Solved: Two identical dipoles, each of charge q and separation a, are a distance x

Conceptual Example 20.1 shows that the gravitational force between an electron and a proton is about 10-40 times weaker than the electric force between them. Since matter consists largely of electrons and protons, why is the gravitational force important at all?

A free neutron is unstable and soon decays to other particles, one of them a proton. Must there be others? If so, what electric properties must it or they have?

Where in Fig. 20.5 could you put a third charge so it would experience no net force? Would it be in stable or unstable equilibrium?

Why should the test charge used to measure an electric field be small?

Equation 20.3 gives the electric field of a point charge. Does the direction of (a) nr or (b) E S depend on whether the charge is positive or negative?

Is the electric force on a charged particle always in the direction of the field? Explain.

Why does a dipole, which has no net charge, produce an electric field?

The ring in Example 20.6 carries total charge Q, and the point P is the same distance r = 2x 2 + a2 from all parts of the ring. So why isnt the electric field of the ring just kQ/r2 ?

A spherical balloon is initially uncharged. If you spread positive charge uniformly over the balloons surface, would it expand or contract? What would happen if you spread negative charge instead?

Under what circumstances is the path of a charged particle a parabola? A circle?

Why should there be a force between two dipoles, which each have zero net charge?

Dipoles A and B are both located in the field of a point charge Q, as shown in Fig. 20.27. Does either experience a net torque? A net force? If each dipole is released from rest, describe qualitatively its subsequent motion.

Suppose the electron and proton charges differed by one part in one billion. Estimate the net charge on your body, assuming it contains equal numbers of electrons and protons.

A typical lightning flash delivers about 25 C of negative charge from cloud to ground. How many electrons are involved?

Protons and neutrons are made from combinations of the two most common quarks, the u quark (charge +2 3 e) and the d quark (charge -1 3 e). How could three of these quarks combine to make (a) a proton and (b) a neutron?

Earth carries a net charge of about -5*105 C. How many more electrons are there than protons on Earth?

As they fly, honeybees may acquire electric charges of about 180 pC. Electric forces between charged honeybees and spider webs can make the bees more vulnerable to capture by spiders. How many electrons would a honeybee have to lose to acquire a charge of +180 pC?

The electron and proton in a hydrogen atom are 52.9 pm apart. Find the magnitude of the electric force between them.

An electron at Earths surface experiences a gravitational force meg. How far away can a proton be and still produce the same force on the electron? (Your answer should show why gravity is unimportant on the molecular scale!)

You break a piece of Styrofoam packing material, and it releases lots of little spheres whose electric charge makes them stick annoyingly to you. If two of the spheres carry equal charges and repel with a force of 21 mN when theyre 15 mm apart, whats the magnitude of the charge on each?

A charge q is at the point x = 1 m, y = 0 m. Write expressions for the unit vectors you would use in Coulombs law if you were finding the force that q exerts on other charges located at (a) x = 1 m, y = 1 m; (b) the origin; and (c) x = 2 m, y = 3 m. Youre not given the sign of q. Why doesnt this matter?

A proton is at the origin and an electron is at the point x = 0.41 nm, y = 0.36 nm. Find the electric force on the proton.

An electron experiences an electric force of 0.61 nN. Whats the field strength at its location?

Find the magnitude of the electric force on a 2.0@C charge in a 100-N/C electric field.

A 68-nC charge experiences a 150-mN force in a certain electric field. Find (a) the field strength and (b) the force that a 35@C charge would experience in the same field.

The electric field inside a cell membrane is 8.0 MN/C. Whats the force on a singly charged ion in this field?

A -1.0@C charge experiences a 10ni@N electric force in a certain electric field. What force would a proton experience in the same field?

The electron in a hydrogen atom is 52.9 pm from the proton. At this distance, whats the strength of the electric field due to the proton?

In Fig. 20.28, point P is midway between the two charges. Find the electric field in the plane of the page (a) 5.0 cm to the left of P, (b) 5.0 cm directly above P, and (c) at P.

The water molecules dipole moment is 6.2*10-30 C# m. What would be the separation distance if the molecule consisted of charges {e? (The effective charge is actually less because H and O atoms share the electrons.)

