A parallel-plate capacitor has fixed charges +Q and ?Q.

Problem 2P How much work does the electric field do in moving a proton from a point with a potential of +125 V to a point where it is ?55 V? Express your answer both in joules and electron volts.

Problem 1Q If two points are at the same potential, does this mean that no net work is done in moving a test charge from one point to the other? Does this imply that no force must be exerted? Explain.

Problem 2Q If a negative charge is initially at rest in an electric field, will it move toward a region of higher potential or lower potential? What about a positive charge? How does the potential energy of the charge change in each instance? Explain.

Problem 3Q State clearly the difference (a) between electric potential and electric field, (b) between electric potential and electric potential energy.

Problem 5Q Is there a point along the line joining two equal positive charges where the electric field is zero? Where the electric potential is zero? Explain.

Problem 6Q Can a particle ever move from a region of low electric potential to one of high potential and yet have its electric potential energy decrease? Explain.

If V = 0 at a point in space, must \(\vec{E}=0\) there? If \(\vec{E}=0\) at some point, must V = 0 at that point? Explain. Give examples for each. Equation Transcription: Text Transcription: \vec E=0 \vec E=0

Draw in a few equipotential lines in Fig. 16-31b.

(II) An alpha particle (which is a helium nucleus. \(Q =+2e, m = 6.64 \times 10^{-27} \ \mathrm {Kg}\)) is emitted in a radioactive decay with KE = 5.53 MeV. What is its speed?

When a battery is connected to a capacitor, why do the two plates acquire charges of the same magnitude? Will this be true if the two conductors are different sizes or shapes?

We have seen that the capacitance C depends on the size, shape and position of the two conductors, as well as on the dielectric constant K. What then did we mean when we said that C is constant in Eq. 17-7?

(II) \(A + 35 \ \mu {C}\) point charge is placed 32 cm from an identical \( + 35 \ \mu {C}\) charge. How much work would be required to move a \(+ 0.50 \ \mu {C}\) test charge from a point midway between them to a point 12 cm closer to either of the charges?

Problem 17P (II) Draw a conductor in the oblong shape of a football. This conductor carries a net negative charge. -Q. Draw in a dozen or so electric field lines and equipotential lines.

(II) (a) What is the electric potential a distance of \(2.5 \times 10^{-15} \ \mathrm {m}\) away from a proton? (b) What is the electric potential energy of a system that consists of two protons \(2.5 \times 10^{-15} \ \mathrm {m}\) apart—as might occur inside a typical nucleus?

(II) Three point charges are arranged at the corners of a square of side L as shown in Fig. 17-25. What is the potential at the fourth corner (point A), taking V = 0 at a great distance.

Problem 23P (II) How much work must be done to bring three electrons from a great distance apart to 1.0 X 10-10 m from one another (at the corners of an equilateral triangle)?

Problem 25P (Ill) How much voltage must be used to accelerate a proton (radius 1.2 X 10-15 m) so that it has sufficient energy to just ‘‘touch" a silicon nucleus? A silicon nucleus has a charge of +14e. and its radius is about 3.6 X 10-15 m. Assume the potential is that for point charges.

(Ill) Two equal but opposite charges are separated by a distance d, as shown in Fig. 17-27. Determine a formula for \(V_{B A}=V_{B}-V_{A}\) for points B and A on the line between the charges. Equation Transcription: Text Transcription: V_B A=V_B-V_A

Problem 27P (Ill) In the Bohr model of the hydrogen atom, an electron orbits a proton (the nucleus) in a circular orbit of radius 0.53 X 10-10 m. (a) What is the electric potential at the electron's orbit due to the proton? (b) What is the kinetic energy of the electron? (c) What is the total energy of the electron in its orbit? (d) What is the ionization energy— that is, the energy required to remove the electron from the atom and take it to r = «*, at rest? Express the results of parts (b), (c), and (d) in joules and eV.

Problem 28P (I) An electron and a proton are 0.53 X 10-10 m apart. What is their dipole moment if they are at rest?

