Chapter 19

Electric Potential of a Point Charge:

The discussion of electric potential is important because we are always looking for convenient sources of energy.  Since any two point charges exert a force of attraction or repulsion on each other, if one charge moves in the field of the other a distance Δr  under an average force F, the work done is equal to FΔr.  This means that if there is just a single charge alone in the entire space, there is potential.  When a second charge is placed in the field of the first charge, then work and energy develop and we think of potential energy.  Recall the "potential energy" concept you learned in Physics I.  The cause of gravitational potential energy is gravity.  The potential energy a rock has in the gravitational field of the Earth is because of gravity.  Gravity attracts mass M with a force F = Mg.  When a rock is lifted to a height h, the work done on it is equivalent to Mgh.  The potential energy stored or consumed is also Mgh. 

A similar concept applies to the electrostatic field that charge +q1 creates at a distance r1 from itself: E = kq1/r12.  If another charge +q2 is placed at r1,  q1 repels it.  As q2 moves away under the varying force F = kq1q2 /r2 to , more and more work will be done on it.  The maximum amount of work that can be done may be calculated by adding all FΔr's from r1 to . 

Now, how can we add all FΔr's?  We may choose all Δr's equal, but as q2 is repelled by q1, and goes farther and farther away from q1,  F becomes weaker.  That is why calculus and integration is needed here.  Using calculus, we can show that the total work that q1 does for us on q2 in pushing it from an initial distance r1 to  is equal to W = kq1q2 /r1.  This is the maximum amount of energy that we can squeeze out of the situation and we call it the potential energy that q2 has if placed at a distance r1 from q1.

If we divide both sides of W by q2, we get:  W/q2 = kq1/r1.  (1)

W/q2 is energy per unit charge.  We name it "electric potential, V."

Let's write (1) as V1 = kq1/r1  and continue with the following definition:

 The electric potential V1 of Point charge q1 at Point P in space at distance r1 from it is defined as:

 Now if another charge like q2 is placed at P,  q2 finds a potential energy equal to

Figure 1

Consequently, we can write  P.E. = V1q2  or, in general,  

Ue = Vq.

In electricity, Ue is most often used for energy or electric potential energy rather than P.E.In SI the unit of electric potential V is J/C.   Joule/Coul is called Volt.   1Volt means  1J/C.

A single charge in space generates different Potentials at different distances from it.  The presence of a second charge is necessary for potential energy to develop.

Example 1: Calculate the electric potential of q1 = 25.0nC at (a) 1.00m, (b) 2.00m, and (c) infinite long distance from it.

Solution: (a) V1 = kq1/r1 [9.00x109(25.0x10-9)/1.00] J/C =  225 Volts.

                  (b) V2 = kq1/r2 [9.00x109(25.0x10-9)/2.00] J/C = 113 Volts.

               (c) V3 = kq1/r3 [9.00x109(25.0x10-9)/  ] J/C  0. 

Example 2:  Calculate the potential energy that another charge q2 = 5.00nC possess when placed at the three different points of the previous example.

Solution:  P.E.1 = V1q2 = 225(J/C) * 5.00x10-9C = 1130nJ.

This means that it takes 1130nJ of energy to push a 5.00-nC of positive charge from infinity to a distance of 1.00m from charge q1.

                P.E.2 = V2q2 = 113(J/C) * 5.00x10-9C = 565nJ.

             It takes 565nJ to push a 5.00-nC charge from infinity to a distance of 2.00m from q1.

                P.E.3 = V3q3  = 0.0 * 5.00x10-9C  =  0.

This means that it takes no effort (energy) to place a charge very far away from q1.

Example 3:  How much energy is needed to place 1.00-μCof charge at each corner of an equilateral triangle 0.250m on each side?  Suppose that each charge is coming from far away (infinity) and that the triangle itself is far away from other electric charges.  See Fig. 2.

Solution:  1) Placing the 1st charge does not require any energy because other corners are empty.   There is no  repelling force against the first charge and it can be done effortlessly (W1 = 0).     2) To bring a 2nd charge from infinity and place it at 0.250m from the 1st charge some work must be done.  The work to be done is equal to the change in P.E. of the 2nd charge in the field of the 1st charge.  It is equal to:

W2 = kq1q2/r = [9.00x109(1.00x10-6)(1.00x10-6)/0.250]J  = 36.0mJ.