The electric field 22 cm from a long wire carrying a uniform line charge density is 1.9 kN/C. Whats the field strength 38 cm from the wire?

Find the line charge density on a long wire if the electric field 45 cm from the wire has magnitude 260 kN/C and points toward the wire.

Find the magnitude of the electric field due to a charged ring of radius a and total charge Q on the ring axis at distance a from the rings center.

In his famous 1909 experiment that demonstrated quantization of electric charge, R. A. Millikan suspended small oil drops in an electric field. With field strength 20 MN/C, what mass drop can be suspended when the drop carries 10 elementary charges?

How strong an electric field is needed to accelerate electrons in an X-ray tube from rest to one-tenth the speed of light in a distance of 5.0 cm?

A proton moving to the right at 3.8*105 m/s enters a region where a 56-kN/C electric field points to the left. (a) How far will the proton get before it momentarily stops? (b) Describe its subsequent motion.

An electrostatic analyzer like that of Example 20.8 has b = 7.5 cm. What value of E0 will enable the device to select protons moving at 84 m/s?

A 2-g ping-pong ball rubbed against a wool jacket acquires a net positive charge of 1 C. Estimate the fraction of the balls electrons that have been removed.

Two charges, one whose magnitude is twice as large as the others, are located 12.5 cm apart and experience an attractive force of 143 N. (a) Whats the magnitude of the larger charge? (b) Can you determine the sign of the larger charge?

A proton is on the x-axis at x = 1.6 nm. An electron is on the y-axis at y = 0.85 nm. Find the net force the two exert on a helium nucleus (charge +2e) at the origin.

A 9.5@C charge is at x = 15 cm, y = 5.0 cm and a -3.2@C charge is at x = 4.4 cm, y = 11 cm. Find the force on the negative charge.

A charge 3q is at the origin, and a charge -2q is on the positive x-axis at x = a. Where would you place a third charge so it would experience no net electric force?

A negative charge -q lies midway between two positive charges +Q. What must Q be such that the electric force on all three charges is zero?

In Fig. 20.29, take q1 = 68 C, q2 = -34 C, and q3 = 15 C. Find the electric force on q3

In Fig. 20.29, take q1 = 25 C and q2 = 20 C. If the force on q1 points in the -x-direction, find (a) q3 and (b) the magnitude of the force on q1.

DNA fragments introduced into an electrophoresis apparatus (see Application, page 360) generally carry negative charges equivalent to two extra electrons per base pair of nucleotides in the fragment. The table below shows the forces on several DNA fragments in an electrophoresis apparatus, as a function of the number of base pairs. Plot these data, establish a best-fit line, and use the resulting slope to determine the strength of the electric field in the electrophoresis apparatus. Base pairs 400 800 1200 2000 3000 5000 Force (pN) 0.235 0.472 0.724 1.15 1.65 2.87

A 65@C point charge is at the origin. Find the electric field at the points (a) x = 50 cm, y = 0 cm; (b) x = 50 cm, y = 50 cm; and (c) x = 25 cm, y = -75 cm

A 1.0@C charge and a 2.0@C charge are 10 cm apart. Find a point where the electric field is zero.

A proton is at the origin and an ion is at x = 5.0 nm. If the electric field is zero at x = -5.0 nm, whats the ions charge?

(a) Find an expression for the electric field on the y-axis due to the two charges q in Fig. 20.7. (b) At what point is the field on the y-axis a maximum?

A dipole lies on the y-axis and consists of an electron at y = 0.60 nm and a proton at y = -0.60 nm. Find the electric field (a) midway between the two charges; (b) at the point x = 2.0 nm, y = 0 nm; and (c) at the point x = -20 nm, y = 0 nm.

Show that the field on the x-axis for the dipole of Example 20.5 is given by Equation 20.6b, for  x  W a.

Youre 1.44 m from a charge distribution that is well under 1 cm in size. You measure an electric field strength of 296 N/C due to this distribution. You then move to a distance of 2.16 m from the distribution, where you measure a field strength of 87.7 N/C. Whats the net charge of the distribution? Hint: Dont try to calculate the charge. Determine instead how the field decreases with distance, and from that infer the charge

Three identical charges q form an equilateral triangle of side a, with two charges on the x-axis and one on the positive y-axis. (a) Find an expression for the electric field at points on the y-axis above the uppermost charge. (b) Show that your result reduces to the field of a point charge 3q for y W a.

