Chapter 18

A charge of 28.0 \(\mathrm{nC}\) is placed in a uniform electric field that is directed vertically upward and that has a magnitude of \(4.00 \times 10^{4} \mathrm{N} / \mathrm{C}\) . What work is done by the electric force when the charge moves (a) 0.450 \(\mathrm{m}\) to the right; (b) 0.670 \(\mathrm{m}\) upward; (c) 2.60 \(\mathrm{m}\) at an angle of \(45.0^{\circ}\) downward from the horizontal?

Two very large charged parallel metal plates are 10.0 \(\mathrm{cm}\) apart and produce a uniform electric field of \(2.80 \times 10^{6} \mathrm{N} / \mathrm{C}\) between them. A proton is fired perpendicular to these plates with an initial speed of 5.20 \(\mathrm{km} / \mathrm{s}\) , starting at the middle of the negative plate and going toward the positive plate. How much work has the electric field done on this proton by the time it reaches the positive plate?

How far from a \(-7.20 \mu \mathrm{C}\) point charge must a \(+2.30 \mu \mathrm{C}\) point charge be placed in order for the electric potential energy of the pair of charges to be \(-0.400 \mathrm{J} ?\) (Take the energy to be zero when the charges are infinitely far apart.)

A point charge \(q_{1}=+2.40 \mu C\) is held stationary at the origin. A second point charge \(q_{2}=-4.30 \mu C\) moves from the point \(x=0.150 \mathrm{m}, y=0,\) to the point \(x=0.250 \mathrm{m},\) \(y=0.250 \mathrm{m} .\) How much work is done by the electric forceon \(q_{2} ?\)

Two stationary point charges of \(+3.00 \mathrm{nC}\) and \(+2.00 \mathrm{nC}\) are separated by a distance of 50.0 \(\mathrm{cm} .\) An electron is released from rest at a point midway between the charges and moves along the line connecting them. What is the electric potential energy for the electron when it is (a) at the midpoint and (b) 10.0 \(\mathrm{cm}\) from the \(+3.00 \mathrm{nC}\) charge?

Three equal \(1.20-\mu \mathrm{C}\) point charges are placed at the corners of an equilateral triangle whose sides are 0.500 \(\mathrm{m}\) long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

When two point charges are a distance \(R\) apart, their potential energy is \(-2.0 \mathrm{J} .\) How far far (in terms of \(R )\) should they be from each other so that their potential energy is \(-6.0 \mathrm{J} ?\)

Two large metal parallel plates carry opposite charges of equal magnitude. They are separated by \(45.0 \mathrm{mm},\) and the potential difference between them is 360 \(\mathrm{V}\) (a) What is the magnitude of the electric field (assumed to be uniform) in the region between the plates? (b) What is the magnitude of the force this field exerts on a particle with charge \(+2.40 \mathrm{nC}\) ?

A potential difference of 4.75 \(\mathrm{kV}\) is established between parallel plates in air. If the air becomes ionized (and hence electrically conducting) when the electric field exceeds \(3.00 \times 10^{6} \mathrm{V} / \mathrm{m},\) what is the minimum separation the plates can have without ionizing the air?

Oscilloscope. Oscilloscopes are found in most science laboratories. Inside, they contain deflecting plates consisting of more-or-less square parallel metal sheets, typically about 2.5 \(\mathrm{cm}\) on each side and 2.0 \(\mathrm{mm}\) apart. In many experiments, the maximum potential across these plates is about 25 \(\mathrm{V}\) . For this maximum potential, (a) what is the strength of the electric field between the plates, and (b) what magnitude of acceleration would this field produce on an electron midway between the plates?

Axons. Neurons are the basic units of the nervous system. They contain long tubular structures called axons that propagate electrical signals away from the axon contains a solution axon contains a solution of potassium ions \(\mathrm{K}^{+}\) and large negative organic ions. The axon membrane prevents the large ions from leaking out, but the smaller \(\mathrm{K}^{+}\) ions are able to penetrate the membrane to some degree. (See Figure 18.39 . ) This leaves an excess negative charge on the inner surface of the axon membrane and an excess of positive charge on the outer surface, resulting in a potential difference across the membrane that prevents further \(\mathrm{K}^{+}\) ions from leaking out. Measurements show that this potential difference is typically about 70 \(\mathrm{mV}\) . The thickness of the axon membrane itself varies from about 5 to \(10 \mathrm{nm},\) so we'll use an average of 7.5 \(\mathrm{nm}\) . We can model the membrane as a large sheet having equal and opposite charge densities on its faces. (a) Find the electric field inside the axon membrane, assuming (not too realistically) that it is filled with air. Which way does it point, into or out of the axon? (b) Which is at a higher potential, the inside surface or the outside surface of the axon membrane?

