A nerve cell in its resting state has a membrane potential

How are electric potential and field related?

What are the properties of conductors?

What are sources of electric potential?

What is a capacitor?

How are capacitors used?

FIGURE 26.2 is a graph of Ex, the x-component of the electric field, versus position along the x-axis. Find and graph V1x2. Assume V = 0 V at x = 0 m.

Which potential graph describes the electric field at the left? E u y x V y (a) V y (b) V y (c) V y (d) V y (e)

In Chapter 23, the electric field inside a capacitor was found to be E u = 1 Q P0A , from positive to negative2 Find the electric potential inside the capacitor. Let V = 0 V at the negative plate.

Which set of equipotential surfaces matches this electric field? E u (a) 0 V 50 V 0 V 50 V (b) 0 V 50 V (c) (d) 50 V 0 V (e) 50 V 0 V (f) 50 V 0 V

In Chapter 25, we found the on-axis potential of a ring of radius R and charge Q to be Vring = 1 4pP0 Q 2z 2 + R2 Find the on-axis electric field of a ring of charge. SOLVE Symmetry requires the electric field along the axis to point straight outward from the ring with only a z-component Ez. The electric field at position z is Ez = - dV dz = - d dz 1 1 4pP0 Q 2z 2 + R2 2 = 1 4pP0 zQ 1z 2 + R2 23/2

Three charged metal spheres of different radii are connected by a thin metal wire. The potential and electric field at the surface of each sphere are V and E. Which of the following is true? a. V1 = V2 = V3 and E1 = E2 = E3 b. V1 = V2 = V3 and E1 7 E2 7 E3 c. V1 7 V2 7 V3 and E1 = E2 = E3 d. V1 7 V2 7 V3 and E1 7 E2 7 E3 e. V3 7 V2 7 V1 and E3 = E2 = E1 f. V3 7 V2 7 V1 and E3 7 E2 7 E1

Figure 26.7 is a graph of the electric potential in a region of space where E u is parallel to the x-axis. Draw a graph of Ex versus x.

What total potential difference is created by these three batteries? 3.0 V 1.0 V 3.0 V

In Figure 26.11 a 1 cm * 1 cm grid is superimposed on a contour map of the potential. Estimate the strength and direction of the electric field at points 1, 2, and 3. Show your results graphically by drawing the electric field vectors on the contour map.

Rank in order, from largest to smallest, the equivalent capacitance (Ceq)a to (Ceq)d of circuits a to d. (a) 5 mF (b) (c) (d) 3 mF 3 mF 3 mF 4 mF 4 mF 3 mF 3 m

The spacing between the plates of a 1.0 mF capacitor is 0.050 mm. a. What is the surface area of the plates? b. How much charge is on the plates if this capacitor is charged to 1.5 V?

Find the charge on and the potential difference across each of the three capacitors in Figure 26.25.

How much energy is stored in a 220 mF camera-flash capacitor that has been charged to 330 V? What is the average power dissipation if this capacitor is discharged in 1.0 ms?

A 5.0 nF parallel-plate capacitor is charged to 160 V. It is then disconnected from the battery and immersed in distilled water. What are (a) the capacitance and voltage of the water-filled capacitor and (b) the energy stored in the capacitor before and after its immersion?

A defibrillator unit contains a 150 mF capacitor that is charged to 2100 V. The capacitor plates are separated by a 0.050-mm-thick insulator with dielectric constant 120. a. What is the area of the capacitor plates? b. What are the stored energy and the energy density in the electric field when the capacitor is charged?

The radiation detector known as a Geiger counter consists of a 25-mm-diameter cylindrical metal tube, sealed at the ends, with a 1.0-mm-diameter wire along its axis. The wire and cylinder are separated by a low-pressure gas whose dielectric strength is 1.0 * 106 V/m. a. What is the capacitance per unit length? b. What is the maximum potential difference between the wire and the tube?

Figure Q26.1 shows the x-component of E u as a function of x. Draw a graph of V versus x in this same region of space. Let V = 0 V at x = 0 m and include an appropriate vertical scale.