Two identical small metal spheres initially carry charges q1 and q2. When theyre 1.0 m apart, they experience a 2.5-N attractive force. Then theyre brought together so charge moves from one to the other until they have the same net charge. Theyre again placed 1.0 m apart, and now they repel with a 2.5-N force. What were the original charges q1 and q2?

Two 38.0@C charges are attached to opposite ends of a spring with spring constant k = 145 N/m and equilibrium length 52.6 cm. By how much does the spring stretch? Hint: Youll need to use a computer or advanced calculator to solve the cubic equation that arises in this problem

A thin rod lies on the x-axis between x = 0 and x = L and carries total charge Q distributed uniformly over its length. Show that the electric field strength for x 7 L is given by E = kQ/3x1x - L24

An electron is moving in a circular path around a long, uniformly charged wire carrying 2.5 nC/m. Whats the electrons speed?

Find the line charge density on a long wire if a 6.8@g particle carrying 2.1 nC describes a circular orbit about the wire with speed 280 m/s.

A dipole with dipole moment 1.5 nC# m is oriented at 30 to a 4.0-MN/C electric field. Find (a) the magnitude of the torque on the dipole and (b) the work required to rotate the dipole until its antiparallel to the field.

You have a job examining patent applications. Youre presented with the device in Fig. 20.30, which its inventor claims will separate isotopes of a particular element. Atoms are first stripped completely of their electrons, then accelerated from rest through an electric field chosen to give the desired isotope exactly the right speed to pass through the electrostatic analyzer (see Example 20.8). Will the device work?

A 5.0@m strand of DNA carries charge +e per nm of length. Treating it as a charged line, whats the electric field strength 25 nm from the DNA, not near either end?

A molecule has its dipole moment aligned with a 1.2-kN/C electric field. If it takes 3.1*10-27 J to reverse the molecules orientation, whats its dipole moment?

Two identical dipoles, each of charge q and separation a, are a distance x apart, as shown in Fig. 20.31. (a) By considering forces between pairs of charges in the different dipoles, calculate the force between the dipoles and show that, in the limit a V x, it has magnitude 6kp2 /x 4 , where p = qa is the dipole moment. (b) Is the force attractive or repulsive?

A dipole with charges {q and separation 2a is located a distance x from a point charge +Q, oriented as shown in Fig. 20.32. Find expressions for the magnitude of (a) the net torque and (b) the net force on the dipole, both in the limit x W a. (c) Whats the direction of the net force?

An electron is at the origin, and an ion with charge +5e is at x = 10 nm. Find a point where the electric field is zero.

Youre taking physical chemistry, and your professor is discussing molecular dipole moments. Water, he says, has a dipole moment of 1.85 debyes, while carbon monoxides dipole moment is only 0.12 debye. Your physics professor wants these moments expressed in SI. She tells you that the atomic separation in these two covalent compounds is about the same, and asks what that indicates about the way shared charge is distributed. What do you tell her?

The electric field on the axis of a uniformly charged ring has magnitude 380 kN/C at a point 5.0 cm from the ring center. The magnitude 15 cm from the center is 160 kN/C; in both cases the field points away from the ring. Find (a) the rings radius and (b) its charge.

An electric quadrupole consists of two oppositely directed dipoles in close proximity. (a) Calculate the field of the quadrupole shown in Fig. 20.33 for points to the right of x = a and (b) show that for x W a the quadrupole field falls off as 1/x 4 .

You measure the electric field on a dipoles axis, at a distance from the dipole thats large compared with the separation of the two charges in the dipole. If you maintain that distance but move to a point located at 45 from the dipole axis, by what factor will the magnitude of the electric field change? Will the field increase or decrease?

A straight wire 10 m long carries 25 C distributed uniformly over its length. (a) Whats the line charge density on the wire? Find the electric field strength (b) 15 cm from the wire axis, not near either end, and (c) 350 m from the wire. Make suitable approximations in both cases.

