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www.GOALias.blogspot.comAlternating CurrentExample 7.10 At an airport, a person is made to walk through thedoorway of a metal detector, for security reasons. If she/he is carryinganything made of metal, the metal detector emits a sound. On whatprinciple does this detector work?Solution The metal detector works on the principle of resonance inac circuits. When you walk through a metal detector, you are,in fact, walking through a coil of many turns. The coil is connected toa capacitor tuned so that the circuit is in resonance. Whenyou walk through with metal in your pocket, the impedance of thecircuit changes – resulting in significant change in current in thecircuit. This change in current is detected and the electronic circuitrycauses a sound to be emitted as an alarm.EXAMPLE 7.107.8 LC OSCILLATIONSWe know that a capacitor and an inductor can store electrical andmagnetic energy, respectively. When a capacitor (initially charged) isconnected to an inductor, the charge on the capacitor andthe current in the circuit exhibit the phenomenon ofelectrical oscillations similar to oscillations in mechanicalsystems (Chapter 14, Class XI).Let a capacitor be charged q m(at t = 0) and connectedto an inductor as shown in Fig. 7.18.The moment the circuit is completed, the charge onthe capacitor starts decreasing, giving rise to current inthe circuit. Let q and i be the charge and current in thecircuit at time t. Since di/dt is positive, the induced emfin L will have polarity as shown, i.e., v b< v a. According toKirchhoff’s loop rule,q di− L = 0(7.39)C dti = – (dq/dt) in the present case (as q decreases, i increases).Therefore, Eq. (7.39) becomes:2d q 12dt+ q = 0(7.40)LCFIGURE 7.18 At theinstant shown, the currentis increasing so thepolarity of induced emf inthe inductor is as shown.2d x 2This equation has the form + ω2 0x = 0 for a simple harmonicdtoscillator. The charge, therefore, oscillates with a natural frequency1LCand varies sinusoidally with time asω0= (7.41)( ω φ)q = q cos t + (7.42)m0where q mis the maximum value of q and φ is a phase constant. Sinceq = q mat t = 0, we have cos φ =1 or φ = 0. Therefore, in the present case,255