The 3rd charge faces resistance from the 1st charge and the 2nd charge.

W3= { kq1q3/0.250 + kq2q3/0.250}mJ = 36.0mJ +36.0mJ = 72.0mJ.

Finally, Wtotal  =  W1+W2+W3  = {0 +36.0+72.0}mJ  = 108.0mJ.

Figure 2

Potential in a Constant Electric Field:

A constant electric field (Fig. 3) is the field in between two parallel sheets equally but oppositely charged.  If the positive sheet is on the left and the negative on the right, as shown, the direction of E is to the right or along the (+x) axis.  Since E is constant, so is F, the force on any point charge q placed in E.  The Work done on +q by the constant F as it is pushed to the right varies linearly with x, simply

W = F  x.

Since F = qE, the work done becomes:

   W = qE  =  qEx   (2)

 

 

Figure 3:     

As the field does more work in pushing q to the right, more of its potential is used up.  As the work W done by the field increases because of the increase in x, the P.E. or Ue of q decreases.

We may write: W = - P.E.  = -Ue.    Equation (2) becomes:

-Ue = qEx    or,      Ue/q = -Ex.     (3)

The left side of (3) is energy per unit charge or potential, V.   Equation (3) becomes:

V = -Ex.

V is a linear function of x when E is constant At x = 0, potential V is maximum and equal to Ex1. Why?  At x = x1, potential is zero. Why?  This makes the V equation to be 

V = -E ( x - x1 ).

Equi-Potentials:

Around a charge distribution, there are points at which potential is the same.  This means that if charge q is placed at any of such points, it will have the same potential energy.  A collection of such points form a surface that is called an "equi-potential surface."  In 2-D (2 dimensions), we show such surfaces as lines and call them "equi-potential lines."  For example, around a positive or a negative point charge, there are infinite concentric spheres that each is an equi-potential surface Of course, the each charge itself is at the center of such sphere.  In the 2-D drawings shown below, for each of a separate (+) or a separate (-) charge, the concentric circles are in fact concentric spheres.

The above charges are of course assumed to be separate and very far from each other and any other charges.  

Figure 4

If a charge stays on any of these circles while moving, its potential energy does not change.  One important point is that anywhere a field line (red or blue) crosses an equi-potential line (each black circle), the angle is 90 degrees; in other words, field lines are necessarily perpendicular the equi-potential lines.  In Lab, it is easy to find the equi-potential lines for a certain charge distribution by using a voltmeter.  Field lines can then be drawn keeping in mind that they must be at right angles to the equi-potential lines.  Below, the field-lines and equi-potential lines are shown for an electric dipole and a parallel-plates capacitor.

Field-lines and Equi-potential Lines of a Dipole

 Figure 5

 

Field-lines and Equi-potential Lines in between the plates of a parallel-plates capacitor

Figure 6

As long as a charge like q2 travels on one equi-potential surface (caused by another charge like q1), its potential energy remains constant.  When q2 travels from one equi-potential surface to another one, its energy changes (Fig. 7 ).  The change in energy is equal to the work done on the charge that can be positive or negative.  The path that q2 takes in going from one equi-potential surface to another, is not important.  What is important is the potential  difference between the two equi-potentials.  In the following figure, two of the infinite equi-potentials around charge q1 are shown.  The energy change that q2 experiences as it goes from equi-potential 1 to equi-potential 2 is the same regardless of the path taken.

Figure 7

Test Yourself 1:   click here.

1) The electric field strength E1 of a point charge q1 at a distance r is (a) E1 = kq1/r2   (b) E1 = kq1/r   (c)  E1 = kq1/r3.

2) The force F of field E1on charge q2 is (a) F = E1q2    (b) F = {kq1/r2}q2   (c)  both a & b.   click here.

3) The electric potential V1 of a point charge q1 at a distance r is (a) V1 = kq1/r  (b) V1 = kq1/r2  (c)  V1 = kq1/r3.

4) The potential energy P.E. of point charge q2  at points in space where the potential is V1 is  (a) P.E. =V1q2     (b) P.E. = (kq1/r) q2    (c) P.E. =  kq1q2 /r    (d) a, b, and c.   click here.