Electrical sensitivity of sharks. Certain sharks can detect an electric field as weak as 1.0\(\mu \mathrm{V} / \mathrm{m} .\) To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary 1.5 \(\mathrm{V}\) A battery across theseplates, how far apart would the plates have to be?

A particle with a charge of \(+4.20 \mathrm{nC}\) is in a uniform electric field \(\vec{\boldsymbol{E}}\) directed to the left. It is released from rest and moves to the left; after it has moved \(6.00 \mathrm{cm},\) its kinetic energy is found to be \(+1.50 \times 10^{-6} \mathrm{J}\) (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the endpoint? (c) What is the magnitude of \(\vec{E} ?\)

A uniform electric field has magnitude \(E\) and is directed in the negative \(x\) -direction. The potential difference between point \(a\) (at \(x=0.60 \mathrm{m} )\) and point \(b\) (at \(x=0.90 \mathrm{m} )\) is 240 \(\mathrm{V}\) . (a) Which point, \(a\) or \(b\) , is at the higher potential? (b) Calculate the value of \(E (\mathrm{c})\) A negative point charge \(q=-0.200 \mu \mathrm{C}\) is moved from \(b\) to \(a\) . Calculate the work done on the point charge by the electric field.

A point charge has a charge of \(2.50 \times 10^{-11} \mathrm{C}\). At what distance from the point charge is the electric potential (a) \(90.0 \mathrm{~V} ?\) (b) \(30.0 \mathrm{~V}\) ? Take the potential to be zero at an infinite distance from the charge.

(a) An electron is to be accelerated from \(3.00 \times 10^{6} \mathrm{m} / \mathrm{s}\right.\) to \(8.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) . Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from \(8.00 \times 10^{6} \mathrm{m} / \mathrm{s}\) to a halt?

A small particle has charge \(-5.00 \mu \mathrm{C}\) and mass \(2.00 \times\) \(10^{-4} \mathrm{kg} .\) It moves from point \(A,\) where the electric potential is \(V_{A}=+200 \mathrm{V},\) to point \(B,\) where the electric potential is \(V_{B}=+800 \mathrm{V} .\) The electric force is the only force acting on the particle. The particle has speed 5.00 \(\mathrm{m} / \mathrm{s}\) at point \(A .\) What is its speed at point \(B ?\) Is it moving faster or slower at \(B\) than at \(A\)? Explain.

A point charge \(Q=+4.60 \mu \mathrm{C}\) is held fixed at the origin. A second point charge \(q=+1.20 \mu \mathrm{C}\) with mass of \(2.80 \times\) \(10^{-4} \mathrm{kg}\) is placed on the \(x\) axis, 0.250 \(\mathrm{m}\) from the origin. (a) What is the electric potential energy \(U\) of the pair of charges? (Take \(U\) to be zero when the charges have infinite separation.) (b) The second point charge is released from rest. What is its speed when its distance from the origin is (i) \(0.500 \mathrm{m} ;\) (ii) 5.00 \(\mathrm{m}\) ; (iii) 50.0 \(\mathrm{m} ?\)

Cathode-ray tube. A cathode-ray tube (CRT) is an evacuated glass tube. Electrons are produced at one end, usually by the heating of a metal. After being focused electromagnetically into a beam, they are accelerated through a potential difference, called the accelerating potential. The electrons then strike a coated screen, where they transfer their energy to the coating through collisions, causing it to glow. CRTs are found in oscilloscopes and computer monitors, as well as in earlier versions of television screens. (a) If an electron of mass \(m\) and charge \(-e\) is accelerated from rest through an accelerating potential \(V,\) show that the speed it gains is \(v=\sqrt{2 e V / m}\) . We are assuming that \(V\) is small enough that the final speed is much less than the speed of light.) (b) If the accelerating potential is \(95 \mathrm{V},\) how fast will the electrons be moving when they hit the screen?

