Deck 2: Electrical Energy and Capacitance

Question 1

When charged particles are separated by an infinite distance, the electric potential energy of the pair is zero. When the particles are brought close, the electric potential energy of a pair with the same sign is positive, whereas the electric potential energy of a pair with opposite signs is negative. Explain.

Answer 1

The electrostatic potential energy between a pair of point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the distance between the charges. Thus, the mathematical expression for the electrostatic potential energy between a pair of point charges is, Here, k is the Coulomb constant, and are the magnitudes of the two charges, and R is the distance between them.
Like charges repel each other. If two positive charges are separated by infinite distance, the potential energy of the system of charges will be zero. In order to bring these positive charges closer, one has to perform work by an external agent. This work is stored as potential energy in the system and hence the potential energy will be positive.
Unlike charges attract each other. If two positive charges are separated by infinite distance, the potential energy of the system of charges will be zero. As these charges with move together from an infinite separation, energy is released. Thus, the potential energy of the system becomes negative.

Question 2

If you were asked to design a capacitor in which small size and large capacitance were required, what would be the two most important factors in your design

Answer 2

The capacitance of a capacitance can be increased by inserting a dielectric between the plates of the capacitor. Choose a dielectric material with large dielectric constant and dielectric strength.
The capacitance of a capacitor is inversely proportional to the plate separation, thus to maximize capacitance one can decrease the plate separation.
The capacitance of a capacitor is directly proportional to the plate area, thus to maximize capacitance one can increase the plate area.
The effective capacitance of a parallel combination of capacitors is greater than the capacitance of any capacitor used in the combination. This could be achieved through "stacking" the plates of the capacitor.

Question 3

Consider the Earth and a cloud layer 800 m above the planet to be the plates of a parallel-plate capacitor. (a) If the cloud layer has an area of 1.0 km 2 = 1.0 × 10 6 m 2 , what is the capacitance (b) If an electric field strength greater than 3.0 × 10 6 N/C causes the air to break down and conduct charge (lightning), what is the maximum charge the cloud can hold

Answer 3

(a)
A conductor that stores electric charge at low potentials is called as capacitor. A capacitor consists of two parallel plates with equal cross-sectional area that are separated by a distance.
The capacitance of the capacitor is given as follows: Here, is the permittivity of free space, A is the cross-sectional area of the plates of the capacitor, and d is the plate separation.
Substitute for , for A , 800 m for d. Therefore, the capacitance of the Earth cloud layer that serves as parallel plate capacitor is .
(b)
The maximum electric potential difference between the Earth and the cloud layer is given follows: Here, is the maximum electric field.
The maximum charge that cloud can hold is, Substitute for C and for . Substitute for , for A , and for . Therefore, the maximum charge that cloud can hold is .

Question 4

A 25.0- F capacitor and a 40.0- F capacitor are charged by being connected across separate 50.0-V batteries. (a) Determine the resulting charge on each capacitor. (b) The capacitors are then disconnected from their batteries and connected to each other, with each negative plate connected to the other positive plate. What is the final charge of each capacitor (c) What is the final potential difference across the 40.0- F capacitor

Question 5

A parallel-plate capacitor with a plate separation d has a capacitance C 0 in the absence of a dielectric. A slab of dielectric material of dielectric constant k and thickness d/ 3 is then inserted between the plates as in Figure P16.57a. Show that the capacitance of this partially filled capacitor is given by
Hint: Treat the system as two capacitors connected in series as in Figure P16.57b, one with dielectric in it and the other one empty.

Question 6

A metal sphere of radius 5.00 cm is initially uncharged. How many electrons would have to be placed on the sphere to produce an electric field of magnitude 1.50 × 10 5 N/C at a point 8.00 cm from the center of the sphere

Question 7

The two charges in Figure P16.12 are separated by d = 2.00 cm. Find the electric potential at (a) point A and (b) point B , which is halfway between the charges.

