Chapter 21

Excess electrons are placed on a small lead sphere with mass 8.00 g so that its net charge is \(-3.20 \times 10^{-9} \mathrm{C}\) (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 \(207 \mathrm{g} / \mathrm{mol} .\)

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 \(20,000 \mathrm{C} / \mathrm{s} ;\) this lasts for 100\(\mu \mathrm{s}\) or less. How much charge flows between the ground and the cloud in this time? How many electrons flow during this time?

Estimate how many electrons there are in your body. Make any assumptions you feel are necessary, but clearly state what they are. (Hint: Most of the atoms in your body have equal numbers of electrons, protons, and neutrons.) What is the combined charge of all these electrons?

Particles in a Gold Ring. You have a pure ( 24 karal) gold ring with mass 17.7 \(\mathrm{g}\) . Gold has an atomic mass of 197 \(\mathrm{g} / \mathrm{mol}\) 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?

An average human weighs about 650 \(\mathrm{N}\) . If two such generic humans each carried 1.0 coulomb 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 - \(\mathrm{N}\) weight?

Two small spheres spaced 20.0 \(\mathrm{cm}\) 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 \(4.57 \times 10^{-21} \mathrm{N} ?\)

Two small plastic spheres are given positive electrical charges. When they are 15.0 \(\mathrm{cm}\) apart, the repulsive force between them has magnitude 0.220 \(\mathrm{N}\) . 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?

Two small aluminum spheres, each having mass 0.0250 \(\mathrm{kg}\) are separated by \(80.0 \mathrm{cm} .\) (a) How many electrons does each sphere contain? (The atomic mass of aluminum is 26.982 \(\mathrm{g} / \mathrm{mol}\) , 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 \(1.00 \times 10^{4} \mathrm{N}\) (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 very small \(8.55-\) g spheres, 15.0 \(\mathrm{cm}\) apart from center to center, are charged by adding equal numbers of electrons to each of them. Disregarding all other forces, how many electrons would you have to add to each sphere so that the two spheres will accelerate at 25.0 \(\mathrm{g}\) when released? Which way will they accelerate?

(a) Assuming that only gravity is acting on it, how far does an electron have to be from a proton so that its acceleration is the same as that of a freely falling object at the earth's surface? (b) Suppose the earth were made only of protons but had the same size and mass it presently has. What would be the acceleration of an electron released at the surface? Is it necessary to consider the gravitational attraction as well as the electrical force? Why or why not?

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

A negative charge \(-0.550 \mu C\) exerts an upward \(0.200-\mathrm{N}\) force on an unknown charge 0.300 \(\mathrm{m}\) directly below it. (a) What is the unknown charge (magnitude and sign)? (b) What are the magnitude and direction of the force that the unknown charge exerts on the \(-0.550-\mu \mathrm{C}\) charge?

Three point charges are arranged on a line. Charge \(q_{3}=+5.00 \mathrm{nC}\) and is at the origin. Charge \(q_{2}=-3.00 \mathrm{nC}\) and is at \(x=+4.00 \mathrm{cm} .\) Charge \(q_{1}\) is at \(x=+2.00 \mathrm{cm} .\) 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 \(q_{1}=+3.00 \mu \mathrm{C}\) is at the origin, and charge \(q_{2}=-5.00 \mu \mathrm{C}\) is at \(x=0.200 \mathrm{m} .\) Charge \(q_{3}=-8.00 \mu \mathrm{C} .\) Where is \(q_{3}\) located if the net force on \(q_{1}\) is 7.00 \(\mathrm{N}\) in the \(-\mathrm{x}\) -direction?

