Sheet 3 Capacitors and electric current

*All students must solve the following problems.

1- In the given circuit, find the current

flowing in each branch, the power

dissipated in the (8 ) resistance and the potential difference between (A) and (B).

2- Two capacitors (A.B), capacitor (A) is charged through a potential difference (50 volt)

and capacitor (B) has a capacitance of ( 30 F) and is charged through a potential difference (20 V). After disconnecting the charging batteries, capacitors (A and B) are connected together in parallel (positive plate to positive plate) and the common voltage is found to be (35 volt). Find the capacitance of (A) and the amount of transferred charge.

3- Two capacitors (A.B), capacitor (A) has a capacitance of ( 20 F) and is charged through a potential difference (50 volt) and capacitor (B) has a capacitance of

( 30 F) and is charged through a potential difference (20 V). After disconnecting the charging batteries, capacitors (A and B) are connected together in parallel (positive plate to negative plate). Find the common voltage and find the charge of each capacitor after connection. Calculate the work done in transferring the charge between capacitors.

4- Capacitor (A) has a capacitance of (12 pF) is charged using a battery (20 v), after

disconnecting the battery it is connected in parallel to another uncharged capacitor (15 pF). Find the common potential for the capacitors and the work done in transferring the charge from the first capacitor to the second.

5- A parallel plate air capacitor has a plate area (2 cm2) and a plate separation (3 mm),

find its capacitance and after introducing a slab of dielectric material with thickness (1 mm) the capacitance increases by a factor of (1.36). Find the relative permittivity for the dielectric material.

6- In the given circuit, find the current flowing in each branch and calculate the potential difference between (A) and (B).

A

B

8

4 V

10 V

3

6 7 V

10 v 4

20

5

12 v

16 v

A

B

7- In the given circuit, find the charge and the potential difference on each capacitor and

the stored energy in the (4F) capacitor.

8- In the given circuit, find the current flowing in each branch and calculate the potential difference between (A) and (B).

9- Two resistances (A,B) are made from the same material. The parallel connection

between them gives an equivalent resistance (40 ) at (0 oC). At temperature (50 oC),

resistance (A) equals (102 ) and resistance (B) equals (68 ). Find the series and parallel combinations of (A,B) at (250 oC).

10- Two resistances (A and B), at (0 oC) resistance (A) is 4 times greater than resistance

(B) and the parallel combination of (A,B) gives (16 ). At temperature (100 oC) the

series combination of (A,B) gives a resistance of (112 ). If the temperature

coefficient of resistance (A) is (A =0.001 oC

-1), find the parallel combination of

(A,B) at temperature (250 oC).

11- Two resistances (A and B) are made from two different materials. The temperature coefficient of resistance for material (A) is twice that for material (B). At temperature

(0 oC) resistance (A) equals (50 ) and the parallel combination of (A, B) gives (20 ).

At temperature (100 oC), the series combination of (A,B) gives (95 ). Calculate the parallel combination of (A, B) at (300 oC).

C4 = 7/3 F 40 V

C1 = 5 F

C2 =4 F

C3 =8 F

A

B

20 v

8 v 18 v

6 9 15

12- In the given circuit, find the charge and

the potential difference for each

capacitor. Calculate also the stored

energy in the (4 F) capacitor.

13- In the given circuit, find the charge and the potential difference for each capacitor.

Calculate also the stored energy in the (18 F) capacitor.

14- In the given circuit, find the charge and the potential difference on each capacitor.

Calculate also the stored energy in capacitor (C4).

15- Two resistances (A and B), are made from the same material. At temperature (0

oC),

the series connection of (A and B) gives a resistance of (50 ). At temperature of

(150 oC) resistance (A) equals (31.8 ) and resistance (B) equals (21.2 ).

Calculate the series and parallel combination of (A and B) at (350 oC).

+

10V

3uF

18uF

8/3uF

10uF

5uF

+

10V

C49uF

C16uF

C38uF

C24uF

4 F

12 F

8/5 F

6 F

10 v

16- In the given circuit, find the electric current in each branch. Calculate also the

potential difference between points (A, B).

17- A parallel plate capacitor shown in Figure, has A=1 cm2, d=2 mm, εr1= 4.9, ε

r2= 5.6 and ε r3=2.1, find its equivalent capacitance.

18- Consider three capacitors having capacitances of 3 F, 6 F and 12 F.

Find their equivalent capacitance if they are connected in series and in parallel.

19- Suppose that the capacitance in the absence of a dielectric is 8.5 pF, and

the capacitor is charged to a potential difference of 12 volt. If the battery is

disconnected and a slab of dielectric constant r= 2.56 is inserted between the

plates, calculate the difference in stored energy U-Uo .