Question 8

(a) When a 9.00-V battery is connected to the plates of a capacitor, it stores a charge of 27.0 C. What is the value of the capacitance (b) If the same capacitor is connected to a 12.0-V battery, what charge is stored

Question 9

(a) Find the equivalent capacitance between points a and b for the group of capacitors connected as shown in Figure P16.42 if C 1 = 5.00 F, C 2 = 10.00 F, and C 3 = 2.00 F. (b) If the potential between points a and b is 60.0 V, what charge is stored on C 3

Question 10

Two capacitors give an equivalent capacitance of C p when connected in parallel and an equivalent capacitance of C s when connected in series. What is the capacitance of each capacitor

Question 11

Suppose you are sitting in a car and a 20-kV power line drops across the car. Should you stay in the car or get out The power line potential is 20 kV compared to the potential of the ground.

Question 12

Is it always possible to reduce a combination of capacitors to one equivalent capacitor with the rules developed in this chapter Explain.

Question 13

An air-filled parallel-plate capacitor has plates of area 2.30 cm 2 separated by 1.50 mm. The capacitor is connected to a 12.0-V battery. (a) Find the value of its capacitance. (b) What is the charge on the capacitor (c) What is the magnitude of the uniform electric field between the plates

Question 14

A 1.00- F capacitor is charged by being connected across a 10.0-V battery. It is then disconnected from the battery and connected across an uncharged 2.00- F capacitor. Determine the resulting charge on each capacitor.

Question 15

A parallel-plate capacitor is constructed using a dielectric material whose dielectric constant is 3.00 and whose dielectric strength is 2.00 × 10 8 V/m. The desired capacitance is 0.250 F, and the capacitor must withstand a maximum potential difference of 4.00 kV. Find the minimum area of the capacitor plates.

Question 16

The potential difference between the accelerating plates of a TV set is about 25 kV. If the distance between the plates is 1.5 cm, find the magnitude of the uniform electric field in the region between the plates.

Question 17

(a) Find the electric potential, taking zero at infinity, at the upper right corner (the corner without a charge) of the rectangle in Figure P16.13. (b) Repeat if the 2.00- C charge is replaced with a charge of 2.00 C.

Question 18

Two conductors having net charges of +10.0 C and 10.0 C have a potential difference of 10.0 V between them. (a) Determine the capacitance of the system. (b) What is the potential difference between the two conductors if the charges on each are increased to +100 C and 100 C

Question 19

Four capacitors are connected as shown in Figure P16.44. (a) Find the equivalent capacitance between points a and b. (b) Calculate the charge on each capacitor, taking V ab = 15.0 V.

Question 20

Two charges of 1.0 C and 22.0 C are 0.50 m apart at two vertices of an equilateral triangle as in Figure P16.60. (a) What is the electric potential due to the 1.0-mC charge at the third vertex, point P (b) What is the electric potential due to the 2.0- C charge at P (c) Find the total electric potential at P. (d) What is the work required to move a 3.0- C charge from infinity to P

Question 21

Why is it important to avoid sharp edges or points on conductors used in high-voltage equipment

Question 22

Explain why a dielectric increases the maximum operating voltage of a capacitor even though the physical size of the capacitor doesn't change.

Question 23

An air-filled capacitor consists of two parallel plates, each with an area of 7.60 cm 2 and separated by a distance of 1.80 mm. If a 20.0-V potential difference is applied to these plates, calculate (a) the electric field between the plates, (b) the capacitance, and (c) the charge on each plate.

Question 24

A 12.0-V battery is connected to a 4.50- F capacitor. How much energy is stored in the capacitor

Question 25

Find the equivalent capacitance of the group of capacitors shown in Figure P16.61.

Question 26

A point charge q = +40.0 C moves from A to B separated by a distance d = 0.180 m in the presence of an external electric field S of magnitude 275 N/C directed toward the right as in Figure P16.6. Find (a) the electric force exerted on the charge, (b) the work done by the electric force, (c) the change in the electric potential energy of the charge, and (d) the potential difference between A and B.

Question 27

Three charges are situated at corners of a rectangle as in Figure P16.13. How much work must an external agent do to move the 8.00- C charge to infinity

Question 28

A 1-megabit computer memory chip contains many 60.0 × 10 15 -F capacitors. Each capacitor has a plate area of 21.0 × 10 12 m 2. Determine the plate separation of such a capacitor. (Assume a parallel-plate configuration.) The diameter of an atom is on the order of 10 10 m 5 1 Å. Express the plate separation in angstroms.