Figure Q26.2 shows the electric potential as a function of x. Draw a graph of Ex versus x in this same region of space.

a. Suppose that E u = 0 u V/m throughout some region of space. Can you conclude that V = 0 V in this region? Explain. b. Suppose that V = 0 V throughout some region of space. Can you conclude that E u = 0 u V/m in this region? Explain

Estimate the electric fields E u 1 and E u 2 at points 1 and 2 in Figure Q26.4. Dont forget that E u is a vector.

Estimate the electric fields E u 1 and E u 2 at points 1 and 2 in Figure Q26.5. Dont forget that E u is a vector.

An electron is released from rest at x = 2 m in the potential shown in Figure Q26.6. Does it move? If so, to the left or to the right? Explain.

Figure Q26.7 shows an electric field diagram. Dashed lines 1 and 2 are two surfaces in space, not physical objects. a. Is the electric potential at point a higher than, lower than, or equal to the electric potential at point b? Explain. b. Rank in order, from largest to smallest, the magnitudes of the potential differences Vab, Vcd, and Vef. c. Is surface 1 an equipotential surface? What about surface 2? Explain why or why not.

Figure Q26.8 shows a negatively charged electroscope. The gold leaf stands away from the rigid metal post. Is the electric potential of the leaf higher than, lower than, or equal to the potential of the post? Explain.

The two metal spheres in Figure Q26.9 are connected by a metal wire with a switch in the middle. Initially the switch is open. Sphere 1, with the larger radius, is given a positive charge. Sphere 2, with the smaller radius, is neutral. Then the switch is closed. Afterward, sphere 1 has charge Q1, is at potential V1, and the electric field strength at its surface is E1. The values for sphere 2 are Q2, V2, and E2. a. Is V1 larger than, smaller than, or equal to V2? Explain. b. Is Q1 larger than, smaller than, or equal to Q2? Explain. c. Is E1 larger than, smaller than, or equal to E2? Explain.

Figure Q26.10 shows a 3 V battery with metal wires attached to each end. What are the potential differences V12 = V2 - V1, V23 = V3 - V2, V34 = V4 - V3, and V41 = V1 - V4?

The parallel-plate capacitor in Figure Q26.11 is connected to a battery having potential difference Vbat . Without breaking any of the connections, insulating handles are used to increase the plate separation to 2d.a. Does the potential difference VC change as the separation increases? If so, by what factor? If not, why not? b. Does the capacitance change? If so, by what factor? If not, why not? c. Does the capacitor charge Q change? If so, by what factor? If not, why not?

Rank in order, from largest to smallest, the potential differences 1VC21 to 1VC24 3 V of the four capacitors in Figure Q26.12. Explain.

What is the potential difference between xi = 10 cm and xf = 30 cm in the uniform electric field Ex = 1000 V/m?

What is the potential difference between yi = -5 cm and yf = 5 cm in the uniform electric field E u = 120,000ni - 50,000nj2 V/m?

Figure EX26.3 is a graph of Ex. What is the potential difference between xi = 1.0 m and xf = 3.0 m?

Figure EX26.4 is a graph of Ex. The potential at the origin is -50 V. What is the potential at x = 3.0 m?

a. Which point in Figure EX26.5, A or B, has a larger electric potential? b. What is the potential difference between A and B?

Two flat, parallel electrodes 2.5 cm apart are kept at potentials of 20 V and 35 V. Estimate the electric field strength between them.

What are the magnitude and direction of the electric field at the dot in Figure EX26.7?

What are the magnitude and direction of the electric field at the dot in Figure EX26.8?

Figure EX26.9 shows a graph of V versus x in a region of space. The potential is independent of y and z. What is Ex at (a) x = -2 cm, (b) x = 0 cm, and (c) x = 2 cm?

Determine the magnitude and direction of the electric field at points 1 and 2 in Figure EX26.10.

Figure EX26.11 is a graph of V versus x. Draw the corresponding graph of Ex versus x.

Figure EX26.12 is a graph of V versus x. Draw the corresponding graph of Ex versus x.

The electric potential in a region of uniform electric field is -1000 V at x = -1.0 m and +1000 V at x = +1.0 m. What is Ex?

