Chapter 19

Suppose that the electric potential outside a living cell is higher than that inside the cell by \(0.070 \mathrm{~V}\). How much work is done by the electric force when a sodium ion (charge \(=+e\) ) moves from the outside to the inside?

During a particular thunderstorm, the electric potential difference between a cloud and the ground is \(V\) cloud \(-V_{\text {ground }}=1.3 \times 10^{8} \mathrm{~V},\) with the cloud being at the higher potential. What is the change in an electron's electric potential energy when the electron moves from the ground to the cloud?

Multiple-Concept Example 4 provides useful background for this problem. Point \(\mathrm{A}\) is at a potential of \(+250 \mathrm{~V}\), and point \(B\) is at a potential of \(-150 \mathrm{~V}\). An \(\alpha\) -particle is a helium nucleus that contains two protons and two neutrons; the neutrons are electrically neutral. An \(\alpha\) -particle starts from rest at \(A\) and accelerates toward \(B\). When the \(\alpha\) -particle arrives at \(B\), what kinetic energy (in electron volts) does it have?

A particle has a charge of \(+1.5 \mu \mathrm{C}\) and moves from point \(A\) to point \(B\), a distance of \(0.20 \mathrm{~m}\). The particle experiences a constant electric force, and its motion is along the line of action of the force. The difference between the particle's electric potential energy at \(A\) and \(B\) is \(\mathrm{EPE}_{A}-\mathrm{EPE}_{B}=+9.0 \times 10^{-4} \mathrm{~J}\). (a) Find the magnitude and direction of the electric force that acts on the particle. (b) Find the magnitude and direction of the electric field that the particle experiences.

Multiple-Concept Example 4 deals with the concepts that are important in this problem. As illustrated in Figure \(19-6 b\), a negatively charged particle is released from rest at point \(B\) and accelerates until it reaches point \(A\). The mass and charge of the particle are \(4.0 \times 10^{-6} \mathrm{~kg}\) and \(-2.0 \times 10^{-5} \mathrm{C},\) respectively. Only the gravitational force and the electrostatic force act on the particle, which moves on a horizontal straight line without rotating. The electric potential at \(A\) is \(36 \mathrm{~V}\) greater than that at \(B\); in other words, \(V_{A}-V_{B}=36 \mathrm{~V}\). What is the translational speed of the particle at point \(A ?\)

A typical \(12-V\) car battery can deliver about \(7.5 \times 10^{5} \mathrm{C}\) of charge before dying. This is not very much. To get a feel for this, calculate the maximum number of kilograms of water \(\left(100^{\circ} \mathrm{C}\right)\) that could be boiled into steam \(\left(100^{\circ} \mathrm{C}\right)\) using energy from this battery.

The potential at location \(A\) is \(452 \mathrm{~V}\). A positively charged particle is released there from rest and arrives at location \(B\) with a speed \(v_{B}\). The potential at location \(C\) is \(791 \mathrm{~V},\) and when released from rest from this spot, the particle arrives at \(B\) with twice the speed it previously had, or \(2 v_{B} .\) Find the potential at \(B\).

A particle is uncharged and is thrown vertically upward from ground level with a speed of \(25.0 \mathrm{~m} / \mathrm{s}\). As a result, it attains a maximum height \(h\). The particle is then given a positive charge \(+q\) and reaches the same maximum height \(h\) when thrown vertically upward with a speed of \(30.0 \mathrm{~m} / \mathrm{s}\). The electric potential at the height \(h\) exceeds the electric potential at ground level. Finally, the particle is given a negative charge \(-q\). Ignoring air resistance, determine the speed with which the negatively charged particle must be thrown vertically upward, so that it attains exactly the maximum height \(h\). In all three situations, be sure to include the effect of gravity.

Two charges \(A\) and \(B\) are fixed in place, at different distances from a certain spot. At this spot the potentials due to the two charges are equal. Charge \(\mathrm{A}\) is \(0.18 \mathrm{~m}\) from the spot, while charge \(\mathrm{B}\) is \(0.43 \mathrm{~m}\) from it. Find the ratio \(q_{\mathrm{B}} / q_{\mathrm{A}}\) of the charges.

