Chapter 21

Excess electrons are placed on a small lead sphere with mass 8.00 g so that its net charge is -3.20 x 10\(^{-9}\)C. (a) Find the number of excess electrons on the sphere. (b) How many excess electrons are there per lead atom? The atomic number of lead is 82, and its atomic mass is 207 g/mol.

Lightning occurs when there is a flow of electric charge (principally electrons) between the ground and a thundercloud. The maximum rate of charge flow in a lightning bolt is about 20,000 C\(/\)s; this lasts for 100 \(\mu\)s or less. How much charge flows between the ground and the cloud in this time? How many electrons flow during this time?

If a proton and an electron are released when they are 2.0 x 10\(^{-10}\) m apart (a typical atomic distance), find the initial acceleration of each particle.

You have a pure (24-karat) gold ring of mass 10.8 g. Gold has an atomic mass of 197 g\(/\)mol and an atomic number of 79. (a) How many protons are in the ring, and what is their total positive charge? (b) If the ring carries no net charge, how many electrons are in it?

\(Neurons\) are components of the nervous system of the body that transmit signals as electric impulses travel along their length. These impulses propagate when charge suddenly rushes into and then out of a part of the neuron called an axon. Measurements have shown that, during the inflow part of this cycle, approximately 5.6 x \(10^{11}\) Na\(^{+}\) (sodium ions) per meter, each with charge \(+e\), enter the axon. How many coulombs of charge enter a 1.5-cm length of the axon during this process?

Two small spheres spaced 20.0 cm apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is 3.33 \(\times\) \(10^{-21}\) N?

An average human weighs about 650 N. If each of two average humans could carry 1.0 C of excess charge, one positive and one negative, how far apart would they have to be for the electric attraction between them to equal their 650-N weight?

Two small aluminum spheres, each having mass 0.0250 kg, are separated by 80.0 cm. (a) How many electrons does each sphere contain? (The atomic mass of aluminum is 26.982 g\(/\)mol, and its atomic number is 13.) (b) How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude 1.00 \(\times\) 10\(^4\) N (roughly 1 ton)? Assume that the spheres may be treated as point charges. (c) What fraction of all the electrons in each sphere does this represent?

Two small plastic spheres are given positive electric charges. When they are 15.0 cm apart, the repulsive force between them has magnitude 0.220 N. What is the charge on each sphere (a) if the two charges are equal and (b) if one sphere has four times the charge of the other?

Suppose you had two small boxes, each containing 1.0 g of protons. (a) If one were placed on the moon by an astronaut and the other were left on the earth, and if they were connected by a very light (and very long!) string, what would be the tension in the string? Express your answer in newtons and in pounds. Do you need to take into account the gravitational forces of the earth and moon on the protons? Why? (b) What gravitational force would each box of protons exert on the other box?

In an experiment in space, one proton is held fixed and another proton is released from rest a distance of 2.50 mm away. (a) What is the initial acceleration of the proton after it is released? (b) Sketch qualitative (no numbers!) acceleration-time and velocity-time graphs of the released proton's motion.

A negative charge of \(-0.550 \space \mu\)C exerts an upward 0.600-N force on an unknown charge that is located 0.300 m directly below the first charge. What are (a) the value of the unknown charge (magnitude and sign); (b) the magnitude and direction of the force that the unknown charge exerts on the \(-\)0.550-\(\mu\)C charge?

Three point charges are arranged on a line. Charge \(q_3 = +\)5.00 nC and is at the origin. Charge \(q_2 = -\)3.00 nC and is at \(x = +\)4.00 cm. Charge \(q_1\) is at \(x = +\)2.00 cm. What is \(q_1\) (magnitude and sign) if the net force on \(q_3\) is zero?

Three point charges are arranged along the \(x\)-axis. Charge \(q_1 = +3.00 \space \mu\)C is at the origin, and charge \(q_2 = -5.00 \space \mu\)C is at \(x =\) 0.200 m. Charge \(q_3 = -8.00 \space \mu\)C. Where is \(q_3\) located if the net force on \(q_1\) is 7.00 N in the \(-\) \(x\)-direction ?

Two point charges are located on the \(y\)-axis as follows: charge \(q_1 = -1.50 \)nC at \(y = -\)0.600 m, and charge \(q_2 = +\)3.20 nC at the origin \((y = 0)\). What is the total force (magnitude and direction) exerted by these two charges on a third charge \(q_3 = +\)5.00 nC located at \(y = -\)0.400 m ?

Two point charges are placed on the \(x\)-axis as follows: Charge \(q_1 = +\)4.00 nC is located at \(x =\) 0.200 m, and charge \(q_2 = +\)5.00 nC is at \(x = -\)0.300 m . What are the magnitude and direction of the total force exerted by these two charges on a negative point charge \(q_3 = -\)6.00 nC that is placed at the origin?