Two point charges are located on the \(y\) -axis as follows: charge \(q_{1}=-1.50 \mathrm{nC}\) at \(y=-0.600 \mathrm{m},\) and charge \(q_{2}=+3.20 \mathrm{nC}\) at the origin \((y=0) .\) What is the total force (magnitude and direction) exerted by these two charges on a third charge \(q_{3}=+5.00 \mathrm{nC}\) located at \(y=-0.400 \mathrm{m} ?\)

Two point charges are placed on the \(x\) -axis as follows: Charge \(q_{1}=+4.00 \mathrm{nC}\) is located at \(x=0.200 \mathrm{m},\) and charge \(q_{2}=+5.00 \mathrm{nC}\) is at \(x=-0.300 \mathrm{m}\) . What are the magnitnde and direction of the total force exerted by these two charges on a negative point charge \(q_{3}=-6.00 \mathrm{nC}\) that is placed at the origin?

A positive point charge \(q\) is placed on the \(+y\) -axis at \(y=a\) and a negative point clarge \(-q\) is placed on the \(-y\) -axis at \(y=-a\) . A negative point charge \(-Q\) is located at some point on the \(+x\) -axis, (a) In a free- body diagram, show the forces that act on the charge \(-Q\) . (b) Find the \(x\) - and \(y\) -components of the net force that the two charges \(q\) and \(-q\) exert on \(-Q .\) (Your answer should involve only \(k, q, Q, a\) and the coordinate \(x\) of the third charge. \()\) (c) What is the net force on the charge \(-Q\) when it is at the origin \((x=0)\) ? (d) Graph the \(y\) -component of the net force on the charge \(-Q\) as a function of \(x\) for values of \(x\) between \(-4 a\) and \(+4 a\) .

Two positive point charges \(q\) are placed on the \(y\) -axis at \(y=a\) and \(y=-a .\) A negative point charge \(-Q\) is located at some point on the \(+x\) -axis. (a) In a free-body diagram, show the forces that act on the charge \(-Q .\) (b) Find the \(x\) -and \(y\) -components of the net force that the two positive charges exert on \(-Q\) . (Your answer \(r\) . should involve only \(k, q, Q, a\) and the coordinate \(x\) of the third charge.) (c) What is the net force on the charge \(-Q\) when it is at the origin \((x=0) ?(\mathrm{d})\) Graph the \(x\) -component of the net force on the charge \(-Q\) as a function of \(x\) for values of \(x\) between \(-4 a\) and \(+4 a\) .

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.

Two charges, one of 2.50\(\mu \mathrm{C}\) and the other of \(-3.50 \mu \mathrm{C}\) , are placed on the \(x\) -axis, one at the origin and the other at \(x=0.600 \mathrm{m}\) , as shown in Fig. 21.36 . Find the position on the \(x\) -axis where the net force on a small charge \(+q\) would be zero. figure can't copy

A proton is placed in a uniform electric field of \(2.75 \times\) \(10^{3} \mathrm{N} / \mathrm{C}\) . 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\(\mu \mathrm{s}\) in the field, assuming it starts from rest.

A particle has charge \(-3.00 \mathrm{nC}\) (a) Find the magnitude and direction of the electric field due to this particle at a point 0.250 \(\mathrm{m}\) directly above it. (b) At what distance from this particle does its electric field have a magnitude of 12.0 \(\mathrm{N} / \mathrm{C} ?\)

A proton is traveling horizontally to the right at \(4.50 \times\) \(10^{6} \mathrm{m} / \mathrm{s}\) (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of \(3.20 \mathrm{cm} .\) (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 \(\mathrm{m}\) 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 \(1.45-\mathrm{g}\) particle be for it to remain stationary when placed in a downward- directed electric field of magnitude 650 \(\mathrm{N} / \mathrm{C} 2\) (b) What is the magnitude of an electric field in which the electric force on a proton is equal in magnitnde to its weight?

(a) What is the electric field of an iron nucleus at a distance of \(6.00 \times 10^{-10} \mathrm{m}\) from the nucleus? The atomic number of iron is \(26 .\) Assume that the nucleus may be treated as a point charge. (b) What is the electric field of a proton at a distance of \(5.29 \times 10^{-11} \mathrm{m}\) from the proton? (This is the radius of the electron orbit in the Bohr model for the ground state of the hydrogen atom.)

