When a series combination of two uncharged capacitors is connected to a 12 V battery

5.  When a series combination of two uncharged capacitors is connected to a 12 V battery, 96 mJ of energy is drawn from the battery.  If one of the capacitors has a capacitance of 4 mF, what is the capacitance of the other?

1. Algebra Based Solution

Given Data: Vb=12V U=96 mJ C1=4mF C1 and C2 are series C2=?

Useful formulas: U=QV/2=CV2/2 C=Q/V 1/Ceq=1/C1+1/C2

Algebra Job U=CeqVb2/2 1/Ceq= Vb2/(2U) 1/C1+1/C2 = Vb2/(2U) 1/C2 = Vb2/(2U) - 1/C1 C2=1/( Vb2/(2U) - 1/C1 ) or 1/C2 = Vb2/(2U) - 1/C1 1/C2 = (Vb2/2 – U/C1) /U C2 = U/(Vb2/2 - U/C1)

Substitution and Calculation:

C₂=1/( V_b²/(2U) - 1/C₁ ) = 1/(  (12V)²/(2·96 mJ) - 1/(4mF)  )

Expression for Google Scientific Calculator

1/( (12Volt)^2/(2*96microjoule) – 1/(4microfarad)) in microfarad

Google result: 2 microfarad

Result: C₂ = 2mF

2. Arithmetic Based Solution

Given Data: Vb=12V U=96 mJ C1=4mF C1 and C2 are series C2=?

Useful formulas: U=QV/2=CV2/2 C=Q/V 1/Ceq=1/C1+1/C2

U=CeqVb2/2 96mJ =Ceq(12V)2/2 (4·4·2·3)mJ =Ceq ·4·3·4·3V2/2 (2)mJ =Ceq ·3V2/2 4/3 (mJ/ V2) = 4/3 mF = Ceq 1/Ceq = ¾  · 1/mF 1/C2=1/Ceq-1/C1 1/C2 = ¾ · 1/mF – ¼ ·1/mF 1/C2 = 2/4 · 1/mF = 1/2  · 1/mF C2= 2mF

3. Another Arithmetic Based Solution

Given Data: Vb=12V U=96 mJ C1=4mF C1 and C2 are series C2=?

Useful formulas: U=QV/2=CV2/2 C=Q/V Q1=Q2=Qeq=Q Vb=V1+ V2

Q=2U/V=2·96 mJ/12V Q= 2·4·4·2·3 /(4·3) mJ/V   Q= 2·4·2 mC V1=Q/C1=2·4·2 mC/4mF=4V V2=Vb -V1=12V-4V=8V C2=Q/V2=2·4·2 mC/8V=2mF C2= 2mF

Homework Quiz 5. Resistance-Capacitance Circuits

1. A series circuit consists of a 1.02 V source of emf, a 2.00 mF capacitor, a 1000 W resistor, and a switch. When the switch is closed, how long does it take for the current to reach one-half its maximum value? 
A. 0.693 s                              B. 1.69 s                C. 0.0250 s                            D. 2.00 s                E. 1.39 s

2.  The ammeter your physics instructor uses for in-class demonstrations has a resistance R_g and requires a current of 1.0 mA for full-scale deflection. The same ammeter can be used to measure currents of much greater values by wiring a shunt resistor of relatively small resistance, R_p , in parallel with the ammeter. The shunt resistance needed to produce a full-scale current of 12 A is 3.5 mW.  What is the resistance of the ammeter, R_g?  
A.  75 W.                         B.  28 W.                              C.  35 W.                             D.  42 W.                             E.  60 W.

3. Two identical circuits have a capacitor in series with a resistor.  In circuit A, the resistor has resistance R, while in circuit B, its resistance is 4R.  The capacitors are identical.  Each capacitor begins fully charged, and each circuit is open.  When each circuit is connected (i.e. the switch in them is closed) the capacitor will fully discharge.  If a time t is required for circuit A’s capacitor to fully discharge, what time will be required for circuit B’s capacitor to do the same?  
A. 0.5t                    B. 2t                       C. 4t                       D. t                          E. 0.25t
4. In the circuit shown, with the switch open the capacitor is completely uncharged. The switch is closed for a long time. the current through the circuit is 0.2 A. Determine the value of the unknown resistor.

A. 61 W              B. 70 W              C. 27 W              D. 10 W              E. 56 W  

5.  A galvanometer reads I = 6 mA full scale and has an internal resistance r = 9.1 ohms. If a resistor, R = 425 W is connected in series with the galvanometer, what is the full scale voltage of the device?