Question 29

Two capacitors, C 1 = 18.0 F and C 2 = 36.0 F, are connected in series, and a 12.0-V battery is connected across them. (a) Find the equivalent capacitance, and the energy contained in this equivalent capacitor. (b) Find the energy stored in each individual capacitor. Show that the sum of these two energies is the same as the energy found in part (a). Will this equality always be true, or does it depend on the number of capacitors and their capacitances (c) If the same capacitors were connected in parallel, what potential difference would be required across them so that the combination stores the same energy as in part (a) Which capacitor stores more energy in this situation, C 1 or C 2

Question 30

A spherical capacitor consists of a spherical conducting shell of radius b and charge Q concentric with a smaller conducting sphere of radius a and charge Q. (a) Find the capacitance of this device. (b) Show that as the radius b of the outer sphere approaches infinity, the capacitance approaches the value a / k e = 4 0 a.

Question 31

Explain why, under static conditions, all points in a conductor must be at the same electric potential.

Question 32

Two point charges Q 1 = +5.00 nC and Q 2 = 3.00 nC are separated by 35.0 cm. (a) What is the electric potential at a point midway between the charges (b) What is the potential energy of the pair of charges What is the significance of the algebraic sign of your answer

Question 33

A parallel-plate capacitor with area 0.200 m 2 and plate separation of 3.00 mm is connected to a 6.00-V battery. (a) What is the capacitance (b) How much charge is stored on the plates (c) What is the electric field between the plates (d) Find the magnitude of the charge density on each plate. (e) Without disconnecting the battery, the plates are moved farther apart. Qualitatively, what happens to each of the previous answers

Question 34

A parallel-plate capacitor has capacitance 3.00 F. (a) How much energy is stored in the capacitor if it is connected to a 6.00-V battery (b) If the battery is disconnected and the distance between the charged plates doubled, what is the energy stored (c) The battery is subsequently reattached to the capacitor, but the plate separation remains as in part (b). How much energy is stored (Answer each part in microjoules.)

Question 35

The immediate cause of many deaths is ventricular fibrillation, an uncoordinated quivering of the heart, as opposed to proper beating. An electric shock to the chest can cause momentary paralysis of the heart muscle, after which the heart will sometimes start organized beating again. A defibrillator is a device that applies a strong electric shock to the chest over a time of a few milliseconds. The device contains a capacitor of a few microfarads, charged to several thousand volts. Electrodes called paddles, about 8 cm across and coated with conducting paste, are held against the chest on both sides of the heart. Their handles are insulated to prevent injury to the operator, who calls "Clear!" and pushes a button on one paddle to discharge the capacitor through the patient's chest. Assume an energy of 300 W s is to be delivered from a 30.0- F capacitor. To what potential difference must it be charged

Question 36

Oppositely charged parallel plates are separated by 5.33 mm. A potential difference of 600 V exists between the plates. (a) What is the magnitude of the electric field between the plates (b) What is the magnitude of the force on an electron between the plates (c) How much work must be done on the electron to move it to the negative plate if it is initially positioned 2.90 mm from the positive plate

Question 37

Three identical point charges each of charge q are located at the vertices of an equilateral triangle as in Figure P16.16. The distance from the center of the triangle to each vertex is a. (a) Show that the electric field at the center of the triangle is zero. (b) Find a symbolic expression for the electric potential at the center of the triangle. (c) Give a physical explanation of the fact that the electric potential is not zero, yet the electric field is zero at the center.

Question 38

A small object with a mass of 350 g carries a charge of 30.0 nC and is suspended by a thread between the vertical plates of a parallel-plate capacitor. The plates are separated by 4.00 cm. If the thread makes an angle of 15.08 with the vertical, what is the potential difference between the plates

Question 39

A certain storm cloud has a potential difference of 1.00 × 10 8 V relative to a tree. If, during a lightning storm, 50.0 C of charge is transferred through this potential difference and 1.00% of the energy is absorbed by the tree, how much water (sap in the tree) initially at 30.08C can be boiled away Water has a specific heat of 4 186 J/kg 8C, a boiling point of 1008C, and a heat of vaporization of 2.26 × 10 6 J/kg.