Point \(A\) is located \(0.25 \mathrm{~m}\) away from a charge of \(-2.1 \times 10^{-9} \mathrm{C}\). Point \(B\) is located \(0.50 \mathrm{~m}\) away from the charge. What is the electric potential difference \(V_{B}-V_{A}\) between these two points?

An electron and a proton are initially very far apart (effectively an infinite distance apart). They are then brought together to form a hydrogen atom, in which the electron orbits the proton at an average distance of \(5.29 \times 10^{-11} \mathrm{~m}\). What is EPE \(_{\text {final }}-\) EPE \(_{\text {initial }}\), which is the change in the electric potential energy?

Location \(A\) is \(3.00 \mathrm{~m}\) to the right of a point charge \(q .\) Location \(B\) lies on the same line and is \(4.00 \mathrm{~m}\) to the right of the charge. The potential difference between the two locations is \(V_{B}-V_{A}=45.0 \mathrm{~V}\). What is the magnitude and sign of the charge?

Two identical point charges are fixed to diagonally opposite corners of a square that is \(0.500 \mathrm{~m}\) on a side. Each charge is \(+3.0 \times 10^{-6} \mathrm{C} .\) How much work is done by the electric force as one of the charges moves to an empty corner?

Identical \(+1.8 \mu \mathrm{C}\) charges are fixed to adjacent corners of a square. What charge (magnitude and algebraic sign) should be fixed to one of the empty corners, so that the total electric potential at the remaining empty corner is \(0 \mathrm{~V} ?\)

A charge of \(-3.00 \mu \mathrm{C}\) is fixed in place. From a horizontal distance of \(0.0450 \mathrm{~m}, \mathrm{a}\) particle of mass \(7.20 \times 10^{-3} \mathrm{~kg}\) and charge \(-8.00 \mu \mathrm{C}\) is fired with an initial speed of \(65.0 \mathrm{~m} / \mathrm{s}\) directly toward the fixed charge. How far does the particle travel before its speed is zero?

Four identical charges \((+2.0 \mu \mathrm{C}\) each \()\) are brought from infinity and fixed to a straight line. The charges are located \(0.40 \mathrm{~m}\) apart. Determine the electric potential energy of this group.

Four identical charges \((+2.0 \mu C\) each \()\) are brought from infinity and fixed to a straight line. The charges are located \(0.40 \mathrm{~m}\) apart. Determine the electric potential energy of this group.

Identical point charges of \(+1.7 \mu \mathrm{C}\) are fixed to diagonally opposite corners of a square. A third charge is then fixed at the center of the square, such that it causes the potentials at the empty corners to change signs without changing magnitudes. Find the sign and magnitude of the third charge.

A positive charge \(+q_{1}\) is located to the left of a negative charge \(-q_{2} .\) On a line passing through the two charges, there are two places where the total potential is zero. The first place is between the charges and is \(4.00 \mathrm{~cm}\) to the left of the negative charge. The second place is \(7.00 \mathrm{~cm}\) to the right of the negative charge. (a) What is the distance between the charges? (b) Find \(q_{1} / q_{2}\), the ratio of the magnitudes of the charges.

Charges \(q_{1}\) and \(q_{2}\) are fixed in place, \(q_{2}\) being located at a distance \(d\) to the right of \(\mathrm{q}_{1} . \mathrm{A}\) third charge \(q_{3}\) is then fixed to the line joining \(q_{1}\) and \(q_{2}\) at a distance \(d\) to the right of \(q_{2}\). The third charge is chosen so the potential energy of the group is zero; that is, the potential energy has the same value as that of the three charges when they are widely separated. Determine the value for \(q_{3}\), assuming that (a) \(q_{1}=q_{2}=q\) and (b) \(q_{1}=q\) and \(q_{2}=-q .\) Express your answers in terms of \(q\)

One particle has a mass of \(3.00 \times 10^{-3} \mathrm{~kg}\) and a charge of \(+8.00 \mu \mathrm{C}\). A second particle has a mass of \(6.00 \times 10^{-3} \mathrm{~kg}\) and the same charge. The two particles are initially held in place and then released. The particles fly apart, and when the separation between them is \(0.100 \mathrm{~m},\) the speed of the \(3.00 \times 10^{-3}-\mathrm{kg}\) particle is \(125 \mathrm{~m} / \mathrm{s} .\) Find the initial separation between the particles.