A proton is placed in a uniform electric field of 2.75 \(\times 10^3 \space N/C\). Calculate (a) the magnitude of the electric force felt by the proton; (b) the proton's acceleration; (c) the proton's speed after 1.00 \(\mu\)s in the field, assuming it starts from rest.

A proton is traveling horizontally to the right at 4.50 \(\times 10^6\) m\(/\)s. (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of 3.20 cm. (b) How much time does it take the proton to stop after entering the field? (c) What minimum field (magnitude and direction) would be needed to stop an electron under the conditions of part (a)?

An electron is released from rest in a uniform electric field. The electron accelerates vertically upward, traveling 4.50 m in the first 3.00 \(\mu\)s after it is released. (a) What are the magnitude and direction of the electric field? (b) Are we justified in ignoring the effects of gravity? Justify your answer quantitatively.

(a) What must the charge (sign and magnitude) of a 1.45-g particle be for it to remain stationary when placed in a downward-directed electric field of magnitude 650 N\(/\)C? (b) What is the magnitude of an electric field in which the electric force on a proton is equal in magnitude to its weight?

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?

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 \(\times \space 10^{-6}\) s. (a) Find the magnitude of the electric field. (b) Find the speed of the proton when it strikes the negatively charged plate.

A point charge is at the origin. With this point charge as the source point, what is the unit vector \(\hat{r}\) in the direction of the field point (a) at \(x = 0, \space y = -\)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 \(\hat{\imath}\) and \(\hat{\jmath}\).

(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 \(\times 10^5\) 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 \(\times 10^4\) m\(/\)s, east. What is the speed of the proton at point \(B\)?

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.

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).

Point charge \(q_1 = -\)5.00 nC is at the origin and point charge \(q_2 = +\)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 \(\overrightarrow{E_1}\) and \(\overrightarrow{E_2}\) at point \(P\) due to the charges \(q_1\) and \(q_2\) . 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.

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 \(\mu\)m in diameter, and measurements show that about 5.6 \(\times \space 10^{11} \space Na^+\)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 \(\mu N/C\). How far from this segment of axon could a shark be and still detect its electric field?

In a rectangular coordinate system a positive point charge \(q = 6.00 \times \space 10^{-9} C\) 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.

A point charge \(q_1 = -\)4.00 nC is at the point \(x =\) 0.600 m, \(y =\) 0.800 m, and a second point charge \(q_2 = +\)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.

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?

A very long, straight wire has charge per unit length \(3.20 \times 10^{-10}\) C/m. At what distance from the wire is the electric-field magnitude equal to 2.50 N\(/\)C?

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 \(q = -2.50 \space \mu\)C is placed at \(P\). What are the magnitude and direction of the force exerted by the charge \(q\) \(on\) the ring?

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.

Point charges \(q_1 = -\)4.5 nC and \(q_2 = +\)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\(^\circ\) with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude \(7.2 \times 10^{-9}\) N \(\cdot\) m ?

The ammonia molecule (NH\(_3\)) has a dipole moment of \(5.0 \times 10^{30}\) C \(\cdot\) m. Ammonia molecules in the gas phase are placed in a uniform electric field \(\overrightarrow{E}\) with magnitude \(1.6 \times 10^6\) N\(/\)C. (a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with respect to \(\overrightarrow{E}\) from parallel to perpendicular? (b) At what absolute temperature \(T\) is the average translational kinetic energy \(\frac{3}{2} kT\) 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.)

An electric dipole with dipole moment \(\overrightarrow{p}\) is in a uniform external electric field \(\overrightarrow{E}\) . (a) Find the orientations of the dipole for which the torque on the dipole is zero. (b) Which of the orientations in part (a) is stable, and which is unstable? (\(Hint:\) Consider a small rotation away from the equilibrium position and see what happens.) (c) Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field.

Four identical charges \(Q\) are placed at the corners of a square of side \(L\). (a) In a free-body diagram, show all of the forces that act on one of the charges. (b) Find the magnitude and direction of the total force exerted on one charge by the other three charges.

A charge \(q_1 = +\)5.00 nC is placed at the origin of an \(xy\)-coordinate system, and a charge \(q_2 = -\)2.00 nC is placed on the positive \(x\)-axis at \(x = \)4.00 cm. (a) If a third charge \(q_3 = +\)6.00 nC is now placed at the point \(x =\) 4.00 cm, \(y =\) 3.00 cm, find the \(x\)- and \(y\)-components of the total force exerted on this charge by the other two. (b) Find the magnitude and direction of this force.