Question 40

When a certain air-filled parallel-plate capacitor is connected across a battery, it acquires a charge of 150 C on each plate. While the battery connection is maintained, a dielectric slab is inserted into, and fills, the region between the plates. This results in the accumulation of an additional charge of 200 C on each plate. What is the dielectric constant of the slab

Question 41

If you are given three different capacitors C 1 , C 2 , and C 3 , how many different combinations of capacitance can you produce, using all capacitors in your circuits

Question 42

The three charges in Figure P16.17 are at the vertices of an isosceles triangle. Let q = 7.00 nC and calculate the electric potential at the midpoint of the base.

Question 43

Given a 2.50- F capacitor, a 6.25- F capacitor, and a 6.00-V battery, find the charge on each capacitor if you connect them (a) in series across the battery and (b) in parallel across the battery.

Question 44

The voltage across an air-filled parallel-plate capacitor is measured to be 85.0 V. When a dielectric is inserted and completely fills the space between the plates as in Figure P16.49, the voltage drops to 25.0 V. (a) What is the dielectric constant of the inserted material Can you identify the dielectric (b) If the dielectric doesn't completely fill the space between the plates, what could you conclude about the voltage across the plates

Question 45

Capacitors C 1 = 6.0 F and C 2 = 2.0 F are charged as a parallel combination across a 250-V battery. The capacitors are disconnected from the battery and from each other. They are then connected positive plate to negative plate and negative plate to positive plate. Calculate the resulting charge on each capacitor.

Question 46

(a) Find the potential difference D Ve required to stop an electron (called a "stopping potential") moving with an initial speed of 2.85 × 10 7 m/s. (b) Would a proton traveling at the same speed require a greater or lesser magnitude potential difference Explain. (c) Find a symbolic expression for the ratio of the proton stopping potential and the electron stopping potential, V p / V e. The answer should be in terms of the proton mass mp and electron mass me.

Question 47

A positive point charge q = + 2.50 nC is located at x = 1.20 m and a negative charge of 2 q = 5.00 nC is located at the origin as in Figure P16.18. (a) Sketch the electric potential versus x for points along the x -axis in the range 1.50 m x 1.50 m. (b) Find a symbolic expression for the potential on the x -axis at an arbitrary point P between the two charges. (c) Find the electric potential at x = 0.600 m. (d) Find the point along the x -axis between the two charges where the electric potential is zero.

Question 48

Two capacitors, C 1 = 5.00 F and C 2 = 12.0 F, are connected in parallel, and the resulting combination is connected to a 9.00-V battery. Find (a) the equivalent capacitance of the combination, (b) the potential difference across each capacitor, and (c) the charge stored on each capacitor.

Question 49

(a) How much charge can be placed on a capacitor with air between the plates before it breaks down if the area of each plate is 5.00 cm 2 (b) Find the maximum charge if polystyrene is used between the plates instead of air. Assume the dielectric strength of air is 3.00 × 10 6 V/m and that of polystyrene is 24.0 × 10 6 V/m.

Question 50

Two positive charges each of charge q are fixed on the y -axis, one at y = d and the other at y = d as in Figure P16.66. A third positive charge 2 q located on the x -axis at x = 2 d is released from rest. Find symbolic expressions for (a) the total electric potential due to the first two charges at the location of the charge 2 q , (b) the electric potential energy of the charge 2 q , (c) the kinetic energy of the charge 2 q after it has moved infinitely far from the other charges, and (d) the speed of the charge 2 q after it has moved infinitely far from the other charges if its mass is m.

Question 51

(a) Describe the motion of a proton after it is released from rest in a uniform electric field. (b) Describe the changes (if any) in its kinetic energy and the electric potential energy associated with the proton.

Question 52

(a) Why is it dangerous to touch the terminals of a highvoltage capacitor even after the voltage source that charged the battery is disconnected from the capacitor (b) What can be done to make the capacitor safe to handle after the voltage source has been removed

Question 53

A proton is located at the origin, and a second proton is located on the x -axis at x = 6.00 fm (1 fm = 10 15 m). (a) Calculate the electric potential energy associated with this configuration. (b) An alpha particle (charge = 2 e , mass = 6.64 × 10 27 kg) is now placed at ( x , y ) = (3.00, 3.00) fm. Calculate the electric potential energy associated with this configuration. (c) Starting with the three-particle system, find the change in electric potential energy if the alpha particle is allowed to escape to infinity while the two protons remain fixed in place. (Throughout, neglect any radiation effects.) (d) Use conservation of energy to calculate the speed of the alpha particle at infinity. (e) If the two protons are released from rest and the alpha particle remains fixed, calculate the speed of the protons at infinity.

