Chapter 25 and 26
Electric Potential:
The discussion of electric potential is important because we are always looking for convenient sources of energy. Since any two electric 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 is done 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 actualize 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 masses with a force F = Mg. When a rock moves through a height h, the work done is Fh or Mgh. The potential energy stored or consumed is also Mgh. A similar concept applies to the electrostatic field that charge q1 creates: E = kq1/r2. If another charge q2 moves toward or away from q1, the change in potential energy of q2 within a distance change Δr is ΔP.E.=q2EΔr. This is similar to P.E. = Mgh. Or, we may write: ΔP.E. = q2( kq1/r2 )Δr.
Dividing both sides of this equation by q2 gives us the potential energy per unit charge that is defined as the electric potential energy, V. We get:
P.E. /q2 = ( kq1/r2 )Δr.
Note that from the units point of view, Δr and 1/r2 simplify and the result is dimensionally equivalent to 1/r . The symbol used for the left side, that is energy per unit charge, is V called the electric potential. the right side is equivalent to kq1/r.
The electric potential V of a point charge q1 at a typical point P in space at a distance r from it is given by :
Now if another charge like q2 is placed at P a distance r from q1 , then q2 finds a potential energy equal to
Consequently, we can write : P.E. = V1 q2
The reason for thinking of a quantity such as potential is that, if for example, q1 and q2 are positive, there is a repulsion force between them and that if q2 is free to move, it can do some work for us or release some potential energy as it moves farther from q1. If we examine the unit of V, we will see that it has units of (energy per charge) or in SI units (Joules / Coulomb). Let's do this examination. Also let [ ] denote "the unit of ".
(Joule / Coul ) is called ( Volt ). 1 Volt means (1 J/Coul. ). The unit is verified. That is why V is called potential.
1 volt is the potential of a point around (+q) that if 1C is placed at that point, 1 Joule of work is produced as (+q) repels the 1C to infinity.
A charge in space generates different Potentials at different distances from it. The presence of a second charge is necessary for Potential Energy to make sense.
Example 1: Calculate the electric potential that of q1 = 25.0nC at distances of 1.00m, 2.00m, and at infinity (far enough from the charge).
Solution: V1 = -kq1 / r1 = -(9.00x109)(25.0x10-9) / (1.00) = -225 J /Coul. or -225 Volts.
V2 = -kq1 / r2 = -(9.00x109)(25.0x10-9) / (2.00) = -113 J /Coul. or -113 Volts.
V3 = -kq1 / r3 = -(9.00x109)(25.0x10-9) / ( ∞ ) = 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 = V1(q2) = -225(J/Coul.) * (5.00x10-9 Coul.) = -1130 nJ
This means that it takes 1130 nJ of energy to push a 5.00-nC of positive charge from infinity to a distance of 1.00m from charge q1 that is also positive.
P.E.2 = V2(q2) = -113(J/Coul.) * (5.00x10-9 Coul.) = -565 nJ
It takes 565 nJ to push a 5.00-nC charge from infinity to a distance of 2.00m from q.
P.E.3 = V3(q3) = (0.0) * (5.00x10-9 Coul.) = 0.0 nJ
This means that it takes no effort (energy) to place a charge very far away from q1.
Note: The ( - ) sign given to the formula is on purpose. The purpose is to be able to give the interpretations such as ones given in the above example.
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 is in space far away from other electric charges.
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 done is equal to the change in P.E. of the 2nd charge in the field of the 1st charge. It is equal to:
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W1 = 0. W2 = -kq1q2 / r = [ (9.00x109)(1.00x10-6)(1.00x10-6) / (0.250) ] J = -36.0 mJ 3. The 3rd charge faces resistance from both the 1st charge and the 2nd charge. W3 = -{kq1 q3 / (0.250)} - {kq2 q3 / (0.250)} = -36.0 mJ -36.0 mJ = - 72.0 mJ Finally, Wtotal = W1 +W2 +W3 = {0 - 36.0 -72.0} mJ = -108.0 mJ
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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 electric 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 because of electric field magnitude E1 is (a) P.E. =V1 q2 . (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) only if E is caused by a point charge. (b) only if E is uniform and caused by a parallel-plate capacitor. (c) whether E is that of a point charge or the field between the plates of a parallel-plate capacitor.
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) "Electric P.E. per unit charge at that point." click here
8) The definition of the electric potential, V at a point is (a) "V = K.E. /q2 at that point." (b) "V =F/q2 at that point." (c) "V = -P.E./q2 at that point." click here
9) The electric potential energy, P.E. is (a) P.E. = Vq2. (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) the 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 other than q. (b) q is placed in the field E of a different charge . (c) both a & b.