5) The first 4 questions and their correct answers apply to point charges  (a) True   (b) False.

6) The force-field formula F = Eq is true (a) if E is caused by a point charge (b) if E is uniform and caused by a parallel-plates capacitor (c) both a and b.

7) The definition of the electric potential, V, at a point is (a) K.E. per unit charge at that point  (b) Force per unit charge at that point  (c)  P.E. per unit charge at that point  click here.

8) The definition of the electric field, E, at a point is (a) E = force per distance at that point  (b) E = force per unit charge at that point (c)  E = kinetic energy per unit charge at that point.   click here.

9) The electric potential energy P.E. is (a) P.E. = Vq   (b) P.E. = Vq22  (c)  P.E. = (K.E.)q2.

10 ) The reason for using q2 in the above questions instead of just q is that (a) q2 is the charge that is placed in the field of q1 (b) potential V,  field E, and force F, in the above formulas are caused by charge q1  (c) both a & b.

11) Even if we do not use q2 instead of simply q, and write F = Eq and P.E. = Vq,  it is understood that (a) E and V are caused by a charge or chrges other than q  (b) q is placed in the field E of a different charge   (c) both a & b.

12) The SI unit of E, the electric filed, is (a) C/m  (b) N/C  (c) N/m.    click here.

13) The SI unit of V, the electric potential, is (a) C/s  (b) Joules/C  (c) Joules/m. 

14) The potential V at 9.0m from a +25μC charge is (a) 2778 J/C  (b) 25000 J/C   (c) 0.

15) If  +1.0C of charge is placed at 9.0m from the charge in Question 14, it finds a potential energy equal to (a) 2778J  (b) 0   (c) 25000J.   click here.

16) If  -1.0Coul. of charge is placed at 9.0m from the charge in Question 14, it finds an energy of (a) -2778J  (b) 0   (c) - 25000J.

17) The reason why the answer in Question 16 is negative is that (a) work has to be done to move the negative charge to infinity under the attraction of the positive charge (b) in moving the negative charge away, displacement is outward while the attraction force is inward  (c) both a & b.   click here.

18) The potential at 3.0m from a -15.0μC charge is (a) -45000J/C  (b) 15000J/C  (c) 45000J/C.

19) If 40.0μC of charge is placed at 3.0m from the charge in Question 18, it finds a potential energy of (a) -4.5J   (b) -1.8J   (c) 9.0J.

20) If -40.0μC of charge is placed at 3.0m from the charge in Question 18, it finds a potential energy of (a) -4.5J    (b) -1.8J    (c) 1.8J.   click here.

21) The energy it takes to place a 4.0μC charge at a corner of an equilateral triangle (2.0m long on each side) that has no charge on it and is far from other charges is (a) zero   (b) 2.0J  (c)  -2.0J.

22) In Question 21, if one corner has that 4.0μC charge, the energy it takes to place another 4.0μC charge at a 2nd corner is  (a) -0.144J  (b) 0.072J  (c)  0.144J.   click here.

23)  In Question 22, to place another 4.0μC charge at the 3rd corner, it takes (a) 0.144J    (b) -0.072J    (c) 0.072J.

24)  Formulas  V1 = kq1/r  and  P.E. =  kq1q2 /r  apply (a) to point charges only  (b) surface charges only  (c) both a & b.

25) In the space between a parallel-plates capacitor, electric field, E is constant.  Potential energy varies with distance from each plate.  The way P.E. varies with (x), its distance from one of the plates, is (a) proportional to x  (b) proportional to 1/x  (c)  proportional to x2.   click here.

 

 Parallel-plates Capacitor: 

Two parallel and metallic plates separated by an insulator form a "parallel-plates capacitor".   Capacitors store electric energy.  If two flat sheets of aluminum foil sandwich a thin sheet of paper, a parallel-plates capacitor is formed.  When aluminum foils are connected to the poles of a battery, electrons from the negative pole flow through the connecting wire and distribute themselves over one foil making it the negative plate.  This negative plate (foil) repels equal number of electrons from the other plate (foil) and causes the other foil to become the positive plate The repelled electrons flow toward the positive pole of the battery where they are wanted and get absorbed by it.  The closer the plates (or the thinner the insulating material, here the paper), the more charge accumulation occurs on them.  However, there is a limit to the amount of positive and equally negative charges that can accumulate themselves on the two plates (foils).  If accumulation exceeds a certain amount, electric discharge takes place via a spark through the insulator.   The internal spark will burn the insulator and the capacitor goes bad.  We will come back to the discussion of parallel-plates capacitors after the following general discussion on capacitors.