A spark plug in an automobile engine consists of two metal conductors that are separated by a distance of \(0.75 \mathrm{~mm}\). When an electric spark jumps between them, the magnitude of the electric field is \(4.7 \times 10^{7} \mathrm{~V} / \mathrm{m}\). What is the magnitude of the potential difference \(\Delta V\) between the conductors?

Two equipotential surfaces surround \(\mathrm{a}+1.50 \times 10^{-8}-\mathrm{C}\) point charge. How far is the 190-V surface from the 75.0 -V surface?

The inner and outer surfaces of a cell membrane carry a negative and positive charge, respectively. Because of these charges, a potential difference of about \(0.070 \mathrm{~V}\) exists across the membrane. The thickness of the cell membrane is \(8.0 \times 10^{-9} \mathrm{~m}\). What is the magnitude of the electric field in the membrane?

When you walk across a rug on a dry day, your body can become electrified, and its electric potential can change. When the potential becomes large enough, a spark of negative charges can jump between your hand and a metal surface. A spark occurs when the electric field strength created by the charges on your body reaches the dielectric strength of the air. The dielectric strength of the air is \(3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}\) and is the electric field strength at which the air suffers electrical breakdown. Suppose a spark \(3.0 \mathrm{~mm}\) long jumps between your hand and a metal doorknob. Assuming that the electric field is uniform, find the potential difference \(\left(V_{\mathrm{knob}}-V_{\text {hand }}\right)\) between your hand and the doorknob.

Refer to Interactive Solution \(\underline{19.33}\) at to review a method by which this problem can be solved. The electric field has a constant value of \(4.0 \times 10^{3} \mathrm{~V} / \mathrm{m}\) and is directed downward. The field is the same everywhere. The potential at a point \(P\) within this region is \(155 \mathrm{~V}\). Find the potential at the following points: (a) \(6.0 \times 10^{-3} \mathrm{~m}\) directly above \(P\), (b) \(3.0 \times 10^{-3} \mathrm{~m}\) directly below \(P,\) (c) \(8.0 \times 10^{-3} \mathrm{~m}\) directly to the right of \(P\).

Equipotential surface \(A\) has a potential of \(5650 \mathrm{~V},\) while equipotential surface \(B\) has a potential of \(7850 \mathrm{~V}\). A particle has a mass of \(5.00 \times 10^{-2} \mathrm{~kg}\) and a charge of \(+4.00 \times 10^{-5} \mathrm{C} .\) The particle has a speed of \(2.00 \mathrm{~m} / \mathrm{s}\) on surface \(A .\) An outside force is applied to the particle, and it moves to surface \(B\), arriving there with a speed of \(3.00 \mathrm{~m} / \mathrm{s}\) How much work is done by the outside force in moving the particle from \(A\) to \(B ?\)

An axon is the relatively long tail-like part of a neuron, or nerve cell. The outer surface of the axon membrane (dielectric constant \(=5,\) thickness \(=1 \times 10^{-8} \mathrm{~m}\) ) is charged positively, and the inner portion is charged negatively. Thus, the membrane is a kind of capacitor. Assuming that an axon can be treated like a parallel plate capacitor with a plate area of \(5 \times 10^{-6} \mathrm{~m}^{2},\) what is its capacitance?