12) The Metric unit of E, the electric filed, is (a) Coul./m . (b) N/Coul.. (c) N/m . click here
13) The Metric unit of V, the electric potential, is (a) Coul/s. (b) Joules/Coul. (c) Joules/m.
14) The potential , V, at 9.0m from a +25μCoul. charge is (a) -2778 J/Coul.. (b) -25000 J/Coul.. (c) 0.
15) If +1.0Coul. of charge is placed at 9.0m from the charge in Question 14, it finds a potential energy of (a) -2778J. (b) 0.0J. (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) 0.0. (c) + 25000J.
17) The reason why the answer in Question 16 is positive is that (a) the negative charge placed at 9.0m will be pulled by the 25.0μC charge and will release energy as it moves toward it. (b) the negative in the formula is provided for this purpose. (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) If in Question 21, 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) 0.072J. 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-plate 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, in a constant E is (a) proportional to x. (b) proportional to 1/x. (c) proportional to x2. click here
Parallel-Plate Capacitor:
Two parallel and metallic plates separated by an insulator form a capacitor that can store electric energy. If two flat sheets of aluminum foil sandwich a thin sheet of paper, a 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 negative. This negative side repels equal number of electrons from the other side and causes the other side to become positive. The repelled electrons flow toward the positive pole of the battery where they get absorbed by it. This process causes the battery to find new poles: the plates of the capacitor. One difference is that the new poles have more charges accumulated on them. 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. If accumulation exceeds a certain value, electric discharge takes place via a spark through the insulator. The capacity of the plates is obviously dependent on the area of each plate. Of course the assumption is that both plates are made equal and completely face each other. The thinner the insulating material ( called the dielectric) the stronger the electric field between the plates and therefore the greater the capacity. The Capacity of a parallel-plate capacitor is given by
In the above formula, A is the area of each plate and d is the gap between the plates or the dielectric thickness. The quantity ( ε ) is called the Permittivity of the material for electric field transmission through it. The permittivity of vacuum (free space) is shown by ( εo ) . These two quantities are related by ( ε = κ εo ) where κ 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 gap between the plates), the capacity increases by a factor of 5.4 compared to vacuum or almost air.
Example 4: Calculate the capacity of a parallel-plate 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:
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C = 3.3(8.85x10-12
(F/m) )* ( 0.200m X 0.300m) / ( 0.00010m) = 17.5x10-9
F = 17.5 nF
Note: 1 Farad of capacity is a very large capacity |
Example 5: Calculate the area of each plate of a 1.00-Farad parallel-plate 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
This means a square of side 2380m = 1.48 miles ( A capacitor 1.48mi by 1.48mi ?)
Charge-Voltage Formula:
For a capacitor, the charge on the capacitor, Q, and the voltage, V, it generates are related by the formula:
The capacity of a capacitor is 1F if it can hold a maximum charge of 1Coul. when connected to a voltage of 1V. Majority of capacitors are of very small capacities because with the normal voltages of a few 10 or 100 volts they can only hold charges of micro- or nano-Coulombs magnitudes. Farad = Coul. / Volt
Example 6: 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) = (30.0 / 12.0) (micro-Coul./ Volts) or C = 2.50 μF.
Example 7: 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 charge, the greater the voltage it makes. Capacity C is constant, anyway. C = (60.0μC) / ( 7.50 volts) = 8.00μF (micro-Farads).
Connection of Capacitors:
It is sometimes necessary to have a capacitor of a certain capacity that is not available in the lab. 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 of connection:
Fig. 1 Fig. 2
| Series: The battery voltage must be 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 positive. Those repelled electrons move to the left side of C2 making it negative while repelling 2 electrons from the right side of it making its right side positive. 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 current is complete. Of course saying "2 electrons" is just an example. In reality some 1013 or 1014 more or less electrons might flow. We end up with equal charges on capacitors that are in series. 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 may write its own charge-voltage relation. Vab = Q / C1 , Vbc = Q / C2 , Vcd = Q / C3. Using (1), results in Q / Ceq = Q / C1 + Q / C2 + Q / C3 Dividing through by Q, yields: 1 / Ceq = 1 / C1 + 1 / C2 + 1 / C3 |
Parallel: The total charge Qtotal that leaves the battery distributes over the three capacitors such that Qtotal = Q1 + Q2 + Q3. 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 the right side plates making them positive. The repelled 18 electrons will be absorbed by the positive pole of the battery and the current is complete. This is just an example, in reality some 1013 or 1014 more or less electrons might flow. Qtotal = Q1 + Q2 + Q3. Equivalent Capacity: Using Q = CeqV for the equivalent capacitor as well as the individual capacitors, yields: CeqV = C1V + C2V + C3V Dividing through by V, yeilds: Ceq = C1 + C2 + C3 |
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 / 30.0 + 1 / 6.00 ; Ceq = 5.00μF ; Make sure you use horizontal fraction bars when verifying the solution.