Charge-to-Voltage Ratio for Capacitors:

Fig. 8 shows an initially empty capacitor that is connected to a battery.  If key K is closed (turned on) at t = 0, the battery starts charging the capacitor.   The voltmeter will show that the capacitor voltage VC keeps increasing with the increase in charge q.   The reason is very obvious.  As time t increases, the capacitor charge accumulation q increases that causes the electric P.E. of the capacitor to increase.   As a result, the capacitor voltage, VC  increases.  In brief, the more charge q on each plate, the greater the capacitor voltage VC.   Experiment verifies that the charge-to-voltage ratio q /VC or simply  q /V for each capacitor is a constant and is called the "capacity C of the capacitor."  We may write this as

A voltmeter is placed across the capacitor to monitor its voltage, VC.  Resistance R controls the flow of charges to the capacitor and avoids sudden charging.  R is like a valve in water systems that if opened slightly the flow will be controlled and small.

Figure 8

The SI Unit of Capacity:

In SI, charge is in Coulomb and voltage in volt; therefore, capacity becomes Coul./volt called "Farad."   The capacity of a capacitor is said to be 1 Farad (1F)  if it can hold a maximum charge of 1C when connected to a voltage of 1V.  Majority of capacitors have very small capacities.  Most of them are built to hold charges of μC or nC amounts.

Example 4:  Calculate the capacity of a capacitor that holds at most 30.0μC of charge when connected to a 12.0V battery.

Solution: C = Q/V ;  C = 30.0μC/12.0volts = 2.50μC/volt  or,  C =2.50μF.

 

Example 5:  When a capacitor is half charged, it has 60.0μC of charge on each plate and the voltage across it is 7.50 volts.  Find its capacity.

Solution:  The charge-voltage ratio is C = Q/V.  The more the accumulated charges, the greater the capacitor voltage.  Capacity C is constant, anyway.  

= 60.0μC/7.50 volts  =  8.00μF  (micro-Farads).

 Back to Parallel-plates Capacitor:

The capacity, C, of a parallel-plates capacitor is directly proportional to the area of each plate (A) and inversely proportional to the insulator thickness (d).   C is also proportional to the "permittivity, ε " of the insulating materialThe symbol is pronounced "epsilon.It is related to how well the insulating material allows the electric field lines to pass through.   The Capacity, C, of a parallel-plates capacitor is therefore given by

   

 The insulating material is also called the "dielectric."  The permittivity of vacuum (free space) is shown by εo.   The ε of any material is compared to that of vacuum by  ε = κεo  where κ pronounced " kappa" is called the dielectric constant of the material.  The value of  κ  for vacuum is 1, for mica is 5.4, and for water is 80.   This means that if mica is used as the insulator, the capacity increases by a factor of 5.4 compared to vacuum or almost air.

Example 6:  Calculate the capacity of a parallel-plates capacitor with rectangular (20.0cm by 30.0cm) aluminum plates separated by a 0.10mm sheet of paper.  The dielectric constant of regular paper is κ = 3.3.

Solution:

 

 

C = 3.3[8.85x10-12F/m](0.2m X 0.3m) /0.00010m = 17.5x10-9F           ;          C =  17.5 nF.

Note: 1 Farad of capacity is a very large capacity.

 

Example 7:  Calculate the area of each square plate of a 1.00-Farad parallel-plates capacitor with an air gap of 0.0500mm.

Solution:  Solving the capacity formula for (A), yields:   A = Cd /ε Substituting yields:  A = (1.00 F)(0.0500x10-3m) / (8.85x10-12 F/m) = 5.65x106 m2 

Taking square root, each side = 2380m = 1.48 miles ? Not practical !!!

