Capacitors, Inductors, and RC and RL Circuits Homework Set

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Leslie Evan King
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1 Problem 1. You have two flat metal plates of area 1.0 m 2 with which to construct a parallel plate capacitor. If the capacitance of the device is to be 1.0 F, what must be the separation between the plates? Could this actually be constructed? Write the general expression for capacitance of an object in terms of its area A and the separation distance, d. Perform algebraic manipulation and calculate the separation distance. Look up and write below, the average radius of a hydrogen atom in meters Compare and discuss the plate separation distance versus the radius of the hydrogen atom.

2 Problem 2. Find the equivalent capacitance in the circuit below. Assume that C 1 = 10.0 μf, C 2 = 5.0 μf and C 3 = 4.0 μf. Find the equivalent capacitance between C 1 and C 2 Find the equivalent capacitance between C 1 /C 2 and C 3

3 Problem 3. In the figure, the battery has a potential difference of 20 V. Find a) the equivalent capacitance of all capacitors Redraw the circuit with the two 3.0 μf capacitors with their centers aligned. Perform the same re-arrangement for the two 2.0 μf capacitors.

4 Find the equivalent capacitance of the two 4.0 μf capacitors Use this equivalent capacitance along with the capacitances of the two 2.0 μf capacitors to find the equivalent capacitance in the upper parallel junction Find the equivalent capacitance of the two 3 μf capacitors. Use the above information to calculate the total capacitance.

5 b) the charge stored on that equivalent capacitance Find the total charge stored in the circuit. (Hint: Q=CV) c) the potential across and the charge stored on capacitor C 1 Use facts about series networks and charge, to calculate voltage on and charge stored on C 1 d) the potential across and the charge stored on capacitor C 3 Calculate the potential difference across the upper parallel network. Find the amount of charge stored on the equivalent capacitance of the two 4 μf capacitors. Use facts about series networks and charge, to calculate voltage on and charge stored on C 3

6 Problem 4. A 3.0 MΩ resistor and a 1.00 μf capacitor are connected in series with an ideal battery of E=4.00 V. At 1.00 s after the connection is made, what are the rates at which a) the charge of the capacitor is increasing? Write the relationship for the charge of a charging capacitor in terms of the voltage applied V, the capacitance C, the resistance R, and the time t. Take the derivative of this expression with respect to time, dq dt. Use the values from the problem and calculate the charging rate.

7 b) the energy is being stored in the capacitor? Write the relationship for the energy U in a capacitor as a function of charge, q and capacitance C. Take the derivative of this expression with respect to time, then d x dt 2 dx = 2x dt du dt. (Hint: if x=f(t) and Use the values from the problem and calculate the energy storage rate. (Hint: the value of the derivative was calculated previously.) c) the thermal energy is appearing in the resistor? The rate at which the capacitor is charging is the same as the current. Write an expression for the power dissipated in a resistor as a function of current and resistance. Insert the numerical values from the problem and solve.

8 d) the energy is being delivered by the battery? Write an expression for the power in the battery as a function of voltage and current. Insert the numerical values from the problem and solve.

9 Problem 5. The figure below shows the circuit of a flashing lamp, like those attached to barrels at highway construction sites. The fluorescent lamp L (of negligible capacitance) is connected in parallel across the capacitor C of an RC circuit. There is current through the lamp only when the potential difference across it reaches the breakdown voltage V L ; in this event, the capacitor discharges completely through the lamp and the lamp flashes briefly. Suppose that two flashes per second are needed. For a lamp with a breakdown voltage of V L =72.0 V, a 95.0 V ideal battery, and a μf capacitor, what should the Write an expression for the voltage across the capacitor in terms of the capacitance C, the resistance R, the emf E, and the time t. Calculate the time between flashes in seconds. Perform algebraic manipulation and solve for the resistance.