The electric potential along the x-axis is V = 100x2 V, where x is in meters. What is Ex at (a) x = 0 m and (b) x = 1 m?

The electric potential along the x-axis is V = 100e-2x V, where x is in meters. What is Ex at (a) x = 1.0 m and (b) x = 2.0 m?

What is the potential difference V34 in Figure EX26.16?

How much work does the charge escalator do to move 1.0 mC of charge from the negative terminal to the positive terminal of a 1.5 V battery?

How much charge does a 9.0 V battery transfer from the negative to the positive terminal while doing 27 J of work?

How much work does the electric motor of a Van de Graaff generator do to lift a positive ion 1q = e2 if the potential of the spherical electrode is 1.0 MV?

Light from the sun allows a solar cell to move electrons from the positive to the negative terminal, doing 2.4 * 10-19 J of work per electron. What is the emf of this solar cell?

Two 3.0@cm@diameter aluminum electrodes are spaced 0.50 mm apart. The electrodes are connected to a 100 V battery. a. What is the capacitance? b. What is the magnitude of the charge on each electrode?

What is the capacitance of the two metal spheres shown in Figure EX26.22?

You need to construct a 100 pF capacitor for a science project. You plan to cut two L * L metal squares and insert small spacers between their corners. The thinnest spacers you have are 0.20 mm thick. What is the proper value of L?

A switch that connects a battery to a 10 mF capacitor is closed. Several seconds later you find that the capacitor plates are charged to {30 mC. What is the emf of the battery?

A 6 mF capacitor, a 10 mF capacitor, and a 16 mF capacitor are connected in series. What is their equivalent capacitance?

A 6 mF capacitor, a 10 mF capacitor, and a 16 mF capacitor are connected in parallel. What is their equivalent capacitance?

What is the equivalent capacitance of the three capacitors in Figure EX26.27?

What is the equivalent capacitance of the three capacitors in Figure EX26.28?

You need a capacitance of 50 mF, but you dont happen to have a 50 mF capacitor. You do have a 30 mF capacitor. What additional capacitor do you need to produce a total capacitance of 50 mF? Should you join the two capacitors in parallel or in series?

You need a capacitance of 50 mF, but you dont happen to have a 50 mF capacitor. You do have a 75 mF capacitor. What additional capacitor do you need to produce a total capacitance of 50 mF? Should you join the two capacitors in parallel or in series?

To what potential should you charge a 1.0 mF capacitor to store 1.0 J of energy?

50 pJ of energy is stored in a 2.0 cm * 2.0 cm * 2.0 cm region of uniform electric field. What is the electric field strength?

A 2.0-cm-diameter parallel-plate capacitor with a spacing of 0.50 mm is charged to 200 V. What are (a) the total energy stored in the electric field and (b) the energy density?

The 90 mF capacitor in a defibrillator unit supplies an average of 6500 W of power to the chest of the patient during a discharge lasting 5.0 ms. To what voltage is the capacitor charged?

Two 4.0 cm * 4.0 cm metal plates are separated by a 0.20-mm-thick piece of Teflon. a. What is the capacitance? b. What is the maximum potential difference between the plates?

Two 5.0 mm * 5.0 mm electrodes are held 0.10 mm apart and are attached to a 9.0 V battery. Without disconnecting the battery, a 0.10-mm-thick sheet of Mylar is inserted between the electrodes. What are the capacitors potential difference, electric field, and charge (a) before and (b) after the Mylar is inserted? Hint: Section 26.7 considered a capacitor with isolated plates. What changes, and what doesnt, when the plates stay connected to the battery?

A typical cell has a layer of negative charge on the inner surface of the cell wall and a layer of positive charge on the outside surface, thus making the cell wall a capacitor. What is the capacitance of a 50-mm-diameter cell with a 7.0-nm-thick cell wall whose dielectric constant is 9.0? Because the cells diameter is much larger than the wall thickness, it is reasonable to ignore the curvature of the cell and think of it as a parallel-plate capacitor.

The electric field in a region of space is Ex = 5000x V/m, where x is in meters. a. Graph Ex versus x over the region -1 m x 1 m. b. Find an expression for the potential V at position x. As a reference, let V = 0 V at the origin. c. Graph V versus x over the region -1 m x 1 m.