Question 54

Find (a) the equivalent capacitance of the capacitors in Figure P16.35, (b) the charge on each capacitor, and (c) the potential difference across each capacitor.

Question 55

Determine (a) the capacitance and (b) the maximum voltage that can be applied to a Teflon-filled parallelplate capacitor having a plate area of 175 cm 2 and an insulation thickness of 0.040 0 mm.

Question 56

Metal sphere A of radius 12.0 cm carries 6.00 C of charge, and metal sphere B of radius 18.0 cm carries 4.00 C of charge. If the two spheres are attached by a very long conducting thread, what is the final distribution of charge on the two spheres

Question 57

A uniform electric field of magnitude 375 N/C pointing in the positive x -direction acts on an electron, which is initially at rest. After the electron has moved 3.20 cm, what is (a) the work done by the field on the electron, (b) the change in potential energy associated with the electron, and (c) the velocity of the electron

Question 58

A 74.0-g block carrying a charge Q = 35.0 C is connected to a spring for which k = 78.0 N/m. The block lies on a frictionless, horizontal surface and is immersed in a uniform electric field of magnitude E = 4.86 × 10 4 N/C directed as shown in Figure P16.9. If the block is released from rest when the spring is unstretched ( x = 0), (a) by what maximum distance does the block move from its initial position (b) Find the subsequent equilibrium position of the block and the amplitude of its motion. (c) Using conservation of energy, find a symbolic relationship giving the potential difference between its initial position and the point of maximum extension in terms of the spring constant k , the amplitude A , and the charge Q.

Question 59

A proton and an alpha particle (charge = 2 e , mass = 6.64 × 10 27 kg) are initially at rest, separated by 4.00 × 10 15 m. (a) If they are both released simultaneously, explain why you can't find their velocities at infinity using only conservation of energy. (b) What other conservation law can be applied in this case (c) Find the speeds of the proton and alpha particle, respectively, at infinity.

Question 60

Two capacitors give an equivalent capacitance of 9.00 pF when connected in parallel and an equivalent capacitance of 2.00 pF when connected in series. What is the capacitance of each capacitor

Question 61

Consider a plane parallel-plate capacitor made of two strips of aluminum foil separated by a layer of paraffincoated paper. Each strip of foil and paper is 7.00 cm wide. The foil is 0.004 00 mm thick, and the paper is 0.025 0 mm thick and has a dielectric constant of 3.70. What length should the strips be if a capacitance of 9.50 × 10 8 F is desired (If, after this plane capacitor is formed, a second paper strip can be added below the foil-paper-foil stack and the resulting assembly rolled into a cylindrical form-similar to that shown in Figure 16.26-the capacitance can be doubled because both surfaces of each foil strip would then store charge. Without the second strip of paper, however, rolling the layers would result in a short circuit.)

Question 62

An electron is fired at a speed 0 = 5.6 × 10 6 m/s and at an angle 0 = 45° between two parallel conducting plates that are D = 2.0 mm apart, as in Figure P16.68. If the voltage difference between the plates is V = 100 V, determine (a) how close, d , the electron will get to the bottom plate and (b) where the electron will strike the top plate.

Question 63

Rank the potential energies of the four systems of particles shown in Figure CQ16.2 from largest to smallest. Include equalities if appropriate.

Question 64

The plates of a capacitor are connected to a battery. (a) What happens to the charge on the plates if the connecting wires are removed from the battery (b) What happens to the charge if the wires are removed from the battery and connected to each other

Question 65

A tiny sphere of mass 8.00 g and charge 2.80 nC is initially at a distance of 1.60 mm from a fixed charge of +8.50 nC. If the 8.00- g sphere is released from rest, find (a) its kinetic energy when it is 0.500 m from the fixed charge and (b) its speed when it is 0.500 m from the fixed charge.

Question 66

For the system of capacitors shown in Figure P16.37, find (a) the equivalent capacitance of the system, (b) the charge on each capacitor, and (c) the potential difference across each capacitor.