Connection of Capacitors:

It is sometimes necessary to come up with a capacitor of a certain capacity that is not available.  By combining two or more of different capacitors, the desired capacity can be made.  In two ways capacitors may be connected:  in series and in parallel.  An equivalent capacity can be calculated for each type of connection.  The following figure shows both types of connection and a formula that calculates the equivalent capacity for each type:

 

Figure 9                                                        Figure 10

  Series:

The battery voltage must equal the sum of voltages across the three capacitors.  We may write:

 Vtotal  = Vab + Vbc + Vcd    (1)

 If 2 electrons flow to the left of C1, they repel 2 electrons from the right plate of C1 making its right plate 2 units positive.  Those repelled electrons move to the left side of C2 making it -2 units while repelling 2 electrons from the right side of it making its right +2 units.  The same happens to C3.  The repelled 2 electrons from the right of C3 will be absorbed by the positive pole of the battery and the flow for those 2 electrons completes.  Of course saying "2 electrons" is just an example.  In reality some 1013 or 1014 more or less electrons might easily flow.  Every capacitor in Fig. 9 (series) ends up with same amount of charge Q.

Equivalent Capacity: The single capacitor that can replace those three capacitors must hold the same amount of charge, simply Q.  For the equivalent capacitor, we may write:

Q = CeqV from which V = Q/Ceq.

For each capacitor we write its q = CV.  

Vab = Q/C1 ;  Vbc = Q/C2 ;  Vcd = Q/C3.  Substituting in (1), results in

  Q/Ceq = Q/C1  +  Q/C2  + Q/C3

Dividing by Q, yields:

1/Ceq= 1/C1 + 1/C2 + 1/C3

 for series capacitors.

  Parallel:

The total charge Qtotal  that leaves the battery distributes over the three capacitors such that

 Qtotal  = Q1 + Q2 + Q3.  Electrons arriving at e divide into 3 branches. (Fig. 10)

 If capacities C1 , C2 , and C3 are proportional to numbers 2, 3, and 4, for example, and say 18 electrons leave the negative pole of the battery, 4 will flow to C1, 6 will flow to C2, and 8 will flow to C3 and settle on their left plates.  Equal number of electrons will be repelled from their right plates making those plates positive.  The repelled 18 electrons will be absorbed by the positive pole of the battery and the flow for those 18 electrons completes.  This is just an example, in reality some 1013 or 1014 more or less electrons could easily flow.  

Qtotal  = Q1 + Q2 + Q3.

Equivalent Capacity:

Using Q = CeqV for the equivalent capacitor as well as the individual capacitors, yields:

   Ceq= C1 +  C2 +  C3V.

Dividing through by V, yeilds:

 

Ceq  = C1 + C2 + C3 

 for parallel capacitors.

Look at the following two simple examples:

 Example 8: A 30.0μF capacitor is in series with a 6.00μF capacitor.  Find the equivalent capacity.

Solution:

1/Ceq  = 1/C1 + 1/C2 ;

1/Ceq=1/30+1/6 ;  Ceq= 5.00μF.

 Make sure you use horizontal fraction bars when verifying the solution.

 

 

 Figure 11

 Example 9: A 30.0μF capacitor is in parallel with a 6.00μC capacitor.  Find the equivalent capacity.

Solution: Ceq = C1 + C2  ;

 Ceq = 30.0 + 6.00 ;

 Ceq  = 36.00μF.

 

Figure 12

 

Example 10:  In the figure shown, find the equivalent capacity.

Solution:  Between a and b, there is a parallel module that simply adds up to:  

Cab = 60.0μF.

Then, Cab and Cbc are in series and their reciprocals add up to give the reciprocal of Cac.

 1/Cac=1/Cab+1/Cbc ; 1/Cac=1/60+1/20  = 1/15.

      Cac = 15.0μF.

Figure 13

 

Example 11:  In the figure shown, find the voltage across and the charge accumulated in each capacitor.

 

Solution:  From the top figure:

Cab = 12.0 + 15.0 = 27.0μ;  from the bottom figure:

 1/Cac= 1/Cab+ 1/Cbc;  1/Cac= 1/27 +1/13.5 

 Cac = 9.00μF.  This is the overall capacity that the 18.0V battery faces.  Since Q = CV; thus ,

Q = (9.00μF)(18.0V)  = 162 μCoul.  This means that each capacitor in the bottom figure accumulates 162μC of charge.   Knowing their capacities, we can calculate their voltages.

Qab = CabVab ; 

Vab = 6.00 Volts.   (Across C1 & C2)

Qbc = CbcVbc  ;  

Vbc =  12.0 Volts.  (Across C3)

 Go to the next column.

