## 21.54 . The ammonia molecule 1NH32 has a dipole moment of 5.0 * 10-30 C # m. Ammonia molecules in the gas phase are placed in a uniform electric field E S with magnitude 1.6 * 106 N>C. (a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with respect to E S from parallel to perpendicular? (b) At what absolute temperature T is the average translational kinetic energy 32 kT of a molecule equal to the change in potential energy calculated in part (a)? (Note: Above this temperature thermal agitation prevents the dipoles from aligning with the electric field.)

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## 21.53 . Point charges q1 = -4.5 nC and q2 = +4.5 nC are separated by 3.1 mm forming an electric dipole. (a) Find the electric dipole moment (magnitude and direction). (b) The charges are in a uniform electric field whose direction makes an angle of 36.9° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude 7.2 * 10-9 N # m ?

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## 21.52 .. A straight nonconducting plastic wire 8.50 cm long carries a charge density of +175 nC>m distributed uniformly along its length. It is lying on a horizontal tabletop. (a) Find the magnitude and direction of the electric field this wire produces at a point 6.00 cm directly above its midpoint. (b) If the wire is now bent into a circle lying flat on the table find the magnitude and direction of the electric field it produces at a point 6.00 cm directly above its center.

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## 21.51 . A ring-shaped conductor with radius a = 2.50 cm has a total positive charge Q = +0.125 nC uniformly distributed around it (see Fig. 21.23). The center of the ring is at the origin of coordinates O. (a) What is the electric field (magnitude and direction) at point P which is on the x-axis at x = 40.0 cm? (b) A point charge q = -2.50 mC is placed at P. What are the magnitude and direction of the force exerted by the charge q on the ring?

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## 21.50 .. A very long straight wire has charge per unit length 3.20 * 10-10 C>m. At what distance from the wire is the electricfield magnitude equal to 2.50 N>C?

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## 21.49 .. A charge of -6.50 nC is spread uniformly over the surface of one face of a nonconducting disk of radius 1.25 cm. (a) Find the magnitude and direction of the electric field this disk produces at a point P on the axis of the disk a distance of 2.00 cm from its center. (b) Suppose that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Find the magnitude and direction of the electric field at point P. (c) If the charge is all brought to the center of the disk find the magnitude and direction of the electric field at point P. (d) Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields?

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## 21.48 .. A point charge q1 = -4.00 nC is at the point x = 0.600 m y = 0.800 m and a second point charge q2 = +6.00 nC is at the point x = 0.600 m y = 0. Calculate the magnitude and direction of the net electric field at the origin due to these two point charges.

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## 21.47 . In a rectangular coordinate system a positive point charge q = 6.00 * 10-9 C is placed at the point x = +0.150 m y = 0 and an identical point charge is placed at x = -0.150 m y = 0. Find the x- and y-components the magnitude and the direction of the electric field at the following points: (a) the origin; (b) x = 0.300 m y = 0; (c) x = 0.150 m y = -0.400 m; (d) x = 0 y = 0.200 m.

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## 21.46 .. BIO Electric Field of Axons. A nerve signal is transmitted through a neuron when an excess of Na+ ions suddenly enters the axon a long cylindrical part of the neuron. Axons are approximately 10.0 mm in diameter and measurements show that about 5.6 * 1011 Na+ ions per meter (each of charge +e) enter during this process. Although the axon is a long cylinder the charge does not all enter everywhere at the same time. A plausible model would be a series of point charges moving along the axon. Consider a 0.10-mm length of the axon and model it as a point charge. (a) If the charge that enters each meter of the axon gets distributed uniformly along it how many coulombs of charge enter a 0.10-mm length of the axon? (b) What electric field (magnitude and direction) does the sudden influx of charge produce at the surface of the body if the axon is 5.00 cm below the skin? (c) Certain sharks can respond to electric fields as weak as 1.0 mN>C. How far from this segment of axon could a shark be and still detect its electric field?

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## 21.45 .. If two electrons are each 1.50 * 10-10 m from a proton (Fig. E21.45) find the magnitude and direction of the net electric force they will exert on the proton.

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## 21.44 .. Point charge q1 = -5.00 nC is at the origin and point charge q2 = +3.00 nC is on the x-axis at x = 3.00 cm. Point P is on the y-axis at y = 4.00 cm. (a) Calculate the electric fields E S 1 and E S 2 at point P due to the charges q1 and q2 . Express your results in terms of unit vectors (see Example 21.6). (b) Use the results of part (a) to obtain the resultant field at P expressed in unit vector form.

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## 21.43 . Two point charges are separated by 25.0 cm (Fig. E21.43). Find the net electric field these charges produce at (a) point A and (b) point B. (c) What would be the magnitude and direction of the electric force this combination of charges would produce on a proton at A?

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## 21.42 .. A point charge is placed at each corner of a square with side length a. All charges have magnitude q. Two of the charges are positive and two are negative (Fig. E21.42). What is the direction of the net electric field at the center of the square due to the four charges and what is its magnitude in terms of q and a?