The membrane that surrounds a certain type of living cell has a surface area 8 of \(5.0 \times 10^{-9} \mathrm{~m}^{2}\) and a thickness of \(1.0 \times 10^{-8} \mathrm{~m}\). Assume that the membrane behaves like a parallel plate capacitor and has a dielectric constant of \(5.0 .\) (a) The potential on the outer surface of the membrane is \(+60.0 \mathrm{mV}\) greater than that on the inside surface. How much charge resides on the outer surface? (b) If the charge in part (a) is due to \(\mathrm{K}^{+}\) ions (charge \(+e\) ), how many such ions are present on the outer surface?

What voltage is required to store \(7.2 \times 10^{-5} \mathrm{C}\) of charge on the plates of a \(6.0-\mu \mathrm{F}\) capacitor?

Refer to Interactive Solution 19.39 at for one approach to this problem. The electronic flash attachment for a camera contains a capacitor for storing the energy used to produce the flash. In one such unit, the potential difference between the plates of an \(850-\mu \mathrm{F}\) capacitor is \(280 \mathrm{~V}\). (a) Determine the energy that is used to produce the flash in this unit. (b) Assuming that the flash lasts for \(3.9 \times 10^{-3} \mathrm{~s},\) find the effective power or "wattage" of the flash.

A capacitor stores \(5.3 \times 10^{-5} \mathrm{C}\) of charge when connected to a 6.0 -V battery. How much charge does the capacitor store when connected to a \(9.0-\mathrm{V}\) battery?

A parallel plate capacitor has a capacitance of \(7.0 \mu \mathrm{F}\) when filled with a dielectric. The area of each plate is \(1.5 \mathrm{~m}^{2}\) and the separation between the plates is \(1.0 \times 10^{-5} \mathrm{~m}\). What is the dielectric constant of the dielectric?

Two capacitors are identical, except that one is empty and the other is filled with a dielectric \((\kappa=4.50) .\) The empty capacitor is connected to a \(12.0-\mathrm{V}\) battery. What must be the potential difference across the plates of the capacitor filled with a dielectric such that it stores the same amount of electrical energy as the empty capacitor?

Interactive LearningWare 19.2 at reviews the concepts pertinent to this problem. What is the potential difference between the plates of a 3.3-F capacitor that stores sufficient energy to operate a 75 -W light bulb for one minute?

An empty parallel plate capacitor is connected between the terminals of a \(9.0-\mathrm{V}\) battery and charged up. The capacitor is then disconnected from the battery, and the spacing between the capacitor plates is doubled. As a result of this change, what is the new voltage between the plates of the capacitor?

Refer to Interactive Solution \(\underline{19.45}\) at and Multiple-Concept Example 10 for help in solving this problem. An empty capacitor has a capacitance of \(3.2 \mu \mathrm{F}\) and is connected to a 12-V battery. A dielectric material \((\kappa=4.5)\) is inserted between the plates of this capacitor. What is the magnitude of the surface charge on the dielectric that is adjacent to either plate of the capacitor? (Hint: The surface charge is equal to the difference in the charge on the plates with and without the dielectric.)

Refer to Interactive Solution 19.45 at and Multiple-Concept Example 10 for help in solving this problem. An empty capacitor has a capacitance of \(3.2 \mu \mathrm{F}\) and is connected to a 12-V battery. A dielectric material \((\kappa=4.5)\) is inserted between the plates of this capacitor. What is the magnitude of the surface charge on the dielectric that is adjacent to either plate of the capacitor? (Hint: The surface charge is equal to the difference in the charge on the plates with and without the dielectric.)

Two point charges, \(+3.40 \mu \mathrm{C}\) and \(-6.10 \mu \mathrm{C},\) are separated by \(1.20 \mathrm{~m} .\) What is the electric potential midway between them?

The work done by an electric force in moving a charge from point \(A\) to point \(B\) is \(2.70 \times 10^{-3} \mathrm{~J}\). The electric potential difference between the two points is \(V_{A}-V_{B}=50.0 \mathrm{~V} .\) What is the charge?

The electric potential energy stored in the capacitor of a defibrillator is \(73 \mathrm{~J}\), and the capacitance is \(120 \mu \mathrm{F}\). What is the potential difference that exists across the capacitor plates?