The electric field in a region of space is Ex = -1000x V/m, where x is in meters. a. Graph Ex versus x over the region -1 m x 1 m. b. What is the potential difference between xi = -20 cm and xf = 30 cm?

An infinitely long cylinder of radius R has linear charge density l. The potential on the surface of the cylinder is V0, and the electric field outside the cylinder is Er = l/2pP0r. Find the potential relative to the surface at a point that is distance r from the axis, assuming r 7 R

FIGURE P26.41 is an edge view of three charged metal electrodes. Let the left electrode be the zero point of the electric potential. What are V and E u at (a) x = 0.5 cm, (b) x = 1.5 cm, and (c) x = 2.5 cm?

Use the on-axis potential of a charged disk from Chapter 25 to find the on-axis electric field of a charged disk.

a. Use the methods of Chapter 25 to find the potential at distance x on the axis of the charged rod shown in FIGURE P26.43. b. Use the result of part a to find the electric field at distance x on the axis of a rod

It is postulated that the radial electric field of a group of charges falls off as Er = C/rn , where C is a constant, r is the distance from the center of the group, and n is an unknown exponent. To test this hypothesis, you make a field probe consisting of two needle tips spaced 1.00 mm apart. You orient the needles so that a line between the tips points to the center of the charges, then use a voltmeter to read the potential difference between the tips. After you take measurements at several distances from the center of the group, your data are as follows:Use an appropriate graph of the data to determine the constants C and n.

Engineers discover that the electric potential between two electrodes can be modeled as V1x2 = V0 ln11 + x/d2, where V0 is a constant, x is the distance from the first electrode in the direction of the second, and d is the distance between the electrodes. What is the electric field strength midway between the electrodes?

The electric potential in a region of space is V = 1150x2 - 200y2 2 V, where x and y are in meters. What are the strength and direction of the electric field at 1x, y2 = 12.0 m, 2.0 m2? Give the direction as an angle cw or ccw (specify which) from the positive x-axis

The electric potential in a region of space is V = 200/2x2 + y2 , where x and y are in meters. What are the strength and direction of the electric field at 1x, y2 = 12.0 m, 1.0 m2? Give the direction as an angle cw or ccw (specify which) from the positive x-axis

Consider a large, thin, electrically neutral conducting plate in the xy-plane at z = 0 and a point charge q on the z-axis at distance zcharge = d. What is the electric field on and above the plate? Although the plate is neutral, electric forces from the point charge polarize the conducting plate and cause it to have some complex distribution of surface charge. The electric field and potential are then a superposition of fields and potentials due to the point charge and the plates surface charge. Thats complicated! However, it is shown in more advanced classes that the field and potential outside the plate 1z 02 are exactly the same as the field and potential of the original charge q plus a mirror image charge q located at zimage = -d. a. Find an expression for the electric potential in the yz-plane for z 0 (i.e., in the space above the plate). b. We know that electric fields are perpendicular to conductors in electrostatic equilibrium, so the field at the surface of the plate has only a z-component. Find an expression for the field Ez on the surface of the plate 1z = 02 as a function of distance y away from the z-axis. c. A +10 nC point charge is 2.0 cm above a large conducting plate. What is the electric field strength at the surface of the plate (i) directly beneath the point charge and (ii) 2.0 cm away from being directly beneath the point charge?

Metal sphere 1 has a positive charge of 6.0 nC. Metal sphere 2, which is twice the diameter of sphere 1, is initially uncharged. The spheres are then connected together by a long, thin metal wire. What are the final charges on each sphere?

The metal spheres in FIGURE P26.50 are charged to {300 V. Draw this figure on your paper, then draw a plausible contour map of the potential, showing and labeling the -300 V, -200 V, -100 V, . . . , 300 V equipotential surfaces.

The potential at the center of a 4.0-cm-diameter copper sphere is 500 V, relative to V = 0 V at infinity. How much excess charge is on the sphere?

The electric potential is 40 V at point A near a uniformly charged sphere. At point B, 2.0 mm farther away from the sphere, the potential has decreased by 0.16 mV. How far is point A from the center of the sphere?