(III) The dipole moment, considered as a vector, points from the negative to the positive charge. The water molecule, Fig. 17-28, has a dipole moment \(\vec{p}\) which can be considered as the vector sum of the two dipole moments, \(\vec{p}_{1} \text { and } \vec{p}_{2}\), as shown. The distance between H and the O is about \(0.96 \times 10^{-10} m\). The lines joining the center of O atom with each H atom make an angle of 104o, as shown, and the net dipole moment has been measured to be \(p=6.1 \times 10^{-10} \mathrm{C} . \mathrm{m}\). Determine the charge q on each H atom. Equation Transcription: Text Transcription: \vec p \vec p1 and \vec p2 0.96 x 10-10m p=6.1 x 10-10C.m

Problem 47P (I) A cardiac defibrillator is used to shock a heart that is beating erratically. A capacitor in this device is charged to 5.0 kV and stores 1200 J of energy. What is its capacitance?

(III) Electrons are accelerated by \(6.0 \mathrm{kV}\) in a CRT. The screen is \(30 \mathrm{~cm}\) wide and is \(34 \mathrm{~cm}\) from the \(2.6-cm\)-long deflection plates. Over what range must the horizontally deflecting electric field vary to sweep the beam fully across the screen?

Problem 57GP (II) There is an electric field near the Earth’s surface whose magnitude is about 150 V/m. How much energy is stored per cubic meter in this field?

An electron is accelerated horizontally from rest in a television picture tube by a potential difference of 5500 V. It then passes between two horizontal plates 6.5 cm long and 1.3 cm apart that have a potential difference of 250 V (Fig. 17-31). At what angle \(\theta\) will the electron be traveling after it passes between the plates? Equation Transcription: Text Transcription: \theta

Problem 67GP (II) To get an idea how big a farad is, suppose you want to make a 1-F air-filled parallel-plate capacitor for a circuit you are building. To make it a reasonable size, suppose you limit the plate area to 1.0 cm2 What would the gap have to be between the plates? Is this practically achievable?

Problem 1P How much work does the electric field do in moving a ?7.7 ?C charge from ground to a point whose potential is +55 V higher?

In a photocell, ultraviolet (UV) light provides enough energy to some electrons in barium metal to eject them from the surface at high speed. See Fig. 17-33. To measure the maximum energy of the electrons, another plate above the barium surface is kept at a negative enough potential that the emitted electrons are slowed down and stopped, and return to the barium surface. If the plate voltage is -3.02 V (compared to the barium) when the fastest electrons are stopped, what was the speed of these electrons when they were emitted?

Problem 3P How much kinetic energy will an electron gain (in joules and eV) if it accelerates through a potential difference of 23,000 V in a TV picture tube?

Problem 4P An electron acquires 7.45 × 10?16 J of kinetic energy when it is accelerated by an electric field from plate A to plate B. What is the potential difference between the plates, and which plate is at the higher potential?

Problem 4Q An electron is accelerated by a potential difference of, say, 0.10 V. How much greater would its final speed be if it is accelerated with four times as much voltage? Explain.

Problem 5P How strong is the electric field between two parallel plates 5.8 mm apart if the potential difference between them is 220 V?

Problem 6P An electric field of 640 V/m is desired between two parallel plates 11.0mm apart. How large a voltage should be applied?

(I) The electric field between two parallel plates connected to a \(45-\mathrm{V}\) battery is \(1500 \mathrm{~V} / \mathrm{m}\). How far apart are the plates?

Problem 7Q Compare the kinetic energy gained by a proton (q = +e) to the energy gained by an alpha particle (q = +2e)accelerated by the same voltage V.

Problem 8P What potential difference is needed to give a helium nucleus (Q = 2e) 65.0 keV of kinetic energy?

Problem 9P Two parallel plates, connected to a 200-V power supply, are separated by an air gap. How small can the gap be if the air is not to become conducting by exceeding its breakdown value of E = 3 × 106 V/m ?

Problem 9Q Can two equipotential lines cross? Explain.

Problem 10P The work done by an external force to move a ?8.50?C charge from point a to point b is 15.0 × 10?4 J. If the charge was started from rest and had 4.82 × 10?4 J of kinetic energy when it reached point b, what must be the potential difference between a and b?