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## 21.41 . Three negative point charges lie along a line as shown in Fig. E21.41. Find the magnitude and direction of the electric field this combination of charges produces at point P which lies 6.00 cm from the -2.00@mC charge measured perpendicular to the line connecting the three charges.

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## 21.40 . Repeat Exercise 21.39 but now let the charge at the origin be -4.00 nC.

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## 21.39 . A +2.00@nC point charge is at the origin and a second -5.00@nC point charge is on the x-axis at x = 0.800 m. (a) Find the electric field (magnitude and direction) at each of the following points on the x-axis: (i) x = 0.200 m; (ii) x = 1.20 m; (iii) x = -0.200 m. (b) Find the net electric force that the two charges would exert on an electron placed at each point in part (a).

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## 21.38 . The two charges q1 and q2 shown in Fig. E21.38 have equal magnitudes. What is the direction of the net electric field due to these two charges at points A (midway between the charges) B and C if (a) both charges are negative (b) both charges are positive (c) q1 is positive and q2 is negative.

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## 21.37 .. Two positive point charges q are placed on the x-axis one at x = a and one at x = -a. (a) Find the magnitude and direction of the electric field at x = 0. (b) Derive an expression for the electric field at points on the x-axis. Use your result to graph the x-component of the electric field as a function of x for values of x between -4a and +4a.

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## 21.36 . Two point charges Q and +q (where q is positive) produce the net electric field shown at point P in Fig. E21.36. The field points parallel to the line connecting the two charges. (a) What can you conclude about the sign and magnitude of Q? Explain your reasoning. (b) If the lower charge were negative instead would it be possible for the field to have the direction shown in the figure? Explain your reasoning.

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## 21.35 .. (a) An electron is moving east in a uniform electric field of 1.50 N>C directed to the west. At point A the velocity of the electron is 4.50 * 105 m>s toward the east. What is the speed of the electron when it reaches point B 0.375 m east of point A? (b) A proton is moving in the uniform electric field of part (a). At point A the velocity of the proton is 1.90 * 104 m>s east. What is the speed of the proton at point B?

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## 21.34 .. A +8.75@mC point charge is glued down on a horizontal frictionless table. It is tied to a -6.50@mC point charge by a light nonconducting 2.50@cm wire. A uniform electric field of magnitude 1.85 * 108 N>C is directed parallel to the wire as shown in Fig. E21.34. (a) Find the tension in the wire. (b) What would the tension be if both charges were negative?

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## 21.33 . A point charge is at the origin. With this point charge as the source point what is the unit vector rn in the direction of the field point (a) at x = 0 y = -1.35 m; (b) at x = 12.0 cm y = 12.0 cm; (c) at x = -1.10 m y = 2.60 m ? Express your results in terms of the unit vectors dn and en.

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## 21.32 .. CP A uniform electric field exists in the region between two oppositely charged plane parallel plates. A proton is released from rest at the surface of the positively charged plate and strikes the surface of the opposite plate 1.60 cm distant from the first in a time interval of 3.20 * 10-6 s. (a) Find the magnitude of the electric field. (b) Find the speed of the proton when it strikes the negatively charged plate.

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## 21.31 .. CP In Exercise 21.29 what is the speed of the electron as it emerges from the field?

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## 21.30 . (a) Calculate the magnitude and direction (relative to the +x@axis) of the electric field in Example 21.6. (b) A -2.5@nC point charge is placed at point P in Fig. 21.19. Find the magnitude and direction of (i) the force that the -8.0@nC charge at the origin exerts on this charge and (ii) the force that this charge exerts on the -8.0@nC charge at the origin.

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## 21.29 .. CP An electron is projected with an initial speed v0 = 1.60 * 106 m>s into the uniform field between two parallel plates (Fig. E21.29). Assume that the field between the plates is uniform and directed vertically downward and that the field outside the plates is zero. The electron enters the field at a point midway between the plates. (a) If the electron just misses the upper plate as it emerges from the field find the magnitude of the electric field. (b) Suppose that the electron in Fig. E21.29 is replaced by a proton with the same initial speed v0 . Would the proton hit one of the plates? If not what would be the magnitude and direction of its vertical displacement as it exits the region between the plates? (c) Compare the paths traveled by the electron and the proton and explain the differences. (d) Discuss whether it is reasonable to ignore the effects of gravity for each particle.

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## 21.28 .. Electric Field of the Earth. The earth has a net electric charge that causes a field at points near its surface equal to 150 N>C and directed in toward the center of the earth. (a) What magnitude and sign of charge would a 60-kg human have to acquire to overcome his or her weight by the force exerted by the earth’s electric field? (b) What would be the force of repulsion between two people each with the charge calculated in part (a) and separated by a distance of 100 m? Is use of the earth’s electric field a feasible means of flight? Why or why not?

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