Question 67

A model of a red blood cell portrays the cell as a spherical capacitor, a positively charged liquid sphere of surface area A separated from the surrounding negatively charged fluid by a membrane of thickness t. Tiny electrodes introduced into the interior of the cell show a potential difference of 100 mV across the membrane. The membrane's thickness is estimated to be 100 nm and has a dielectric constant of 5.00. (a) If an average red blood cell has a mass of 1.00 × 10 12 kg, estimate the volume of the cell and thus find its surface area. The density of blood is 1 100 kg/m 3. (b) Estimate the capacitance of the cell by assuming the membrane surfaces act as parallel plates. (c) Calculate the charge on the surface of the membrane. How many electronic charges does the surface charge represent

Question 68

A proton is released from rest in a uniform electric field of magnitude 385 N/C. Find (a) the electric force on the proton, (b) the acceleration of the proton, and (c) the distance it travels in 2.00 s.

Question 69

On planet Tehar, the free-fall acceleration is the same as that on the Earth, but there is also a strong downward electric field that is uniform close to the planet's surface. A 2.00-kg ball having a charge of 5.00 C is thrown upward at a speed of 20.1 m/s. It hits the ground after an interval of 4.10 s. What is the potential difference between the starting point and the top point of the trajectory

Question 70

The metal sphere of a small Van de Graaffgenerator illustrated in Figure 15.23 has a radius of 18 cm. When the electric field at the surface of the sphere reaches 3.0 × 10 6 V/m, the air breaks down, and the generator discharges. What is the maximum potential the sphere can have before breakdown occurs

Question 71

Consider the combination of capacitors in Figure P16.38. (a) Find the equivalent single capacitance of the two capacitors in series and redraw the diagram (called diagram 1) with this equivalent capacitance. (b) In diagram 1 find the equivalent capacitance of the three capacitors in parallel and redraw the diagram as a single battery and single capacitor in a loop. (c) Compute the charge on the single equivalent capacitor. (d) Returning to diagram 1, compute the charge on each individual capacitor. Does the sum agree with the value found in part (c) (e) What is the charge on the 24.0- F capacitor and on the 8.00- F capacitor Compute the voltage drop across (f) the 24.0- F capacitor and (g) the 8.00- F capacitor.

Question 72

When a potential difference of 150 V is applied to the plates of an air-filled parallel-plate capacitor, the plates carry a surface charge density of 3.00 × 10 10 C/cm 2. What is the spacing between the plates

Question 73

A parallel-plate capacitor is charged by a battery, and the battery is then disconnected from the capacitor. Because the charges on the capacitor plates are opposite in sign, they attract each other. Hence, it takes positive work to increase the plate separation. Show that the external work done when the plate separation is increased leads to an increase in the energy stored in the capacitor.

Question 74

Rank the electric potentials at the four points shown in Figure CQ16.11 from largest to smallest.

Question 75

In Rutherford's famous scattering experiments that led to the planetary model of the atom, alpha particles (having charges of +2 e and masses of 6.64 × 10 27 kg) were fired toward a gold nucleus with charge +79 e. An alpha particle, initially very far from the gold nucleus, is fired at 2.00 × 10 7 m/s directly toward the nucleus, as in Figure P16.23. How close does the alpha particle get to the gold nucleus before turning around Assume the gold nucleus remains stationary.

Question 76

Find the charge on each of the capacitors in Figure P16.39.

Question 77

Three parallel-plate capacitors are constructed, each having the same plate area A and with C 1 having plate spacing d 1 , C 2 having plate spacing d 2 , and C 3 having plate spacing d 3. Show that the total capacitance C of the three capacitors connected in series is the same as a capacitor of plate area A and with plate spacing d = d 1 + d 2 + d 3.

Question 78

A potential difference of 90 mV exists between the inner and outer surfaces of a cell membrane. The inner surface is negative relative to the outer surface. How much work is required to eject a positive sodium ion (Na + ) from the interior of the cell

Question 79

An electron is at the origin. (a) Calculate the electric potential V A at point A , x = 0.250 cm. (b) Calculate the electric potential V B at point B , x = 0.750 cm. What is the potential difference V B V A (c) Would a negatively charged particle placed at point A necessarily go through this same potential difference upon reaching point B Explain

Question 80

Four point charges each having charge Q are located at the corners of a square having sides of length a. Find symbolic expressions for (a) the total electric potential at the center of the square due to the four charges and (b) the work required to bring a fifth charge q from infinity to the center of the square.