(II) What is the speed of an electron with kinetic energy (a) \(750-\mathrm{eV}\), and \((b) 3.2-\mathrm{keV}\) ?

Problem 11Q What can you say about the electric field in a region of space that has the same potential throughout?

A satellite orbits the Earth along a gravitational equipotential line. What shape must the orbit be?

Problem 12P What is the speed of a proton whose kinetic energy is 3.2 keV?

When dealing with practical devices, we often take the ground (the Earth) to be 0 V. If, instead, we said the ground was ?10 V, how would this affect (a) the potential V, and (b) the electric field E, at other points?

Problem 14P What is the electric potential 15.0 cm from a 4.00 ?C point charge?

Problem 15P A point charge Q creates an electric potential of + 125 V at a distance of 15 cm. What is Q?

(II) An electron starts from rest 32.5 cm from a fixed point charge with \(Q=-0.125 \mu \mathrm{C}\). How fast will the electron be moving when it is very far away?

(II) Two identical \(+9.5 \mu \mathrm{C}\) point charges are initially \(3.5 \mathrm{~cm}\) from each other. If they are released at the same instant from rest, how fast will each be moving when they are very far away from each other? Assume they have identical masses of \(1.0 \mathrm{mg}\).

Problem 22P Two point charges, 3.0 ?C and ?2.0 ?C, are placed 5.0 cm apart on the x axis. At what points along the x axis is (a) the electric field zero and (b) the potential zero? Let V = 0 at r = ?.

Consider point a which is 72cm north of \(a-3.8 \mu C\) point charge, and point b which is 88 cm west of the charge (Fig. 17–26). Determine (a) \(V_{b a}=V_{b}-V_{a}\) and (b) \(\vec{E}_{b}-\vec{E}_{a}\) (magnitude and direction). Equation Transcription: Text Transcription: a-3.8 \mu C V_b a=V_b -V_a \vec E_b -\vec E_a

Problem 29P Calculate the electric potential due to a dipole whose dipole moment is 4.8 × 10?30 C·m at a point 1.1 × 10?9 m away if this point is (a) along the axis of the dipole nearer the positive charge; (b) 45° above the axis but nearer the positive charge; (c) 45° above the axis but nearer the negative charge.

(I) The two plates of a capacitor hold \(+2500 \ \mu C\) and \(-2500 \ \mu C\) of charge, respectively, when the potential difference is 850 V. What is the capacitance?

(I) A \(9500-\mathrm{pF}\) capacitor holds plus and minus charges of \(16.5 \times 10^{-8} \mathrm{C}\). What is the voltage across the capacitor?

Problem 33P The potential difference between two short sections of parallel wire in air is 120 V.They carry equal and opposite charge of magnitude 95 pC. What is the capacitance of the two wires?

Problem 34P How much charge flows from each terminal of a 12.0-V battery when it is connected to a 7.00-?F capacitor?

Problem 35P A 0.20-F capacitor is desired. What area must the plates have if they are to be separated by a 2.2-mm air gap?

Problem 36P The charge on a capacitor increases by 18 ?C when the voltage across it increases from 97 V to 121 V. What is the capacitance of the capacitor?

(II) An electric field of \(8.50 \times 10^5 \mathrm{~V} / \mathrm{m}\) is desired between two parallel plates, each of area \(35.0 \mathrm{~cm}^2\) and separated by \(2.45 \mathrm{~mm}\) of air. What charge must be on each plate?

Problem 38P If a capacitor has opposite 5.2?C charges on the plates, and an electric field of 2.0kV/mm is desired between the plates, what must each plate’s area be?

Problem 39P How strong is the electric field between the plates of a 0.80-?F air-gap capacitor if they are 2.0 mm apart and each has a charge of 72 ?C?