X-ray tube. An X-ray tube is similar to a cathode-ray tube. (See previous problem.) Electrons are accelerated to high speeds at one end of the tube. If they are moving fast enough when they hit the target at the other end, they give up their energy as X-rays (a form of nonvisible light). (a) Through what potential difference should electrons be accelerated so that their speed is 1.0\(\%\) of the speed of light when they hit the target? (b) What potential difference would be needed to give protons the same kinetic energy as the electrons? (c) What speed would this potential difference give to protons? Express your answer in \(\mathrm{m} / \mathrm{s}\) and as a percent of the speed of light.

A gold nucleus has a radius of \(7.3 \times 10^{-15} \mathrm{m}\) and a charge of \(+79 e .\) Through what voltage must an \(\alpha\) -particle, with its charge of \(+2 e,\) be accelerated so that it has just enough energy to reach a distance of \(2.0 \times 10^{-14} \mathrm{m}\) from the surface of a gold nucleus? (Assume the gold nucleus remains stationary and can be treated as a point charge.)

A parallel-plate capacitor having plates 6.0 \(\mathrm{cm}\) apart is connected across the terminals of a 12 \(\mathrm{V}\) battery. (a) Being as quantitative as you can, describe the location and shape of the equipotential surface that is at a potential of \(+6.0 \mathrm{V}\) relative to the potential of the negative plate. Avoid the edges of the plates. (b) Do the same for the equipotential surface that is at \(+2.0 \mathrm{V}\) relative to the negative plate. (c) What is the potential gradient between the plates?

(a) \(A+5.00\) pC charge is located on a sheet of paper. (a) Draw to scale the curves where the equipotential surfaces due to these charges intersect the paper. Show only the surfaces that have a potential (relative to infinity) of \(1.00 \mathrm{V}, 2.00 \mathrm{V}\) \(3.00 \mathrm{V}, 4.00 \mathrm{V},\) and 5.00 \(\mathrm{V} .\) (b) The surfaces are separated equally in potential. Are they also separated equally in distance? (c) In words, describe the shape and orientation of the surfaces you just found.

A metal sphere carrying an evenly distributed charge will have spherical equipotential surfaces surrounding it. Suppose the sphere's radius is 50.0 \(\mathrm{cm}\) and it carries a total charge of \(+1.50 \mu \mathrm{C}\) (a) Calculate the potential of the sphere's surface.(b) You want to draw equipotential surfaces at intervals of 500 \(\mathrm{V}\) outside the sphere's surface. Calculate the distance between the first and the second equipotential surfaces, and between the 20 \(\mathrm{th}\) and 21 \(\mathrm{st}\) equipotential surfaces. (c) What does the changing spacing of the surfaces tell you about the electric field?

Dipole. A dipole is located on a sheet of paper. (a) In the plane of that paper, carefully sketch the electric field lines for this dipole. (b) Use your field lines in part (a) to sketch the equipotential curves where the equipotential surfaces intersect the paper.

In a particular Millikan oil-drop apparatus, the plates are 2.25 \(\mathrm{cm}\) apart. The oil used has a density of \(0.820 \mathrm{g} / \mathrm{cm}^{3},\) and the atomizer that sprays the oil drops produces drops of diameter \(1.00 \times 10^{-3} \mathrm{mm}\) . (a) What strength of electric field is needed to hold such a drop stationary against gravity if the drop contains five excess electrons? (b) What should be the potential difference across the plates to produce this electric field? (c) If another drop of the same oil requires a plate potential of 73.8 \(\mathrm{V}\) to hold it stationary, how many excess electrons did it contain?

(a) If an electron and a proton each have a kinetic energy of 1.00 eV, how fast is each one moving? (b) What would be their speeds if each had a kinetic energy of 1.00 \(\mathrm{keV}\) ? (c) If they were each traveling at 1.00\(\%\) the speed of light, what would be their kinetic energies in keV?

(a) You find that if you place charges of \(\pm 1.25 \mu \mathrm{C}\) on two separated metal objects, the potential difference between them is 11.3 \(\mathrm{V}\) . What is their capacitance? (b) A capacitor has a capacitance of 7.28\(\mu \mathrm{F}\) . What amount of excess charge must be placed on each of its plates to make the potential difference between the plates equal to 25.0 \(\mathrm{V}\) ?

