Question

Fig.1 9. A series RCL circuit contains a 148- 2 resistor, a 1.50-uF capacitor, and a 35.7-mH inductor. The generator has a frequency of 512 Hz and an ms voltage of 25.0 V. Determine resonance frequency in the circuit. (a) 512 Hz (5) 688 H2 (c) 800Hz (d) 1024 Hz (d) None 10. Find the inductance in a coil if the magnitude of the average induced emf is 19.5 mV and the current in changes from 0.20 A to 1.5 A in 0.20 s. (a) 10.5 mH (b) 12.5 mH (c) 5.4 mH 1 (d) 3.0 mH (d) None 11. How much energy is stored in a 70.0-mH inductor at an instant when the current is 2.00 A? u (b) 0.40 J (c) 0.64 J (d) 0.91 J (d) None 12. A 15.0-H inductor is connected in series with a 12.0-V battery, a switch, and a 3.0-0 resistor as indicated in Fig.2. After closing the switch and completing the circuit, what is the estimated length of time required for the current to reach a quarter of its maximum value? (a) 4.56 s (b) 7.52 s (c) 6.93 s (d) 1.4 s (e) None

Answer 1

9) the resonance frequency of series LCR circuit is given by

$f_0 = \frac{1}{2\pi \sqrt{L C}} = \frac{1}{2(3.14) \sqrt{(1.50\times 10^{-6})(35.7\times 10^{-3})}} \\ \\ = \frac{1}{(1.453\times 10^{-3})} = 688~Hz$

10) from the definition of inductance,

$Emf = -L \frac{\Delta I}{\Delta t}\\ \\ \Rightarrow L = - Emf \frac{\Delta t}{\Delta I} = -(19.5\times 10^{-3})\frac{0.20}{0.20-1.50} = 3~mH$

11) the energy stored in an inductor is given by

$E = \frac{1}{2}L I^2 = \frac{1}{2}(70\times 10^{-3})(2.00)^2 = 140\times 10^{-3} = 0.14~J$

12) we have

$\tau = \frac{L}{R} = \frac{15}{3} = 5~s$

$I = I_0 (1-e^{-(R/L)t})\\ \\ \Rightarrow \frac{I}{I_0} = (1-e^{-(R/L)t}) = \frac{1}{4}\\ \\ \Rightarrow e^{-(R/L)t} = \frac{3}{4}\\ \\ or~~~~ -(R/L)t = ln(3/4) = -0.287\\ \\ \Rightarrow t = 0.287\times (L/R) = 0.287(\tau) = 0.287(5) = 1.4~s$