C R. Consider from point of view of energy! Consider the RC and LC series circuits shown:

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1 ircuits onsider the R and series circuits shown: R Suppose that the circuits are formed at t with the capacitor charged to value. There is a qualitative difference in the time development of the currents produced in these two cases. Why?? onsider from point of view of energy! In the R circuit, any current developed will cause energy to be dissipated in the resistor. In the circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor!

2 Energy in the Electric and Magnetic Fields Energy stored in a capacitor... U V energy density... u E ε E electric Energy stored in an inductor. U I energy density... u B magnetic B μ

3 I R R/ ircuits I R: current decays exponentially I I : current oscillates t t

4 Oscillations (qualitative) I + I I I +I I

5 ecture 8, Act At t, the capacitor in the circuit shown has a total charge. At t t, the capacitor is uncharged. A + + t tt - - What is the value of V ab V b -V a, the voltage across the inductor at time t? (a) V ab < (b) V ab (c) V ab > a b B What is the relation between U, the energy stored in the inductor at tt, and U, the energy stored in the capacitor at tt? (a) U < U (b) U U (c) U > U

6 ecture 8, Act At t, the capacitor in the circuit shown has a total charge. At t t, the capacitor is uncharged. A + + t tt - - What is the value of V ab V b -V a, the voltage across the inductor at time t? (a) V ab < (b) V ab (c) V ab > a b V ab is the voltage across the inductor, but it is also (minus) the voltage across the capacitor! Since the charge on the capacitor is zero, the voltage across the capacitor is zero!

7 ecture 8, Act At t, the capacitor in the circuit shown has a total charge. At t t, the capacitor is uncharged. What is the relation between U, B the energy stored in the inductor at tt, and U, the energy stored in the capacitor at tt? t tt a b (a) U < U (b) U U (c) U > U At tt, the charge on the capacitor is zero. At tt, the current is a maximum. U U I >

8 Oscillations (quantitative, but only for R) What is the oscillation frequency ω? Begin with the loop rule: d + dt I Guess solution: (just harmonic oscillator!) cos( ω t + φ) remember: where φ, determined from initial conditions m dx dt + kx Procedure: differentiate above form for and substitute into loop equation to find ω. Note: Dimensional analysis ω

9 Oscillations (quantitative) General solution: cos( ω t + φ) Differentiate: d dt ω sin( ω t + φ) Substitute into loop eqn: d dt ω cos( ωt + φ) d + dt ( ) ω cos( ) ( cos( )) ω t + φ + ω t + φ ω + Therefore, ω which we could have determined from the mass on a spring result: k / ω m

10 ecture 8, Act 3 t At t the capacitor has charge ; the resulting oscillations have frequency ω. The maximum current in the circuit during these oscillations has value I. 3A What is the relation between ω and ω, the frequency of oscillations when the initial charge? (a) ω / ω (b) ω ω (c) ω ω 3B What is the relation between I and I, the maximum current in the circuit when the initial charge? (a) I I (b) I I (c) I 4I

11 ecture 8, Act 3 t At t the capacitor has charge ; the resulting oscillations have frequency ω. The maximum current in the circuit during these oscillations has value I A What is the relation between ω and ω, the frequency of oscillations when the initial charge? (a) ω / ω (b) ω ω (c) ω ω determines the amplitude of the oscillations (initial condition) The frequency of the oscillations is determined by the circuit parameters (, ), just as the frequency of oscillations of a mass on a spring was determined by the physical parameters (k, m)!

12 ecture 8, Act 3 t At t the capacitor has charge ; the resulting oscillations have frequency ω. The maximum current in the circuit during these oscillations has value I. 3B What is the relation between I and I, the maximum current in the circuit when the initial charge? (a) I I (b) I I (c) I 4I The initial charge determines the total energy in the circuit: U / The maximum current occurs when! At this time, all the energy is in the inductor: U / I o Therefore, doubling the initial charge quadruples the total energy. To quadruple the total energy, the max current must double!

13 Preflight 8: The current in a circuit is a sinusoidal oscillation, with frequency ω. 5) If the inductance of the circuit is increased, what will happen to the frequency ω? a) increase b) decrease c) doesn t change 6) If the capacitance of the circuit is increased, what will happen to the frequency? a) increase b) decrease c) doesn t change

14 Oscillations Energy heck Oscillation frequency ω has been found from the loop equation. The other unknowns (, φ ) are found from the initial conditions. E.g., in our original example we assumed initial values for the charge ( i ) and current (). For these values: i, φ. uestion: Does this solution conserve energy? U E U B ( t) ( t) cos ( ω t + φ) ( t) i ( t) ω sin ( ω t + φ)

15 Energy heck Energy in apacitor U E ( t) cos ( ω t + φ) Energy in Inductor U B U B ( t) ω sin ( ω t + φ) ω ( t) sin ( ω t + φ ) U E U B t Therefore, U E ( t) + U B ( t) t

16 Inductor-apacitor ircuits Solving a circuit problem; Suppose ω/sqrt()3 and given the initial conditions, t 5 Solve find and φ, to get complete solution using, I and we find, I ( ) ( t ) 5A ( ) 5 cos( + φ ) ( t ) 5 ω sin( + φ ) 3 sin( + φ ) t 5 3 () 5 + sin ( φ ) + cos ( φ ) 5 φ inv.tan, φ 5 3 ( t) cos( ωt + φ ) [ ] 45 I( t) o ω, sin( ωt + φ ) 5

17 Mathematical Insert The following are all equally valid solutions t t t t () ( ) () ( ) () ( ( ) ( ) ( ) ( )) cos ωt + φ sin Acos ωt ωt + φ cos ωt + cos Bsin φ ωt ( ) ( ) ( ) sin ωt sin The circuit eqn is the analog of the spring force eqn d dt ω + m d dt x ω k m Kx φ

18 Inductor-apacitor-Resistor ircuit + RI + d dt d dt + R d dt + Second order homogeneous differential equation. Solution can give rather different behaviors depending on values of circuit components

19 Inductor-apacitor-Resistor ircuit 3 solutions, depending on,r, values R > 4 R R 4 < 4

20 Inductor-apacitor-Resistor ircuit Solving for all the terms Solution for underdamped circuit; t () Ae cos( ω' t + φ) Ae α R t R αt cos andω' R 4 R 4 > t R + φ 4 For other solutions, use starting form, solve for λ and λ, t λt λ' t ( ) Ae + Be

21 Alternating urrents (hap 3.) In this chapter we study circuits where the battery is replaced by a sinusoidal voltage or current source. V ( t) V cos( ωt) or I( t) I cos( ωt) The circuit symbol is, An example of an R circuit connected to sinusoidal source is,

22 Alternating urrents (hap 3.) Since the currents & voltages are sinusoidal, their values change over time and the averages are zero. A more useful description of sinusoidal currents and voltages are given by considering the average of the square of this quantities. We define the RMS (root mean square), which is the I ( t) I cos ωt square root of the average of, ( ( )) I ( ) cos ω ω I t ( I ( t) ) I + ( t) I I ( t) RMS I ( cos )

23 Alternating urrents ; Phasors A convenient method to describe currents and voltages in A circuits is Phasors. Since currents and voltages in circuits with capacitors & inductors have different phase relations, we introduce a phasor diagram. For a current, We can represented this by a vector rotating about the origin. The angle of the vector is given by ωt and the magnitude of the current is its projection on the X-axis. If we plot simultaneously currents & voltages of different components we can display different phases. i I cos ( ωt) Note this method is equivalent to imaginary numbers approach where we take the real part (x-axis projection) for the magnitude