(III) A \(7.7-\mu F\) capacitor is charged by a 125-V battery (Fig. 17–29a) and then is disconnected from the battery. When this capacitor \(\left(C_{1}\right)\) is then connected (Fig. 17–29b) to a second (initially uncharged) capacitor, \(C_{2}\), the final voltage on each capacitor is 15 V. What is the value of \(C_{2}\)? [Hint: charge is conserved.] Equation Transcription: Text Transcription: 7.7-\mu F (C_1) C_2 C_2

(III) A \(2.50-\mu \mathrm {F}\) capacitor is charged to 857 V and a \(6.80-\mu \mathrm {F}\) capacitor is charged to 652 V. These capacitors are then disconnected from their batteries. Next the positive plates are connected to each other and the negative plates are connected to each other. What will be the potential difference across each and the charge on each? [Hint: charge is conserved.]

Problem 42P What is the capacitance of two square parallel plates 5.5 cm on a side that are separated by 1.8 mm of paraffin?

Problem 43P What is the capacitance of a pair of circular plates with a radius of 5.0 cm separated by 3.2 mm of mica?

(II) A 3500-pF air-gap capacitor is connected to a 22-V battery. If a piece of mica is placed between the plates, how much charge will flow from the battery?

Problem 45P The electric field between the plates of a paper-separated (K = 3.75) capacitor is 8.24 × 104V/m. The plates are 1.95 mm apart, and the charge on each plate is 0.775 ?C. Determine the capacitance of this capacitor and the area of each plate.

Problem 46P 650 V is applied to a 2200-pF capacitor. How much energy is stored?

(II) How much energy is stored by the electric field between two square plates, \(8.0 \mathrm{~cm}\) on a side, separated by a 1.5-mm air gap? The charges on the plates are equal and opposite and of magnitude \(420 \mu \mathrm{C}\).

Problem 49P A homemade capacitor is assembled by placing two 9-in. pie pans 5 cm apart and connecting them to the opposite terminals of a 9-V battery. Estimate (a) the capacitance, (b) the charge on each plate, (c) the electric field halfway between the plates, and (d) the work done by the battery to charge the plates. (e) Which of the above values change if a dielectric is inserted?

(II) A parallel-plate capacitor has fixed charges +Q and ?Q. The separation of the plates is then doubled. (a) By what factor does the energy stored in the electric field change? (b) How much work must be done in doubling the plate separation from d to 2d? The area of each plate is A.

Problem 51P How does the energy stored in a capacitor change if (a) the potential difference is doubled, and (b) the charge on each plate is doubled, as the capacitor remains connected to a battery?

(III) A \(2.70-\mu F\) capacitor is charged by a 12.0-V battery. It is disconnected from the battery and then connected to an uncharged \(4.00-\mu F\) capacitor (Fig. 17–29). Determine the total stored energy (a) before the two capacitors are connected, and (b) after they are connected. (c) What is the change in energy? Equation Transcription: Text Transcription: 2.70-\mu F 4.00-\mu F

(III) In a given CRT, electrons are accelerated horizontally by 7.0 kV. They then pass through a uniform electric field E for a distance of 2.8 cm, which deflects them upward so they reach the screen top 22 cm away, 11 cm above the center. Estimate the value of E.

Problem 55GP An electron starting from rest acquires 6.3 keV of KE in moving from point A to point B. (a) How much KE would a proton acquire, starting from rest at B and moving to point A? (b) Determine the ratio of their speeds at the end of their respective trajectories.

Problem 56GP A lightning flash transfers 4.0 C of charge and 4.2 MJ of energy to the Earth. (a) Across what potential difference did it travel? (b) How much water could this boil and vaporize, starting from room temperature?

Problem 58GP In a television picture tube, electrons are accelerated by thousands of volts through a vacuum. If a television set were laid on its back, would electrons be able to move upward against the force of gravity? What potential difference, acting over a distance of 3.0 cm, would be needed to balance the downward force of gravity so that an electron would remain stationary? Assume that the electric field is uniform.

A huge 4.0-F capacitor has enough stored energy to heat 2.5 kg of water from \(21^\circ {C}\) to \(95^\circ {C}\). What is the potential difference across the plates?

Problem 60GP An uncharged capacitor is connected to a 24.0-V battery until it is fully charged, after which it is disconnected from the battery. A slab of paraffin is then inserted between the plates. What will now be the voltage between the plates?

Problem 61GP Dry air will break down if the electric field exceeds 3.0 × 106 V/m. What amount of charge can be placed on a parallel-plate capacitor if the area of each plate is 56 cm2?