Two 2.0 cm * 2.0 cm metal electrodes are spaced 1.0 mm apart and connected by wires to the terminals of a 9.0 V battery. a. What are the charge on each electrode and the potential difference between them? The wires are disconnected, and insulated handles are used to pull the plates apart to a new spacing of 2.0 mm. b. What are the charge on each electrode and the potential difference between them?

Two 2.0 cm * 2.0 cm metal electrodes are spaced 1.0 mm apart and connected by wires to the terminals of a 9.0 V battery. a. What are the charge on each electrode and the potential difference between them? While the plates are still connected to the battery, insulated handles are used to pull them apart to a new spacing of 2.0 mm. b. What are the charge on each electrode and the potential difference between them?

Find expressions for the equivalent capacitance of (a) N identical capacitors C in parallel and (b) N identical capacitors C in series.

What are the charge on and the potential difference across each capacitor in FIGURE P26.56?

What are the charge on and the potential difference across each capacitor in FIGURE P26.57?

What are the charge on and the potential difference across each capacitor in FIGURE P26.58?

You have three 12 mF capacitors. Draw diagrams showing how you could arrange all three so that their equivalent capacitance is (a) 4.0 mF, (b) 8.0 mF, (c) 18 mF, and (d) 36 mF.

Six identical capacitors with capacitance C are connected as shown in FIGURE P26.60. a. What is the equivalent capacitance of these six capacitors? b. What is the potential difference between points a and b?

Initially, the switch in FIGURE P26.61 is in position A and capacitors C2 and C3 are uncharged. Then the switch is flipped to position B. Afterward, what are the charge on and the potential difference across each capacitor

A battery with an emf of 60 V is connected to the two capacitors shown in FIGURE P26.62. Afterward, the charge on capacitor 2 is 450 mC. What is the capacitance of capacitor 2?

Capacitors C1 = 10 mF and C2 = 20 mF are each charged to 10 V, then disconnected from the battery without changing the charge on the capacitor plates. The two capacitors are then connected in parallel, with the positive plate of C1 connected to the negative plate of C2 and vice versa. Afterward, what are the charge on and the potential difference across each capacitor?

An isolated 5.0 mF parallel-plate capacitor has 4.0 mC of charge. An external force changes the distance between the electrodes until the capacitance is 2.0 mF. How much work is done by the external force?

An ideal parallel-plate capacitor has a uniform electric field between the plates, zero field outside. By superposition, half the field strength is due to one plate and half due to the other. a. The plates of a parallel-plate capacitor are oppositely charged and attract each other. Find an expression in terms of C, VC, and the plate separation d for the force one plate exerts on the other. b. What is the attractive force on each plate of a 100 pF capacitor with a 1.0 mm plate spacing when charged to 1000 V?

High-frequency signals are often transmitted along a coaxial cable, such as the one shown in FIGURE P26.66. For example, the cable TV hookup coming into your home is a coaxial cable. The signal is carried on a wire of radius R1 while the outer conductor of radius R2 is grounded (i.e., at V = 0 V). An insulating material fills the space between them, and an insulating plastic coating goes around the outside. a. Find an expression for the capacitance per meter of a coaxial cable. Assume that the insulating material between the cylinders is air. b. Evaluate the capacitance per meter of a cable having R1 = 0.50 mm and R2 = 3.0 mm

The flash unit in a camera uses a 3.0 V battery to charge a capacitor. The capacitor is then discharged through a flashlamp. The discharge takes 10 ms, and the average power dissipated in the flashlamp is 10 W. What is the capacitance of the capacitor?

The label rubbed off one of the capacitors you are using to build a circuit. To find out its capacitance, you place it in series with a 10 mF capacitor and connect them to a 9.0 V battery. Using your voltmeter, you measure 6.0 V across the unknown capacitor. What is the unknown capacitors capacitance?

A capacitor being charged has a current carrying charge to and away from the plates. In the next chapter we will define current to be dQ/dt, the rate of charge flow. What is the current to a 10 mF capacitor whose voltage is increasing at the rate of 2.0 V/s?

