Exercises 781

23.31 . A very long wire carries a uniform linear charge densityUsing a voltmeter to measure potential difference, you find that

when one probe of the meter is placed from the wire andthe other probe is farther from the wire, the meter reads

(a) What is (b) If you now place one probe at from the wire and the other probe farther away, will thevoltmeter read If not, will it read more or less than Why? (c) If you place both probes from the wire but

from each other, what will the voltmeter read?23.32 .. A very long insulating cylinder of charge of radius

carries a uniform linear density of If you putone probe of a voltmeter at the surface, how far from the surfacemust the other probe be placed so that the voltmeter reads 23.33 .. A very long insulating cylindrical shell of radius

carries charge of linear density spread uni-formly over its outer surface. What would a voltmeter read if itwere connected between (a) the surface of the cylinder and a point

above the surface, and (b) the surface and a point from the central axis of the cylinder?23.34 . A ring of diameter is fixed in place and carries acharge of uniformly spread over its circumference. (a) How much work does it take to move a tiny chargedball of mass from very far away to the center of the ring?(b) Is it necessary to take a path along the axis of the ring? Why?(c) If the ball is slightly displaced from the center of the ring, whatwill it do and what is the maximum speed it will reach?23.35 .. A very small sphere with positive charge is released from rest at a point 1.50 cm from a very long line ofuniform linear charge density . What is thekinetic energy of the sphere when it is 4.50 cm from the line ofcharge if the only force on it is the force exerted by the line ofcharge?23.36 . Charge is distributed uniformly over thevolume of an insulating sphere that has radius . Asmall sphere with charge and mass is projected toward the center of the large sphere from an initiallarge distance. The large sphere is held at a fixed position and thesmall sphere can be treated as a point charge. What minimumspeed must the small sphere have in order to come within 8.00 cmof the surface of the large sphere?23.37 . BIO Axons. Neu-rons are the basic units of thenervous system. They containlong tubular structures calledaxons that propagate electricalsignals away from the ends of the neurons. The axon con-tains a solution of potassium

ions and large negativeorganic ions. The axon membrane prevents the large ions fromleaking out, but the smaller ions are able to penetrate the mem-brane to some degree (Fig. E23.37). This leaves an excess negativecharge on the inner surface of the axon membrane and an excesspositive charge on the outer surface, resulting in a potential differ-ence across the membrane that prevents further ions from leak-ing out. Measurements show that this potential difference istypically about 70 mV. The thickness of the axon membrane itselfvaries from about 5 to 10 nm, so we’ll use an average of 7.5 nm.We can model the membrane as a large sheet having equal andopposite charge densities on its faces. (a) Find the electric fieldinside the axon membrane, assuming (not too realistically) that it isfilled with air. Which way does it point: into or out of the axon?

K+

K+

1K+2

6.00 * 10-5 kgq = +3.00 mCR = 12.0 cm

Q = 5.00 mC

l = +3.00 mC/m

q = +8.00 mC

1.50 g+3.00-mC

+5.00 mC8.00 cm

1.00 cm4.00 cm

8.50 mC>m6.00 cm

175 V?

15.0 nC>m.2.50 cm

17.0 cm3.50 cm

575 V?575 V?1.00 cm

3.50 cml?575 V.1.00 cm

2.50 cml.

(b) Which is at a higher potential: the inside surface or the outsidesurface of the axon membrane?23.38 . CP Two large, parallel conducting plates carrying oppo-site charges of equal magnitude are separated by (a) If thesurface charge density for each plate has magnitude what is the magnitude of in the region between the plates? (b) What is the potential difference between the two plates? (c) Ifthe separation between the plates is doubled while the surfacecharge density is kept constant at the value in part (a), what happensto the magnitude of the electric field and to the potential difference?23.39 . Two large, parallel, metal plates carry opposite charges ofequal magnitude. They are separated by and the poten-tial difference between them is (a) What is the magnitudeof the electric field (assumed to be uniform) in the region betweenthe plates? (b) What is the magnitude of the force this field exertson a particle with charge (c) Use the results of part (b) to compute the work done by the field on the particle as it movesfrom the higher-potential plate to the lower. (d) Compare the resultof part (c) to the change of potential energy of the same charge,computed from the electric potential.23.40 . BIO Electrical Sensitivity of Sharks. Certain sharkscan detect an electric field as weak as To grasp howweak this field is, if you wanted to produce it between two parallelmetal plates by connecting an ordinary 1.5-V AA battery acrossthese plates, how far apart would the plates have to be?23.41 .. (a) Show that for a spherical shell of radius that hascharge distributed uniformly over its surface, is the same as fora solid conductor with radius and charge (b) You rub aninflated balloon on the carpet and it acquires a potential that is

lower than its potential before it became charged. If thecharge is uniformly distributed over the surface of the balloon andif the radius of the balloon is what is the net charge on theballoon? (c) In light of its potential difference relative toyou, do you think this balloon is dangerous? Explain.23.42 .. (a) How much excess charge must be placed on a coppersphere in diameter so that the potential of its center, rela-tive to infinity, is (b) What is the potential of the sphere’ssurface relative to infinity?23.43 . The electric field at the surface of a charged, solid, coppersphere with radius is directed toward the cen-ter of the sphere. What is the potential at the center of the sphere, ifwe take the potential to be zero infinitely far from the sphere?

Section 23.4 Equipotential Surfaces andSection 23.5 Potential Gradient23.44 . A very large plastic sheet carries a uniform charge densityof on one face. (a) As you move away from the sheetalong a line perpendicular to it, does the potential increase ordecrease? How do you know, without doing any calculations? Doesyour answer depend on where you choose the reference point forpotential? (b) Find the spacing between equipotential surfaces thatdiffer from each other by What type of surfaces are these?23.45 . CALC In a certain region of space, the electric potential is

where and are positiveconstants. (a) Calculate the and -components of the electricfield. (b) At which points is the electric field equal to zero?23.46 . CALC In a certain region of space the electric potential isgiven by where V/m3 and8.00 V/m3. Calculate the magnitude and direction of the electricfield at the point in the region that has coordinates m,

m, and z = 0.y = 0.400x = 2.00

B =A = 5.00V = +Ax2y - Bxy2,

zy-,x-,CA, B,V(x, y, z) = Axy - Bx2 + Cy,

1.00 V.

-6.00 nC>m2

3800 N>C,0.200 m

1.50 kV?25.0 cm

1200-V15 cm,

1560 V

q.RVq

R,V

1.0 mV/m.

+2.40 nC?

360 V.45.0 mm,

ES

47.0 nC>m2,2.20 cm.

Axonmembrane

7.5 nm

K1 K1

K1

2

2

2

2

2

K1 K1

Figure E23.37