'''Discussion Question 11B ''' '''Physics 212 week 11''' '''''LC Circuits''''' At right, we see the classic '''LC circuit''', consisting of an inductor in series with a capacitor. To be precise, this is an '''undriven''' LC circuit, since there is no battery driving the flow of current. Nevertheless, this simple circuit has an amazing and highly useful property: it supports a resonant oscillation of current. In the ideal case shown here where there is ''no'' resistance in the circuit, the oscillation can continue ''indefinitely'' without any external source of EMF to drive it. LC circuits often appear as parts of larger networks which are designed to operate at a particular frequency. The most familiar example is a radio, whose circuitry can be tuned to receive (i.e. respond to) incoming electromagnetic waves of a particular frequency. '''(a) What is the angular frequency''' '''''''₀ of the current maintained by the circuit? What is the linear frequency ''f''₀?''' An ''undriven'' LC circuit can ''only'' operate in a stable way at its natural resonant frequency. As for linear versus angular frequency, the conversion is easy if you remember the units: ''f'' is in Hz = 1/sec, '''' is in rad/sec. '''(b) Let’s set our clock so that the current in the circuit is at its maximum value ''I''_max at time ''t'' = 0. Write down an expression for the time-dependence of the current ''I''(''t'') in terms of ''I''_max and the frequency''' '''₀.''' It’s either a sine or a cosine … (c) Starting from your expression for ''I''(''t''), determine the voltages ''V_L''(''t'') and ''V_C''(''t'') across the inductor and capacitor. As part of your solution, find the peak values ''V_L''_,max and ''V_C''_,max in terms of ''I''_max. [[File:media/image1.emf|141x114px]] You’ll need your familiar formulas for the voltage across ''R'', ''C'', and ''L'' ... and remember that, by definition, and so '''.''' '''(d) To visualize what’s going on, sketch your functions ''I''(''t''), ''V_L''(''t''), and ''V_C''(''t'').''' Don’t worry about the amplitudes of your curves, just their shapes and phases. [[File:media/image4.emf|362x194px]] '''e) Have a look at your plot → does ''V_L'' lead or lag the current? How about ''V_C''?''' Leading and lagging can be tricky concepts. Think of it this way: which one “gets there” (i.e. reaches its maximum value) first, ''V'' or ''I''? The one that “gets there first” leads the other. Congratulations! You’ve just derived the '''master relations''' between current and voltage for inductors and capacitors in an AC circuit! Notice how all the peak-value formulas look like good old “''V'' = ''IR''” → the '''reactances''' ''X_L'' and ''X_C'' describe the effective resistance of inductors and capacitors in AC circuits. We now set the circuit into oscillation by “stimulating” it with a brief pulse from some external source of EMF. The result is that the peak voltage across the capacitor is ''V_C''_,max = 120 V. '''(f) What is the peak current ''I''_max? ''' '''(g) What are the maximum energies ''U_L''_,max and ''U_C''_,max stored in the inductor and capacitor respectively?''' '''(h) Determine time-dependent expressions ''U_L''(''t'') and ''U_C''(''t'') for the two stored energies, and add them together to find the total stored energy ''U''(''t''). Finally, plot your results for all three functions.''' [[File:media/image5.emf|444x238px]] '''(i) How are ''U_L''_,max, ''U_C''_,max, and ''U''_max related to each other?''' Knowing this relation is extremely helpful in solving LC circuit problems!