 

Figure 14

Going to the ab-portion of the top figure, we may find how the two parallel capacitors divide the 162μC of charge.  They divide it as (12/27) and (15/27) proportions.

 Q1 = (12/27)(162μC ) = 72.C.

 Q2 = (15/27)(162μC ) = 90.C.  

Of course,  Q3 = 162μC.

 

Energy Stored in a Capacitor:

Although, product QV has unit of energy; however, we cannot just multiply a capacitor's voltage V by its stored charge Q and believe it to be the energy stored in it!

The energy stored in a capacitor is equal to the energy it takes to charge it up. When a battery of voltage V is used to charge a capacitor, it faces no resistance at the beginning.  The charge on an empty capacitor varies from 0 to Q.  Although the battery voltage is V at all time; however, the capacitor voltage starts from 0 and gradually approaches V. 

On the average, the capacitor voltage is (0 + V )/2 or V/2.  We may think that the battery pushed charge Q into the capacitor at a constant voltage of V/2 from the beginning to the end.  This means that the energy stored in a capacitor is simply

Ue = QV/2.

There are two other versions of Ue .  Since Q = CV, we get Ue= (1/2)CV2 Also,   Ue = Q2/(2V).   Verify.

Example 12:  A 15μF-capacitor is connected to a 9.6-V battery.  Calculate (a) the charge accumulation and (b) the energy stored in it.

Solution:  (a)  Q = CV    ;    Q = (15μF)(9.6 V)  =  144μC.

                (b)  Ue = (1/2)QV    ;    Ue = (1/2)(144μC )(9.6V) = 690μJ.

 

Test Yourself 2:

1) A capacitor is a device that stores (a) kinetic energy  (b) electric energy  (c) elastic potential energy.   click here.

2) The capacity C of a parallel-plates capacitor is proportional to (a) the area of one of its plates A  (b) the reciprocal of the gap between its plates, 1/d   (c) to the dielectric constant, κ of the the material between the plates  (d) a, b, and c.

3) The dielectric constant, κ of the material between the plates of a capacitor is (a) the ratio of the permittivity of that material, ε to the permittivity of vacuum εo   (b) such that we may write: ε =κεo  (c) both a & b.  click here.

4) The value of εo, the permittivity of vacuum for the electric field effect, is equal to (a) 8.85x10-12 Farad/meter   (b) 8.85x10-12 Coul.2/(Nm2)    (c) 1/(4πk) where k is the Coulomb's constant.   (d) a, b, &c.   click here.

5) Capacity is defined as (a) the charge-to-voltage ratio of a capacitor   (b) charge -to-distance ratio of a capacitor  (c)  charge-to-energy ratio.  click here.

6) When an empty capacitor is connected to a battery, the very first voltage across the capacitor is (a) zero  (b) exactly equal to the battery voltage  (c)  half of the battery voltage.

7) When an empty (deflated) basketball is connected to an air pump, the very first gauge pressure in the basketball is (a) zero   (b) equal to the pump's or the compressor's pressure  (c) half of the compressor's pressure.

8) When the capacitor in Question 6, is half-charged, the voltage across it is (a) equal to the battery's voltage   (b) equal to 1/2 of the battery's voltage  (c) fluctuates.   click here.

9) When the basketball in Question 7, is halfway filled, the air pressure in it is (a) equal to the pump's pressure   (b) equal to 1/2 of the pump's pressure  (c) fluctuates.

10) When the capacitor is fully charged after sufficient time has elapsed, the voltage across it (a) is almost equal to the battery's voltage   (b) is zero because it does not accept any more charges   (c) is neither a nor b.      click here.

11) When the basketball in Question 7, is fully inflated to where the pump cannot inflate it anymore, the pressure in it (a) is equal to the pressure that the pump can generate   (b) is zero because it does not accept any more air   (c)  neither a nor b.

12) The above questions lead to (a) the proportionality of charge q on each plate of a capacitor to V the developed voltage across it   (b) the fact that capacity C is the proportionality constant  (c) Q = CV.   (d) a, b, & c.

13) The voltage across a 12-μF capacitor is 5.0V.   Each of its plates carry a charge of (a)  |60.μC|   (b) |2.4μC|   (c) 0. 