Figure 20.34 shows a thin rod of length L carrying charge Q distributed uniformly over its length. (a) Whats the line charge density on the rod? (b) Modify the calculation of Example 20.7 to find an expression for the electric field at a point P a distance y along the perpendicular bisector. (c) Show that your result reduces the the field of a point charge when  y  W L.

Figure 20.35 shows a thin, uniformly charged disk of radius R. Imagine the disk divided into rings of varying radii r, as suggested in the figure. (a) Show that the area of such a ring is very nearly 2pr dr. (b) If the disk carries surface charge density s, use the result of part (a) to write an expression for the charge dq on an infinitesimal ring. (c) Use the result of (b) along with the result of Example 20.6 to write the infinitesimal electric field dE of this ring at a point on the disk axis, taken to be the positive x-axis. (d) Integrate over all such rings to show that the net electric field on the axis has magnitude

Use the result of Problem 73 to show that the field of an infinite, uniformly charged flat sheet is 2pks, where s is the surface charge density. (This result is independent of distance from the sheet.)

Use the binomial theorem to show that, for x W R, the result of Problem 73 reduces to the field of a point charge whose total charge is the charge density times the disk area.

A semicircular loop of radius a carries positive charge Q distributed uniformly. Find the electric field at the loops center (point P in Fig. 20.36). (Hint: Divide the loop into charge elements dq as shown, write dq in terms of the angle du, then integrate over u.)

In Example 20.2, find the position on the y-axis where Q will experience the greatest force.

A thin rod carries charge Q distributed uniformly over its length L and is situated on the x-axis between x = {L/2. (a) Find the electric field at an arbitrary point 1x, y2. (Youll have to do separate integrals for the x- and y-components.) (b) Show that your result reduces to that of Problem 72 when x = 0 and to that of Problem 57 when y = 0 and x 7 L/2

A thin rod extends along the x-axis from x = 0 to x = L and carries line charge density l = l01x/L22 , where l0 is a constant. Find the electric field at x = -L.

A rod of length 2L lies on the x-axis, centered at the origin, and carries line charge density l = l01x/L2, where l0 is a constant. (a) Find an expression for the electric field strength at points on the x-axis for x 7 L. (b) Show that for x W L your result has the 1/x 3 dependence of a dipole field, and determine the dipole moment of the rod.

Youre working on the design of an ink-jet printer. Ink drops of mass m, speed v, and charge q will enter a region of uniform electric field E between two charged plates (Fig. 20.37). The drops enter midway between the plates, and the electric field deflects them toward the correct place on the page. Find an expression for the maximum electric field for which drops can still get through without hitting either plate.

BIO The human heart consists largely of elongated muscle cells, some 100 m long and 15 m in diameter. In its resting state, a cell contains two concentric layers of charge, which confine the electric field to the cell membrane (Fig. 20.38a). When the heart contracts, a wave of depolarization sweeps through, depleting charge and giving each cell a dipole moment (Fig. 20.38b). As a result, the entire organ acts like an electric dipole, producing an external field, which is indirectly detected by electrocardiography. Although the direction of the hearts dipole moment varies, Fig. 20.38c is typical. In answering the questions that follow, consider the heart in isolationdont concern yourself with the effect of surrounding tissues on its electric field. At a distance r, far from the heart, the hearts electric field a. falls off as 1/r. b. falls off as 1/r2 . c. falls off as 1/r3 . d. falls off as 1/r4

At a given distance, far from the heart compared with its size, the electric field a. is weaker along an extension of the line shown in Fig. 20.38c than on a perpendicular line. b. is stronger along a an extension of the line shown in Fig. 20.38c than on a perpendicular line. c. has the same value at p

The difference between Figs. 20.38a and 20.38b that results in an external electric field in one case but not the other is that a. theres no net charge in Fig. 20.38a but there is a net charge in Fig. 20.38b. b. the total charge is greater in Fig. 20.38a. c. the charge is distributed in Fig. 20.38b so theres more negative charge to the left and more positive charge to the right

At the instant shown in Fig. 20.38c, theres an electric field within the heart that points approximately a. in the direction of the dipole moment vector p ! . b. opposite the dipole moment vector p ! . c. perpendicular to the dipole moment vector p ! .