10. Question 2 <ul><li>(a) The equations F = kx and V =Q/C are sometimes referred to as analogous mathematical models for a spring and a capacitor respectively. </li></ul><ul><li>(i) State what physical quantities are represented by F , x , V and Q in the equations above. </li></ul><ul><li>(ii) The energy stored in a capacitor is given by </li></ul><ul><ul><li>W c = ½ Q V and also by W c = ½ CV 2 . </li></ul></ul><ul><li>Write down, by analogy, the two equivalent expressions for the energy W s stored in a spring. </li></ul><ul><li>Show that W s can also be expressed as ½ kx 2 </li></ul><ul><li>( i) (Extending) force, extension [Not displacement, length, distance] [Accept l - l 0 implied] (1) Potential difference/voltage, charge (1) </li></ul><ul><li>W s = ½ Fx Ws = ½ F 2 /k (1) </li></ul><ul><li>Ws = ½ kx 2 (since F=kx) (1) </li></ul>