\(\bullet\) The plates of a parallel-plate capacitor are 3.28 \(\mathrm{mm}\) apart, and each has an area of 12.2 \(\mathrm{cm}^{2} .\) Each plate carries a charge of magnitude \(4.35 \times 10^{-8} \mathrm{C}\) . The plates are in vacuum. (a) What is the capacitance? (b) What is the potential difference between the plates? (c) What is the magnitude of the electric field between the plates?

The plates of a parallel-plate capacitor are 2.50 \(\mathrm{mm}\) apart, and each carries a charge of magnitude 80.0 \(\mathrm{nC}\) . The plates are in vacuum. The electric field between the plates has a magnitude of \(4.00 \times 10^{3} \mathrm{Vm}\) (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the capacitance?

A parallel-plate air capacitor has a capacitance of 500.0 \(\mathrm{pF}\) and a charge of magnitude 0.200\(\mu \mathrm{C}\) on each plate. The plates are 0.600 \(\mathrm{mm}\) apart. (a) What is the potential difference between the plates? (b) What is the area of each plate? (c) What is the electric-field magnitude between the plates? (d) What is the surface charge density on each plate?

Capacitance of an oscilloscope. Oscilloscopes have parallel metal plates inside them to deflect the electron beam. These plates are called the deflecting plates. Typically, they are squares 3.0 \(\mathrm{cm}\) on a side and separated by \(5.0 \mathrm{mm},\) with vacuum in between. What is the capacitance of these deflecting plates and hence of the oscilloscope? (This capacitance can sometimes have an effect on the circuit you are trying to study and must be taken into consideration in your calculations.)

A 10.0\(\mu \mathrm{F}\) parallel-plate capacitor with circular plates is connected to a 12.0 \(\mathrm{V}\) battery. (a) What is the charge on each plate? (b) How much charge would be on the plates if their separation were doubled while the capacitor remained connected to the battery? (c) How much charge would be on the plates if the capacitor were connected to the 12.0 \(\mathrm{V}\) battery after the radius of each plate was doubled without changing their separation?

A 10.0\(\mu\) F parallel-plate capacitor is connected to a 12.0 \(\mathrm{V}\) battery. After the capacitor is fully charged, the battery is disconnected without loss of any of the charge on the plates. (a) A volt-meter is connected across the two plates without discharging them. What does it read? (b) What would the voltmeter read if (i) the plate separation were doubled; (ii) the radius of each plate was doubled, but the separation between the plates was unchanged?

You make a capacitor by cutting the \(15.0-\mathrm{cm}\) -diameter bottoms out of two aluminum pie plates, separating them by 3.50 \(\mathrm{mm},\) and connecting them across a \(6.00-\mathrm{V}\) battery. (a) What's the capacitance of your capacitor? (b) If you disconnect the battery and separate the plates to a distance of \(3.50 \mathrm{~cm}\) without discharging them, what will be the potential difference between them?

A 5.00 pF parallel-plate air-filled capacitor with circular plates is to be used in a circuit in which it will be subjected to potentials of up to \(1.00 \times 10^{2} \mathrm{V}\) . The electric field between he plates is to be no greater than \(1.00 \times 10^{4} \mathrm{N} / \mathrm{C} .\) As a budding electrical engineer for Live- Wire Electronics, your tasks are to (a) design the capacitor by finding what its physical dimensions and separation must be and (b) find the maximum charge these plates can hold.

A parallel-plate capacitor \(C\) is charged up to a potential \(V_{0}\) with a charge of magnitude \(Q_{0}\) on each plate. It is then disconnected from the battery, and the plates are pulled apart to twice their original separation. (a) What is the new capacitance in terms of \(C ?\) (b) How much charge is now on the plates in terms of \(Q_{0} ?(\mathrm{c})\) What is the potential difference across the plates in terms of \(V_{0} ?\)

Electric eels. Electric eels and electric fish generate large potential differences that are used to stun 9 enemies and prey. These potentials are produced by cells that each can generate 0.10 V. We can plausibly model such cells as charged capacitors. (a) How should these cells be connected \((\mathrm{in}\) series or in parallel) to produce a total potential of more than 0.10 \(\mathrm{V} ?\) (b) Using the connection in part (a), how many cells must be connected together to produce the 500 \(\mathrm{V}\) surge of the electric eel?