20- A parallel plate capacitor is filled with two dielectrics as shown in the

figures, find the capacity of each case.

A B

2 ohms

+

12 V

+

6 V

+

4 V

5 ohms

7 ohms

ε 1

ε 2

ε 3

L/2

d/2

L/2

d/2

21- Four capacitors are connected

as shown. Find the equivalent

capacitance between points A and B.

22- Find the charge and potential

difference on each capacitor in the

circuit shown.

23- For the system of capacitors shown in Figure, find (a) the equivalent

capacitance of the system, (b) the potential difference across each capacitor, (c)

the charge on each capacitor, and (d) the total energy stored by the group.

r1

r2

A/2 A/2

d (a)

r1

r2

d/2

d/2

A

(b)

A

6 µF 3 µF

B

A

4 µF

3µF

10 V

A

15 µF 10 µF

B

A

4 µF

20 µF

24- Calculate the power delivered to each of the resistors shown in the

following figure.

25- For the circuit shown in the figure shown below, calculate (a) the current

in the( 2.00-) resistor and (b) the potential difference between points a and b.

26- Determine the current in each branch of the circuit shown in the following

figure.

Answers

1- The currents: 1.444, 1.611, 0.1667 A, the power 0.222 W and ( VB-VA )= 5.66 v.

2- CA = 30 F, charge transferred 450 C. 3- The common voltage after connection is (8 v). The charge after connection (Q1

’ = 160

C, Q2’ = 240 C) and the work done equals (29400 J).

4- The common potential is (8.889 v), work done to transfer the charges is (1333.3 pJ). 5- The capacitor before introducing the dielectric:

After introducing the dielectric material

The capacitor can be considered as three capacitors in series: First capacitor (C1) is an air capacitor with separation x in mm. Second capacitor (C2) is a dielectric capacitor with separation 1 mm. Third capacitor (C3) is an air capacitor with separation (3-1-x) in mm.

d = 3 mm

air A = 2 cm

2

1

air air slab

r

x (3-1-x) Dims. in mm

6- The currents are: (0.32 A, 1.22 A, 0.9 A) and the potential difference is (-8.4 v).

7- The charge and potential for each capacitor are:

Capacitor (1)

Capacitor (2)

Capacitor (3)

Capacitor (4)

Charge 100 C 160/3 C 160/3 C 140/3 C

Potential difference 20 v 40/3 v 20/3 v 20 v

The stored energy in (4 F) capacitor is (355.56 J).

8- The currents in the circuit are: (0.495 A, 0.107 A, 0.602 A) and VB-VA = 17.03 v.

9- Series resistance = 183.3 , Parallel resistance = 44 .

10- Parallel resistance = 23.08 .

11- Parallel resistance = 28.4 .

12-

Capacitor (1)

Capacitor (2)

Capacitor (3)

Capacitor (4)

Charge 30 C 7.5 C 18 C 18 C

Potential difference 2.5 v 7.5 v 3 v 4.5 v

13- Stored energy in capacitor = 9 J

Capacitor

(5 F)

Capacitor

(10 F)

Capacitor

(8/3 F)

Capacitor

(18 F)

Capacitor

(3 F)

Charge 10 C 10 C 8 C 18 C 18 C

Potential difference 2 v 1 v 3 v 1 v 6 v

14- Stored energy in C4 = 42.6 J

Capacitor (1)

Capacitor (2)

Capacitor (3)

Capacitor (4)

Charge 360/13

C 120/13 C 240/13

C

360/13

C

Potential difference 60/13 v 30/13 v 30/13 v 40/13 v

15- Series resistance = 57 , Parallel resistance = 13.68 .

16- The currents are: (0.576 A, 2.407 A, 2.983 A), the potential difference (-0.034 v).

17- Ct = 1.76 pF

18- 1.7 F and 21 F

19- 373 pJ

20- a-

b-

21- Ct = 20/3 F

22-

Capacitor (1)

Capacitor (2)

Capacitor (3)

Capacitor (4)

Charge 20/3 C 20/3 C 40/3 C 20 C

Potential difference 10/9 v 20/9 v 20/6 v 20/3 v

23-

CT = 3.333 F, WT = 13500 J.

Capacitor (1)

Capacitor (2)

Capacitor (3)

Capacitor (4)

Charge 180 C 180 C 120 C 120 C

Potential difference 30 v 60 v 30 v 60 v

24-

25- Currents in the circuit are: (1.636 A, 2.545 A, 0.909 A) and potential difference equals (1.818 v). 26- Currents in the circuit are: (0.462 A, 1.307 A, 0.846 A).

Current = 20 A, Power = 800 W

Current = 2.5 A, Power = 25 W

Current = 2.5 A, Power = 25 W Current = 15 A, Power = 450 W