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| 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
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Example 10: In the figure shown, find the equivalent capacity.
| Solution: Between a and b, there
is a parallel set that simply add up. 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 = 1/60 + 1/20 = 1/15 Cac = 15.0μF |
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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 = 27.0μF ; and 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 18.0-V 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 (They are in series). Knowing their capacities, we can calculate their voltages. Qab = CabVab ; Vab = 6.00 Volts (Same for C1 & C2) Qbc = CbcVbc ; Vbc = 12.0 Volts (For C3) 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. Go to the next column.
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Q1 = (12/27)(162μC ) = 72.0μC Q2 = (15/27)(162μC ) = 90.0μC. Of course, Q3 = 162μC
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Energy Stored in a Capacitor:
Note that the product QV has unit of energy. Q is in Coul., and V is in Joules/Coul.. The product QV has unit of Joule in (SI). Keeping this in mind, let's calculate the energy stored in a capacitor that has charge Q on it causing a voltage V across it.
As the charge accumulation on a capacitor varies from 0 to Q, the voltage across it increases from 0 to V. The electric energy Ue is not equal to QV. In the charging process, a battery of voltage V pushes electric charge toward the capacitor. Since charge varies from 0 to Q, it is like the work done by the battery is V times the average charge (0 + Q)/2 or simply Q/2. Therefore, the energy stored is (1/2)QV. We may write:
Ue = (1/2)QV.
There are other versions of this formula. Since Q = CV, we get Ue = (1/2)CV2. Also, Ue = (1/2)Q2/V.
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) = 690 μJ
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 capacitor is proportional to (a) the area of one of its plates, A. (b) the 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 insulating 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 passage of the electric field effect, is (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 also defined as (a) the charge-to-voltage ratio of the 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 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) fluctuating. 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) fluctuating.
10) When the capacitor is fully charged after sufficient time has elapsed, the voltage across it (a) is 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 accumulation, Q on a capacitor's plate to the voltage developed, V 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 has a charge of (a) 60.μCoul. (b) 2.4μCoul. (c) 0.
14) The charge on and the voltage across a capacitor are 85μCoul. and 5.0 volts, 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.1 above. 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? (a) They jump into air. (b) They go to the nearest plate of the middle capacitor and make it 5 trillion electrons negative. (c) the return to the negative plate the same way they came in.
17) What happens to the opposite plate of the middle capacitor? (a) It becomes 5 trillion protons positive. (b) It receives 5 trillion electrons. (c) Neither a no b. click here
18) Is it correct to say that the third capacitor experiences the same process as the middle on? (a) Yes. (b) No.
19) What happens to the repelled electrons from the third capacitor? (a) They go to the positive pole of the battery and get absorbed by it. (b) They 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 (knowing C1>C2) in series with a battery, (a) Q1>Q2. (b) Q1< Q2. (c) Q1= Q2 .
23) For capacitors C1 and C2 (knowing C1>C2) in parallel with a battery, (a) Q1>Q2. (b) Q1< Q2. (c) Q1= Q2.
24) The total capacitance, Ceq for C1= 25.0μF and C2 = 5.00μF connected in parallel is (a) 4.25μF. (b)30.0μF. (c) 125μF.
25) The total capacitance, Ceq for C1= 25.0μF and C2 = 5.00μF connected in series is (a) 4.17μF. (b)30.0μF. (c) 125μF.
26) The total capacitance, Ceq for C1= 15μF and C2 = 52μF connected in series is (a) more than 67μF. (b) less than 15μF. (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 fully 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 = 1/2QV may be written as (a) Ue = 1/2CV2. (b) Ue = 1/2Q2/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 problem is (a) 300.μCoul. . (b)12.0μCoul. . (c) 120μCoul. .
36) If you now use Ue = 1/2QV to calculate the energy again, you get (a) 1500μJ. (b) 3000μJ. (c) 750μJ. click here