10 Problem 6. The circuit below shows a capacitor C, two ideal batteries, two resistors and a switch S. Initially S has been open for a long time. If it is then closed for a long time, by how much does the charge on the capacitor change? Assume C=10.0 μf, E 1 = 1.0 V, E 2 =3.0 V, R 1 =0.20 Ω and R 2 =0.4 Ω. Explain what happens to a capacitor which has been attached to a battery for a long time. Use your explanation to calculate the charge on the capacitor while the switch is open.

11 Once the switch has been closed a long time, redraw below the equivalent circuit. (Hint: is current flowing through the capacitor?) From the equivalent circuit, use the loop rule to calculate the current below.

12 Take a partial loop traversal across E 2 and R 2 and calculate the potential difference across them. This potential difference is the same as the voltage across the capacitor. Calculate the charge on the capacitor. Calculate the change in charge from switch closed to switch open.

13 Problem 7. In the circuit below, E=10 V, R 1 = 5 Ω, R 2 = 10 Ω, and L= 5.0 H. For the two separate conditions (I) switch S has just closed and (II) switch S closed for a long time, calculate: a) the current i 1 through R 1 Condition I How much current is the inductor allowing to flow through it? Condition II How much current is the inductor allowing to flow through it?

14 Condition I Condition II Based on this logic, draw the resultant circuit below. Based on this logic, draw the resultant circuit below. Use the loop rules to solve for i 1 Use the loop rules to solve for i 1

15 b) the current i 2 through R 2 Condition I Based on the previous logic, use the loop rules to solve for i 2 Condition II Based on the previous logic, use se the loop rules to solve for i 2 c) the current i through the switch Condition I Condition II Sum the currents i 1 and i 2. Sum the currents i 1 and i 2.

16 d) the potential difference across R 2 Condition I Based on the current in the inductor, estimate the voltage across the inductor. Condition II Based on the current in the inductor, estimate the voltage across the inductor. The voltage across R 2 can be found as the difference in the voltage of the battery and the voltage across the inductor. Find that difference. The voltage across R 2 can be found as the difference in the voltage of the battery and the voltage across the inductor. Find that difference.

17 e) the potential difference across L Condition I Based on the logic in the previous section, estimate the voltage across the inductor. Condition II Based on the logic in the previous section, estimate the voltage across the inductor. di2 f) the rate of change dt Write an expression for voltage across an inductor with its inductance L and the rate of change of current though it (di/dt).

18 Condition I Based on the logic in the previous section and the above expression, calculate the rate of change of i 2. Condition II Based on the logic in the previous section and the above expression, calculate the rate of change of i 2.

19 Problem 8. A coil with an inductance of 2.0 H and a resistance of 10 W is suddenly connected to a resistanceless battery of E=100 V. At 0.10 s after the connection is made, what are the rates at which a) the energy is stored in the magnetic field? Write the expression for the energy stored in an inductor of inductance L with current i. Take the derivative of this expression with respect to time, d x dt 2 dx = 2x ) dt du dt. (Hint: if x=f(t) and then Write the expression for current in an inductor as a function of the emf E, the resistance R, the inductance L and the time t. Also write its derivative with respect to time.

20 Use the numerical values of the problem and find the current and its rate of change. Apply these values in the rate of change of energy expression.

21 b) the thermal energy is appearing in the resistance? Write an expression for energy dissipation in a resistance in terms of its resistance R and the current through it i. Use the numerical values from the previous section to calculate the energy dissipation.

22 c) the energy is being delivered by the battery? Write an expression for power supplied by a battery in terms of its voltage V and the current through it i. Use the numerical values from the previous section to calculate the power supplied by the battery. Demonstrate from these values that the conservation of energy is obeyed.

EE301 Lesson 14 Reading: 10.1-10.4, 10.11-10.12, 11.1-11.4 and 11.11-11.13

EE301 Lesson 14 Reading: 10.1-10.4, 10.11-10.12, 11.1-11.4 and 11.11-11.13 CAPACITORS AND INDUCTORS Learning Objectives EE301 Lesson 14 a. Define capacitance and state its symbol and unit of measurement. b. Predict the capacitance of a parallel plate capacitor. c. Analyze how