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(a) Derive the relationship between the peak and the rms value of current in an a.c. circuit. (b) Describe briefly, with the help of a labelled diagram, working of a step-up transformer. A step-up transformer converts a low voltage into high voltage. Does it not violate the principal of conservation of energy? Explain. |
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Answer» Solution :(a) Root Mean Square (r.m.s) or virtual value of A.C. : It is that steady current, which when passed through a resistance for a GIVEN time will produce the same amount of heat as the alternating current does in the same resistance and in the same time. It is denoted by `I_("rms") or I_(V)`. Derivation of r.m.s value of current : The instantaneous value of A. C. passing through a resistance R is given by, `I = I_(0) sin omega t` The alternating current changes continuously with time. Suppose that the current through the resistance remains constant for an INFINITESIMALLY small time dt. Then, small amount of heat produced the reistance R in time dt is given by `dH = I^(2) R dt = (I_(0) sin omega r)^(2) R dt` `= I_(0) R sin^(2) omega t dt` The amount of heat produced in the resistance in time T/2 is `H = underset(0)overset(.^(T)//_(2))int I_(0)^(2) R sin^(2) omega t dt = I_(0)^(2) R underset(0)overset(.^(T)//_(2))int (1 - cos 2 omega t)/(2) dt` `H = (I_(0)^(2)R)/(2) [t - (sin 2 omega t)/(2 omega)]_(0)^(.^(T)//_(2)) H = (i_(0)^(2)R)/(2) [(T)/(2) = (sin 2 omega t.(T)/(2))/(2 omega) - 0]` `H = (I_(0)^(2)R)/(2) [(T)/(2) - (sin 2.(2pi)/(T).(T)/(2))/(2 omega] h = (I_(0)^(2)R)/(2) [(T)/(2) - (sin 2 pi)/(2 omega)]` `H = (I_(0)^(2) R)/(2).(T)/(2) " " ..(i) { :' sin 2pi = 0]` If `I_(rms)` be the r.m.svalue of a.c., then by definition `H = I_(rms)^(2) R (T)/(2)`...(ii) From equations (i) and (ii), we have `I_(rms)^(2) R (T)/(2) = (I_(0)^(2)R)/(2).(T)/(2)` `I_(rms)^(2) = (I_(0)^(2))/(2) rArr I_(rms) = (I_(0))/(sqrt2) = 0.707 I_(0)` (b) Transformer : It is a device which converts high A.C. voltage into low A.C. voltage and vice versa. It is based upon the principle of MUTUAL induction. When alternating current passed through a coil, an iduced e.m.f. is set up in the NEIGHBOUR coil. A transformer when increases voltage, the current is reduced so that input power and output power remains same. So, it does not violate conservation of ENERGY principle |
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