Electric Field of the Rarth. The earth has a net electric charge that causes a field at points near its surface equal to 150 \(\mathrm{N} / \mathrm{C}\) and directed in toward the center of the earth. (a) What magnitude and sign of charge would a \(60-\mathrm{kg}\) 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 \(\mathrm{m} ?\) Is use of the earth's electric field a feasible means of flight? Why or why not?

Point charge \(q_{1}=-5.00 \mathrm{nC}\) is at the origin and point charge \(q_{2}=+3.00 \mathrm{nC}\) is on the \(x\) -axis at \(x=3.00 \mathrm{cm} .\) Point \(P\) is on the \(y\) -axis at \(y=4.00 \mathrm{cm}\) . (a) Calculate the electric fields \(\vec{E}_{1}\) and \(\overrightarrow{\boldsymbol{E}}_{2}\) 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 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 \(\mathrm{cm}\) distant from the first, in a time interval of \(1.50 \times 10^{-6} \mathrm{s}\) . (a) Find the magnitude of the electric field. (b) Find the speed of the proton when it strikes the negatively charged plate.

A point change is at the origin. With this point charge as the source point, what is the unit vector \(\hat{r}\) in the direction of \((a)\) the field point at \(x=0, y=-1.35 \mathrm{m} ;(b)\) the field point at \(x=\) \(12.0 \mathrm{cm}, y=12.0 \mathrm{cm} ;(c)\) the field point at \(x=-1.10 \mathrm{m}, y=\) 2.60 \(\mathrm{m} ?\) Express your results in terms of the unit vectors \(\hat{i}\) and \(\hat{j}\) .

(a) An electron is moving east in a uniform electric field of 1.50 \(\mathrm{N} / \mathrm{C}\) directed to the west. At point \(A,\) the velocity of the electron is \(4.50 \times 10^{5} \mathrm{m} / \mathrm{s}\) toward the east. What is the speed of the electron when it reaches point \(B, 0.375 \mathrm{m}\) east of point \(A ?(\mathrm{b}) \mathrm{A}\) proton is moving in the uniform electric field of part (a). At point \(A .\) the velocity of the proton is \(1.90 \times 10^{4} \mathrm{m} / \mathrm{s}\) , 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 \(-4 a\) and \(+4 a\) .

Two particles having charges \(q_{1}=0.500 \mathrm{nC}\) and \(q_{2}=8.00 \mathrm{nC}\) are separated by a distance of 1.20 \(\mathrm{m}\) . At what point along the line connecting the two charges is the total electric field due to the two charges equal to zero?

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

A positive point charge \(q\) is placed at \(x=a,\) and a negative point charge \(-q\) is placed 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 \(-4 a\) and \(+4 a\) .

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

A point charge \(q_{1}=-4.00 \mathrm{nC}\) is at the point \(x=0.600 \mathrm{m},\) \(y=0.800 \mathrm{m},\) and a second point charge \(q_{2}=+6.00 \mathrm{nC}\) is at the point \(x=0.600 \mathrm{m}, y=0 .\) Calculate the magnitude and direction of the net electric field at the origin due to these two point charges.

A very long, straight wire has charge per unit length \(1.50 \times 10^{-10} \mathrm{C} / \mathrm{m}\) . At what distance from the wire is the electric- field magnitude equal to 2.50 \(\mathrm{N} / \mathrm{C} ?\)

A straight, nonconducting plastic wire 8.50 \(\mathrm{cm}\) long carries a charge density of \(+175 \mathrm{nC} / \mathrm{m}\) 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 \(\mathrm{cm}\) 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 \(\mathrm{cm}\) directly above its center.

A charge of \(-6.50 \mathrm{nC}\) is spread uniformly over the surface of one face of a nonconducting disk of radius \(1.25 \mathrm{cm} .\) (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 \(\mathrm{cm}\) 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?