You are working on an electronics project requiring a variety of capacitors, but have only a large supply of 100 nF capacitors available. Show how you can connect these capacitors to produce each of the following equivalent capacitances: (a) \(50 \mathrm{nF},\) (b) \(450 \mathrm{nF},(\mathrm{c}) 25 \mathrm{nF},\) (d) 75 \(\mathrm{nF.}\)

A 4.00\(\mu \mathrm{F}\) and a 6.00\(\mu \mathrm{F}\) capacitor are connected in series, and this combination is connected across a 48.0 \(\mathrm{V}\) potential difference. Calculate (a) the charge on each capacitor and (b) the potential difference across each of them.

How much charge does a 12 \(\mathrm{V}\) battery have to supply to fully charge a 2.5\(\mu \mathrm{F}\) capacitor and a 5.0\(\mu \mathrm{F}\) capacitor when they're (a) in parallel, (b) in series? (c) How much energy does the battery have to supply in each case?

A 5.80\(\mu\) F parallel-plate air capacitor has a plate separation of 5.00 mm and is charged to a potential difference of 400 \(\mathrm{V}\) . Calculate the energy density in the region between the plates, in units of \(\mathrm{J} / \mathrm{m}^{3} .\)

(a) How much charge does a battery have to supply to a 5.0\(\mu \mathrm{F}\) capacitor to create a potential difference of 1.5 \(\mathrm{V}\) across its plates? How much energy is stored in the capacitor in this case? (b) How much charge would the battery have to supply to store 1.0 \(\mathrm{J}\) of energy in the capacitor? What would be the potential across the capacitor in that case?

In the text, it was shown that the energy stored in a capacitor \(C\) charged to a potential \(V\) is \(U=\frac{1}{2} Q V\) . Show that this energy can also be expressed as (a) \(U=Q^{2} / 2 C\) and (b) \(U=\frac{1}{2} C V^{2}\)

A parallel-plate vacuum capacitor has 8.38 J of energy stored in it. The separation between the plates is 2.30 \(\mathrm{mm}\) . If the separation is decreased to \(1.15 \mathrm{mm},\) what is the energy stored (a) if the capacitor is disconnected from the potential source so the charge on the plates remains constant, and (b) if the capacitor remains connected to the potential source so the potential difference between the plates remains constant?

A \(\mathrm{A} 20.0 \mu \mathrm{F}\) capacitor is charged to a potential difference of 800 \(\mathrm{V} .\) The terminals of the charged capacitor are then connected to those of an uncharged 10.0\(\mu \mathrm{F}\) capacitor. Compute (a) the onginal charge of the system, (b) the final potential difference across each capacitor, (c) the final energy of the system, and (d) the decrease in energy when the capacitors are connected.

A parallel-plate air capacitor has a capacitance of 920 pF. The charge on each plate is 2.55\(\mu \mathrm{C}\) . (a) What is the potential difference between the plates? (b) If the charge is kept constant, what will be the potential difference between the plates if the separation is doubled? (c) How much work is required to double the separation?

A parallel-plate capacitor has capacitance \(C_{0}=5.00 \mathrm{pF}\) when there is air between the plates. The separation between the plates is 1.50 \(\mathrm{mm}\) (a) What is the maximum magnitude of charge \(Q\) that can be placed on each plate if the electric field in the region between the plates is not to exceed \(3.00 \times 10^{4} \mathrm{V} / \mathrm{m}\) (b) A dielectric with \(K=2.70\) is inserted between the plates of the capacitor, completely filling the volume between the plates. Now what is the maximum magnitude of charge on each plate if the electric field between the plates is not to exceed \(3.00 \times 10^{4} \mathrm{V} / \mathrm{m} ?\)

A parallel-plate capacitor is to be constructed by using, as a dielectric, rubber with a dielectric constant of 3.20 and a dielectric strength of 20.0 \(\mathrm{MV} / \mathrm{m}\) . The capacitor is to have a capacitance of 1.50 \(\mathrm{nF}\) and must be able to withstand a maximum potential difference of 4.00 \(\mathrm{kV} .\) What is the minimum area the plates of this capacitor can have?

A \(\mathrm{A} 12.5 \mu \mathrm{F}\) capacitor is connected to a power supply that keeps a constant potential difference of 24.0 \(\mathrm{V}\) across the plates. A piece of material having a dielectric constant of 3.75 is placed between the plates, completely filling the space between them. (a) How much energy is stored in the capacitor before and after the dielectric is inserted? (b) By how much did the energy change during the insertion? Did it increase or decrease?