Two small spheres with mass \(m =\) 15.0 g are hung by silk threads of length \(L =\) 1.20 m from a common point (Fig. P21.62). When the spheres are given equal quantities of negative charge, so that \(q_1 = q_2 = q\), each thread hangs at \(\theta = 25.0^\circ\) from the vertical. (a) Draw a diagram showing the forces on each sphere. Treat the spheres as point charges. (b) Find the magnitude of \(q\). (c) Both threads are now shortened to length \(L =\) 0.600 m, while the charges \(q_1\) and \(q_2\) remain unchanged. What new angle will each thread make with the vertical? (\(Hint\): This part of the problem can be solved numerically by using trial values for \(\theta\) and adjusting the values of \(\theta\) until a self-consistent answer is obtained.)

Two identical spheres are each attached to silk threads of length \(L =\) 0.500 m and hung from a common point (Fig. P21.62). Each sphere has mass \(m =\) 8.00 g. The radius of each sphere is very small compared to the distance between the spheres, so they may be treated as point charges. One sphere is given positive charge \(q_1\) , and the other a different positive charge \(q_2\) ; this causes the spheres to separate so that when the spheres are in equilibrium, each thread makes an angle \(\theta = 20.0^\circ\) with the vertical. (a) Draw a free-body diagram for each sphere when in equilibrium, and label all the forces that act on each sphere. (b) Determine the magnitude of the electrostatic force that acts on each sphere, and determine the tension in each thread. (c) Based on the given information, what can you say about the magnitudes of \(q_1\) and \(q_2\)? Explain. (d) A small wire is now connected between the spheres, allowing charge to be transferred from one sphere to the other until the two spheres have equal charges; the wire is then removed. Each thread now makes an angle of 30.0\(^\circ\) with the vertical. Determine the original charges. (\(Hint\): The total charge on the pair of spheres is conserved.)

Point charge \(q_1 = -6.00 \times \space 10^{-6}\) C is on the \(x\)-axis at \(x = -0.200\space \mathrm{m}\). Point charge \(q_2\) is on the \(x\)-axis at \(x = +0.400 \space \mathrm{m}\). Point charge \(q_3 = +3.00 \times \space 10^{-6}\) C is at the origin. What is \(q_2\) (magnitude and sign) (a) if the net force on \(q_3\) is \(6.00 \mathrm{N}\) in the \(+x-\mathrm{direction}\); (b) if the net force on \(q_3\) is \(6.00 \mathrm{N}\) in the \(-x-\mathrm{direction}\)?

Two particles having charges \(q_1 =\) 0.500 nC and \(q_2 =\) 8.00 nC are separated by a distance of 1.20 m. At what point along the line connecting the two charges is the total electric field due to the two charges equal to zero?

A \(-\)3.00-nC point charge is on the \(x\)-axis at \(x =\) 1.20 m. A second point charge, \(Q,\) is on the \(x\)-axis at -0.600 m. What must be the sign and magnitude of \(Q\) for the resultant electric field at the origin to be (a) 45.0 N\(/\)C in the \(+\)x-direction, (b) 45.0 N\(/\)C in the \(-\)x-direction?

A charge \(+Q\) is located at the origin, and a charge \(+Q\) is at distance \(d\) away on the \(x\)-axis. Where should a third charge, \(q\), be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium?

A charge of \(-\)3.00 nC is placed at the origin of an \(xy\)-coordinate system, and a charge of 2.00 nC is placed on the \(y\)-axis at \(y =\) 4.00 cm. (a) If a third charge, of 5.00 nC, is now placed at the point \(x =\) 3.00 cm, \(y =\) 4.00 cm, find the \(x\)- and \(y\)-components of the total force exerted on this charge by the other two charges. (b) Find the magnitude and direction of this force.

Three identical point charges \(q\) are placed at each of three corners of a square of side \(L\). Find the magnitude and direction of the net force on a point charge \(-3q\) placed (a) at the center of the square and (b) at the vacant corner of the square. In each case, draw a free-body diagram showing the forces exerted on the \(-3q\) charge by each of the other three charges.

Consider a model of a hydrogen atom in which an electron is in a circular orbit of radius \(r = 5.29 \times \space 10^{-11}\) m around a stationary proton. What is the speed of the electron in its orbit?

The earth has a downward-directed electric field near its surface of about 150 N\(/\)C. If a raindrop with a diameter of 0.020 mm is suspended, motionless, in this field, how many excess electrons must it have on its surface?

A proton is projected into a uniform electric field that points vertically upward and has magnitude \(E\). The initial velocity of the proton has a magnitude \(v_0\) and is directed at an angle \(\alpha\) below the horizontal. (a) Find the maximum distance \(h_{max}\) that the proton descends vertically below its initial elevation. Ignore gravitational forces. (b) After what horizontal distance d does the proton return to its original elevation? (c) Sketch the trajectory of the proton. (d) Find the numerical values of \(h_{max}\) and \(d\) if \(E =\) 500 N\(/\)C, \(v_0 =\) 4.00 \(\times 10^5\) m\(/\)s, and \(\alpha =\) 30.0\(^\circ\).