14) The charge on and the voltage across a capacitor are 85μC and 5.0volts, respectively.   Its capacity is (a) 425μF   (b) 425 Farad   (c) 17μF.  click here.

Problem:  Draw a battery and three parallel-plate capacitors connected to it in series as shown in Fig.9.  During the very first moments, suppose 5 trillion negative charges travel from the negative pole of the battery and distribute evenly over the nearest plate they can reach.

15)  What happens to the other plate of that capacitor? (a) It receives 5 trillion electrons  (b) It loses 5 trillion electrons  (c) It becomes 5 trillion protons positive  (d) b & c.  click here.

16) Where do the repelled electrons of the first capacitor go?  They (a) jump into air   (b) go to the nearest plate of the middle capacitor and make it 5 trillion electrons negative   (c) they return to the negative plate the same way they came in.

17) What happens to the opposite plate of the middle capacitor?   It becomes (a) 5 trillion protons positive  (b) 5 trillion electrons negative  (c) Neither a no b.  click here.

18) Is it correct to say that the third capacitor experiences the same process as the middle one? (a) Yes  (b) No

19) What happens to the repelled electrons from the third capacitor?  They (a) go to the positive pole of the battery and get absorbed by it   (b) complete the flow of electrons in the circuit  (c) both a & b.  click here.

20) If we name the capacitor's charges Q1, Q2, and Q3,  then (a) Q1= Q2= Q3   (b) Q1= Q2+ Q3   (c) neither a nor b.

21) We may say the capacitors in series accumulate the same amount of charge. (a) True   (b) False  click here.

22) For capacitors C1 and C2 in series (C1 > C2)  with a battery   (a) Q1>Q2   (b) Q1< Q2   (c) Q1= Q2.

23) For capacitors C1 and C2  (C1 > C2) in parallel with a battery   (a) Q1>Q2   (b) Q1< Q2   (c) Q1= Q2.

24)  The equivalent capacity, Ceq for C1= 25.F and C2 = 5.00μF connected in parallel is (a) 4.25μF   (b)30.0μF  (c) 125μF.

25) The equivalent capacity, Ceq for C1= 25.F and C2 = 5.00μF connected in series is (a) 4.17μF  (b)30.0μF (c) 125μF.

26) The equivalent capacity, Ceq for C1= 15μF and C2 = 52μF connected in series is (a) more than 67μFy  (b) less than 15μFy  (c) equal to 67μF.  click here.

27)  Two capacitors C1= 8.0μF and C2 = 16μF are connected in parallel to a 4.0-V battery. The accumulated charges are: (a) 32μCoul. and 64μCoul.   (b) 2.0μCoul. and 4.0μCoul.  (c) neither a nor b.

28)  Two capacitors C1= 8.0μF and C2 = 24μF are connected in series to a 4.0-V battery. The accumulated charges are: (a) 32μCoul. and 96μCoul.   (b) 2.0μCoul. and 6.0μCoul.  (c) 24μCoul. and 24μCoul..  click here.

29) The voltages across C1 and C2 above are (a) 3.0V and 1.0V  (b) 4.0V and 4.0V  (c) 24.0V and 24.0V .

30) The product QV has unit of  (a) force  (b) power  (c)  energy.  click here.

31) When a capacitor is charged, it can give back the accumulated charge, Q, on it (a) as it keeps the same voltage V  (b) as the voltage across it decreases with gradual charge loss  (c) as the voltage across it increases with gradual charge loss.

32)  Based on the previous questions, the energy stored in a capacitor, Ue, is the product (a) QV   (b) (1/2)QV   (c) 2QV.

33) Because Q = CV, the stored energy,  Ue = (1/2)QV may be written as (a) Ue = (1/2)CV2   (b) Ue = (1/2)Q2/C   (c) both a & b.

34) The energy stored in a 60.0-μF-capacitor when the voltage across it is 5.00V is (a) 1500μJ  (b) 3000μJ  (c) 750μJ.

35) The charge accumulation on the capacitor of the previous question is (a) 300μCoul.   (b)12.0μCoul.   (c) 120μCoul.. 

36) If you now use Ue = (1/2)QV to calculate the energy again, you get  (a) 1500μJ  (b) 3000μJ  (c) 750μJ.   click here.