Two horizontal, infinite, plane sheets of charge are separated by a distance \(d\) . The lower sheet has negative charge with uniform surface charge density \(-\sigma<0 .\) The upper sheet has positive charge with uniform surface charge density \(\sigma>0 .\) What is the electric field (magnitude, and direction if the field is nonzero) (a) above the upper sheet, (b) below the lower sheet, (c) between the sheets?

Infinite sheet \(A\) carries a positive uniform charge density \(\sigma\) , and sheet \(B\) , which is to the right of \(A\) and parallel to it, carries a uniform negative charge density \(-2 \sigma .\) (a) Sketch the electric field lines for this pair of sheets. Include the region between the sheets as well as the regions to the left of \(A\) and to the right of \(B\) . (b) Repeat part (a) for the case in which sheet \(B\) carries a charge density of \(+2 \sigma .\)

Sketch the electric field lines for a disk of radius \(R\) with a positive uniform surface charge density \(\sigma\) . Use what you know about the electric field very close to the disk and very far from the disk to make your sketch.

(a) Sketch the electric field lines for an infinite line of charge. You may find it helpful to show the field lines in a plane containing the line of charge in one sketch and the field lines in a a plane perpendicular to the line of charge in a second sketch. (b) Explain how your sketches show (i) that the magnitude \(E\) of the electric field depends only on the distance \(r\) from the line of charge and (ii) that \(E\) decreases like \(1 / r .\)

Point charges \(q_{1}=-4.5 \mathrm{nC}\) and \(q_{2}=+4.5 \mathrm{nC}\) are separated by \(3.1 \mathrm{mm},\) forming an electric dipole. (a) Find the electric dipole moment (magnitude and dircction). (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 \(7.2 \times 10^{-9} \mathrm{N} \cdot \mathrm{m} ?\)

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

A dipole consisting of charges \(\pm e, 220 \mathrm{nm}\) apart, is placed between two very large (essentially infinite) sheets carrying equal but opposite charge densities of 125\(\mu \mathrm{C} / \mathrm{m}^{2}\) (a) What is the maximum potential energy this dipole can have due to the sheets, and how should it be oriented relative to the sheets to attain this value? (b) What is the maximum torgue the sheets can exert on the dipole, and how should it be oriented relative to the sheets to attain this value? (c) What net force do the two sheets exert on the dipole?

A charge \(q_{1}=+5.00 \mathrm{nC}\) is placed at the origin of an \(x y\) - coordinate system, and a charge \(q_{2}=-2.00 \mathrm{nC}\) is placed on the positive \(x\) -axis at \(x=4.00 \mathrm{cm} .\) (a) If a third charge \(q_{3}=+6.00 \mathrm{nC}\) is now placed at the point \(x=4.00 \mathrm{cm}, y=3.00 \mathrm{cm},\) 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 positive point charges \(Q\) are held fixed on the \(x\) -axis at \(x=a\) and \(x=-a .\) A third positive point charge \(q,\) with mass \(m,\) is placed on the \(x\) -axis away from the origin at a coordinate \(x\) such that \(|x| \ll a\) . The charge \(q\) , which is free to move along the \(x\) -axis, is then released. (a) Find the frequency of oscillation of the charge \(q\) .(Hint: Review the definition of simple harmonic motion in Section 13.2 Use the binomial expansion \((1+z)^{n}=1+\) \(n z+n(n-1) z^{2} / 2+\cdots,\) valid for the case \(|z|<1 . .\) (b) Suppose instead that the charge \(q\) were placed on the \(y\) -axis at a coordinate \(y\) such that \(|y| \ll a,\) and then released. If this charge is free to move anywhere in the \(x y\) -plane, what will happen to it? Explain your answer.

Two identical spheres are each attached to silk threads of length \(L=0.500\) in and hung from a common point (Fig. 21.44\()\) . Each sphere has mass \(m=8.00 \mathrm{g}\) . The radius of each sphere isvery 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 afree-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 information you have been given, what can you say about the magnitudes of \(q_{1}\) and \(q_{2} ?\)Explain your answers. (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.)