Three charges are at the corners of an equilateral triangle (side L) as shown in Fig. 17–30. Determine the potential at the midpoint of each of the sides.

A \(3.4 \mu \mathrm{C}\) and \(\mathrm{a}-2.6 \mu \mathrm{C}\) charge are placed \(1.6 \mathrm{~cm}\) apart. At what points along the line joining them is (a) the electric field zero, and (b) the electric potential zero?

Problem 64GP A 2600-pF air-gap capacitor is connected to a 9.0-V battery. If a piece of Pyrex glass is placed between the plates, how much charge will then flow from the battery?

A capacitor of capacitance \(C_{1}\) carries a charge \(Q_{0}). It is then connected directly to a second, uncharged, capacitor of capacitance \(C_{2}\), as shown in Fig. 17–32. What charge will each carry now? What will be the potential difference across each? Equation Transcription: Text Transcription: C1 Qo C2

Problem 68GP Near the surface of the Earth there is an electric field of about 150 V/m which points downward. Two identical balls with mass m = 0.540 kg are dropped from a height of 2.00 m, but one of the balls is positively charged with q1 = 650 ?C, and the second is negatively charged with q2 = ?650 ?C. Use conservation of energy to determine the difference in the speed of the two balls when they hit the ground. (Neglect air resistance.)

The power supply for a pulsed nitrogen laser has a \(0.050-\mu \mathrm {F}\) capacitor with a maximum voltage rating of 30 kV. (a) Estimate how much energy could be stored in this capacitor. (b) If 12% of this stored electrical energy is converted to light energy in a pulse that is 8.0 microseconds long, what is the power of the laser pulse?

Problem 70GP In lightning storms, the potential difference between the Earth and the bottom of the thunderclouds can be as high as 35,000,000 V. The bottoms of the thunderclouds are typically 1500 m above the Earth, and can have an area of 110 km2. Modeling the Earth-cloud system as a huge capacitor, calculate (a) the capacitance of the Earth-cloud system, (b) the charge stored in the “capacitor,” and (c) the energy stored in the “capacitor.”

A \(+33 \mu C\) point charge is placed 36 cm from an identical \(+33 \mu C\) charge. A \(-1.5 \mu C\) charge is moved from point a to point b in Fig. 17–34. What is the change in potential energy? Equation Transcription: Text Transcription: +33 \mu C +33 \mu C -1.5 \mu C

Problem 73GP A capacitor is made from two 1.1-cm-diameter coins separated by a 0.15-mm-thick piece of paper (K = 3.7). A 12-V battery is connected to the capacitor. How much charge is on each coin?

Problem 74GP A +4.5 ?C charge is 23 cm to the right of a ?8.2?C charge. At the midpoint between the two charges, (a) what are the potential and (b) the electric field?

Problem 75GP A parallel-plate capacitor with plate area 2.0 cm2 and air-gap separation 0.50 mm is connected to a 12-V battery, and fully charged. The battery is then disconnected. (a) What is the charge on the capacitor? (b) The plates are now pulled to a separation of 0.75 mm. What is the charge on the capacitor now? (c) What is the potential difference across the plates now? (d) How much work was required to pull the plates to their new separation?

Problem 76GP A 2.5-?F capacitor is fully charged by a 6.0-V battery. The battery is then disconnected. The capacitor is not ideal and the charge slowly leaks out from the plates. The next day, the capacitor has lost half its stored energy. Calculate the amount of charge lost.

Problem 77GP Two point charges are fixed 4.0 cm apart from each other. Their charges are Q1 = Q2 = 5.0 ?C, and their masses are m1 = 1.5 mg and m2 = 2.5 mg. (a) If Q1 is released from rest, what will be its speed after a very long time? (b) If both charges are released from rest at the same time, what will be the speed of Q1 after a very long time?

Two charges are placed as shown in Fig. 17–35 with \(q_{1}=1.5 \mu C \text { and } q_{2}=-3.3 \mu C\). Find the potential difference between points A and B. Equation Transcription: Text Transcription: q_1=1.5 \mu C and q_2=-3.3 \mu C