A.  2.55 V                       B.  7.15 V                              C.  7.10 V                              D.  4.23 V                             E.  2.60 V

6. A capacitor is connected in series with a resistor and a switch. With the switch open, the capacitor is charged to 9.0 V. When the switch is closed, how long will it take for the voltage across the capacitor to drop to 6.0 V if the time constant of the circuit is 4.0 s? 
A. 2.7 s                   B. 1.6 s                   C. 12 s                    D. 1.2 s                   E. 0.3 s

7. In an RC circuit with an initially uncharged capacitor, the time constant is the time that is required for the charge on the capacitor to reach what percentage of its final value?
A. 100%                B. 63%                   C. 37%                   D. 50%                   E. 90%                    

8. In the circuit shown in the figure, the switches S1 and S2 are initially open and the 20-mF capacitor has a charge of 100 mC. About how long will it take after switch S1 is closed (switch S2 remains open) for the charge on the capacitor to drop to 5 mC?

A. 3.0 s                   B. 9.0 s                   C. 6.0 s                   D. 18 s                    E. 12 s                   9.  A series circuit consists of a 12.0 V source of emf, a 2.00 mF capacitor, a 1000 W resistor, and a switch. What is the time constant for this circuit?  
A.  10.0 s                             B.  0.693 s                           C.  0.0825 s                         D.  1.00 ms                         E.  2.00 s  

10.  A 9.0000-V battery is used to produce a current through two identical resistors in series, each having a resistance of 150.00 kW. A handheld digital multimeter (DMM) is used to measure the potential difference across the first resistor by placing probe leads at each end of this resistor. Typical DMMs have an internal resistance of 10 MW. What is the current in the circuit with the DMM connected?  
A.  30.33 mA                    B.  30.00 mA                    C.  29.78 mA                    D.  30.14 mA                    E.  30.22 mA           

11. Two parallel plate capacitors, C₁ and C₂ are connected in series to a 60-V battery, a switch S, and a 200-KW resistor as shown in the figure. Both capacitors are air gap capacitors and have plates with an area of 2.0 cm² and a separation of 0.10 mm. The capacitors are initially uncharged and switch S is open. After switch S is closed, how long will it take for the capacitors to charge to 50% of their maximum value?

A. 1.2 ´ 10^-6 s                   B. 8.9 ´ 10^-7 s                   C. 1.8 ´ 10^-6 s                   D. 2.5 ´ 10^-6 s                   E. 4.9 ´ 10^-6 s 

12. When being used to measure current, an ammeter should be connected in 
A. series with the circuit.       B. series with the voltmeter.     C. parallel with the circuit.    D. parallel with the voltmeter. 

13.  What is the ratio of the time for the capacitor in an RC series circuit to reach 90% of capacity from an empty state, to that for the circuit to discharge from a ‘full’ state to 90% capacity?   
A.  1/9 as long       B.  Answer depends on the particular values of R and C.      C.  The same time is required                      
D.  1/22 times as long                               E.  22 times as long                   F.  9 times as long  

14. In the circuit shown, R₁ = 10 W, R₂ = 4 W, R₃ = 10 W, e = 10 V, and capacitor has a capacitance C = 2 mF. Determine the energy stored in the capacitor when switch S has been closed for a long time.

  A. 4.94 µJ                              B. 3.27 µJ                              C. 3.55 mJ                             D. 1.02 mJ                              

15. In an RC circuit with an initially uncharged capacitor, the time constant is the time that is required for the current through the resistor to reach what percentage of its initial value?  A. 63%               B. 37%              C. 100%             D. 50%        E. 90% 

16. In the figure shown, the time constant for the discharge of the capacitor is

 A. 40 sec.                     B. 20 sec.                        C. 50 sec.                        D. 10 sec.                               E. 30 sec.

17. In a series circuit consisting of a 12.0 emf, a 2.00 mF capacitor, a 1.00 kW resistor and a switch, what is the voltage across the capacitor at a time t after the switch is closed? 
A. 6.32 V                               B. 7.58 V                               C. .632 V                               D. 8.65 V                                

18.  A voltmeter must be placed in parallel with the circuit element whose voltage is to be measured. An ideal voltmeter therefore has  
A.  zero resistance.                               B.  infinite resistance.                          D.  None are correct.  

19. A series circuit consists of a 12.0 V source of emf, a 2.00 mF capacitor, a 1000 W resistor, and a switch. When the switch is closed, how long does it take for the current to reach one-tenth its maximum value? 
A. 4.61 s                B. 0.693 s                              C. 18.0 s                D. 2.30 s                E. 9.00 s 

20.  A capacitor in an RC circuit is charged, storing energy 1.25mJ.  What is the time required to discharge this capacitor, as compared to the time required to discharge an identical capacitor in an identical RC circuit which has stored 2.5mJ?   
A.  twice the time                 B.  the answer depends on the values of R and C.        C.  half the time   D.  the same time