The current that charges a capacitor transfers energy that is stored in the capacitors electric field. Consider a 2.0 mF capacitor, initially uncharged, that is storing energy at a constant 200 W rate. What is the capacitor voltage 2.0 ms after charging begins?

A typical cell has a membrane potential of -70 mV, meaning that the potential inside the cell is 70 mV less than the potential outside due to a layer of negative charge on the inner surface of the cell wall and a layer of positive charge on the outer surface. This effectively makes the cell wall a charged capacitor. Because a cells diameter is much larger than the wall thickness, it is reasonable to ignore the curvature of the cell and think of it as a parallel-plate capacitor. How much energy is stored in the electric field of a 50-mm-diameter cell with a 7.0-nm-thick cell wall whose dielectric constant is 9.0?

A nerve cell in its resting state has a membrane potential of -70 mV, meaning that the potential inside the cell is 70 mV less than the potential outside due to a layer of negative charge on the inner surface of the cell wall and a layer of positive charge on the outer surface. This effectively makes the cell wall a charged capacitor. When the nerve cell fires, sodium ions, Na+, flood through the cell wall to briefly switch the membrane potential to +40 mV. Model the central body of a nerve cellthe somaas a 50-mm-diameter sphere with a 7.0-nm-thick cell wall whose dielectric constant is 9.0. Because a cells diameter is much larger than the wall thickness, it is reasonable to ignore the curvature of the cell and think of it as a parallel-plate capacitor. How many sodium ions enter the cell as it fires?

Derive Equation 26.33 for the induced surface charge density on the dielectric in a capacitor

A vacuum-insulated parallel-plate capacitor with plate separation d has capacitance C0. What is the capacitance if an insulator with dielectric constant k and thickness d/2 is slipped between the electrodes without changing the plate separation?

In Problems 75 through 77 you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). b. Finish the solution of the problem.2az V/m = - dV dz , where a is a constant with units of V/m2 V1z = 02 = 10 V

In Problems 75 through 77 you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). b. Finish the solution of the problem.400 nC = 1100 V2C C = 18.85 * 10-12 C2 /Nm2 210.10 m * 0.10 m2

In Problems 75 through 77 you are given the equation(s) used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation(s). b. Finish the solution of the problem.3 mF + 1 6 mF 2 -1 + C = 4 mF

Two 5.0-cm-diameter metal disks separated by a 0.50-mmthick piece of Pyrex glass are charged to a potential difference of 1000 V. What are (a) the surface charge density on the disks and (b) the surface charge density on the glass?

An electric dipole at the origin consists of two charges {q spaced distance s apart along the y-axis. a. Find an expression for the potential V1x, y2 at an arbitrary point in the xy-plane. Your answer will be in terms of q, s, x, and y. b. Use the binomial approximation to simplify your result of part a when s V x and s V y. c. Assuming s V x and y, find expressions for Ex and Ey, the components of E u for a dipole. d. What is the on-axis field E u ? Does your result agree with Equation 23.10? e. What is the field E u on the bisecting axis? Does your result agree with Equation 23.11?

Charge is uniformly distributed with charge density r inside a very long cylinder of radius R. Find the potential difference between the surface and the axis of the cylinder

Consider a uniformly charged sphere of radius R and total charge Q. The electric field Eout outside the sphere 1r R2 is simply that of a point charge Q. In Chapter 24, we used Gausss law to find that the electric field Ein inside the sphere 1r R2 is radially outward with field strength Ein = 1 4pP0 Q R3 r a. The electric potential Vout outside the sphere is that of a point charge Q. Find an expression for the electric potential Vin at position r inside the sphere. As a reference, let Vin = Vout at the surface of the sphere. b. What is the ratio Vcenter /Vsurface? c. Graph V versus r for 0 r 3R

a. Find an expression for the capacitance of a spherical capacitor, consisting of concentric spherical shells of radii R1 (inner shell) and R2 (outer shell). b. A spherical capacitor with a 1.0 mm gap between the shells has a capacitance of 100 pF. What are the diameters of the two spheres?

Each capacitor in Figure CP26.83 has capacitance C. What is the equivalent capacitance between points a and b?