Refer to Multiple-Concept Example 3 to review the concepts that are needed here. A cordless electric shaver uses energy at a rate of \(4.0 \mathrm{~W}\) from a rechargeable \(1.5-\mathrm{V}\) battery. Each of the charged particles that the battery delivers to the shaver carries a charge that has a magnitude of \(1.6 \times 10^{-19} \mathrm{C}\). A fully charged battery allows the shaver to be used for its maximum operation time, during which \(3.0 \times 10^{22}\) of the charged particles pass between the terminals of the battery as the shaver operates. What is the shaver's maximum operation time?

A capacitor has a capacitance of \(2.5 \times 10^{-8} \mathrm{~F}\). In the charging process, electrons are removed from one plate and placed on the other plate. When the potential difference between the plates is \(450 \mathrm{~V}\), how many electrons have been transferred?

Two points, \(A\) and \(B\), are separated by \(0.016 \mathrm{~m}\). The potential at \(A\) is \(+95 \mathrm{~V}\), and that at \(B\) is \(+28 \mathrm{~V}\). Find the magnitude and direction of the constant electric field between the points.

Two hollow metal spheres are concentric with each other. The inner sphere has a radius of \(0.1500 \mathrm{~m}\) and a potential of \(85.0 \mathrm{~V}\). The radius of the outer sphere is \(0.1520 \mathrm{~m}\) and its potential is \(82.0 \mathrm{~V}\). If the region between the spheres is filled with Teflon, find the electric energy contained in this space.

Two particles each have a mass of \(6.0 \times 10^{-3} \mathrm{~kg}\). One has a charge of \(+5.0 \times 10^{-6} \mathrm{C}\), and the other has a charge of \(-5.0 \times 10^{-6} \mathrm{C}\). They are initially held at rest at a distance of \(0.80 \mathrm{~m}\) apart. Both are then released and accelerate toward each other. How fast is each particle moving when the separation between them is one-third its initial value?

The potential difference between the plates of a capacitor is \(175 \mathrm{~V}\). Midway between the plates, a proton and an electron are released. The electron is released from rest. The proton is projected perpendicularly toward the negative plate with an initial speed. The proton strikes the negative plate at the same instant that the electron strikes the positive plate. Ignore the attraction between the two particles, and find the initial speed of the proton.

A positive charge of \(+q_{1}\) is located \(3.00 \mathrm{~m}\) to the left of a negative charge \(-q_{2}\). The charges have different magnitudes. On the line through the charges, the net electric field is zero at a spot \(1.00 \mathrm{~m}\) to the right of the negative charge. On this line there are also two spots where the potential is zero. Locate these two spots relative to the negative charge.

An electron and a proton, starting from rest, are accelerated through an electric potential difference of the same magnitude. In the process, the electron acquires a speed \(v_{\mathrm{e}}\), while the proton acquires a speed \(v_{\mathrm{p}}\). (a) As each particle accelerates from rest, it gains kinetic energy. Does it gain or lose electric potential energy? (b) Does the electron gain more, less, or the same amount of kinetic energy as the proton does? (c) Is \(v_{\mathrm{e}}\) greater than, less than, or equal to \(v_{\mathrm{p}}\) ? Justify your answers. Find the ratio \(v_{\mathrm{e}} / v_{\mathrm{p}}\). Verify that your answer is consistent with your answers to the Concept Questions.

Concept Questions An electron and a proton, starting from rest, are accelerated through an electric potential difference of the same magnitude. In the process, the electron acquires a speed \(v_{\mathrm{e}},\) while the proton acquires a speed \(v_{\mathrm{p}}\). (a) As each particle accelerates from rest, it gains kinetic energy. Does it gain or lose electric potential energy? (b) Does the electron gain more, less, or the same amount of kinetic energy as the proton does? (c) Is \(v_{\mathrm{e}}\) greater than, less than, or equal to \(v_{\mathrm{p}}\) ? Justify your answers. Find the ratio \(v_{\mathrm{e}} / v_{\mathrm{p}}\). Verify that your answer is consistent with your answers to the Concept Questions.