Chapter 23 Solutions. electrons mol ( ) 2. )( kg) 2 ( m) 2 = N. k e N m 2 /C 2 = 8.

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1 Chapter 3 Solutions 10.0 grams 3.1 (a) N = grams mol 6.0 atoms electrons = mol atom # electrons added = Q e = C = C electron or.38 electrons for every 10 9 already present 3. (a) F e = k e q 1 q r = ( N m /C ) C ( m) = N (repulsion) F g = Gm 1 m ( N m kg )( kg) r = ( m) = N The electric force is larger by times (c) If k e q 1 q r = G m 1m r with q 1 = q = q and m 1 = m = m, then q m = G = N m /kg k e N m /C = C/kg 3.3 If each person has a mass of 70 kg and is (almost) composed of water, then each person contains N 70, 000 grams 18 grams mol 6.0 molecules protons mol molecule protons With an excess of 1% electrons over protons, each person has a charge q = (0.01)( C)( )= C q So F = k 1 q e r = ( ) ( ) 0.6 N = N ~ 10 6 N This force is almost enough to lift a "weight" equal to that of the Earth: Mg = ( kg)(9.8 m s )= N~ 10 6 N 000 by Harcourt, Inc. All rights reserved.

2 Chapter 3 Solutions 3.4 We find the equal-magnitude charges on both spheres: F = k e q 1q r = k e q r so q = r F k e = 1.00 m The number of electron transferred is then N xfer = ( C) ( C/e )= electrons The whole number of electrons in each sphere is N N m /C = C 10.0 g N tot = g / mol ( atoms / mol) ( 47 e / atom)= e The fraction transferred is then f = N xfer = N tot = =.51 charges in every billion 3.5 F = k e q 1 q r = ( N m C ) C ( ) [ ( m) ] = 514 kn *3.6 (a) The force is one of attraction. The distance r in Coulomb's law is the distance between centers. The magnitude of the force is ( C) F = k e q 1 q r = N m C C (0.300 m) = N The net charge of C will be equally split between the two spheres, or C on each. The force is one of repulsion, and its magnitude is ( C) F = k eq 1 q r = N m C C (0.300 m) = N

3 Chapter 3 Solutions F 1 = k e q 1 q r = ( N m /C )( C)( C) (0.500 m) = N F = k e q 1 q r = ( N m /C )( C)( C) (0.500 m) = 1.01 N F x = ( ) cos 60.0 = N F y = ( ) sin 60.0 = N F = (0.755 N)i (0.436 N)j = 0.87 N at an angle of 330 Goal Solution Three point charges are located at the corners of an equilateral triangle as shown in Figure P3.7. Calculate the net electric force on the 7.00 µc charge. G : Gather Information: The 7.00 µc charge experiences a repulsive force F 1 due to the.00 µc charge, and an attractive force F due to the 4.00 µc charge, where F = F 1. If we sketch these force vectors, we find that the resultant appears to be about the same magnitude as F and is directed to the right about 30.0 below the horizontal. O : Organize : We can find the net electric force by adding the two separate forces acting on the 7.00 µc charge. These individual forces can be found by applying Coulomb s law to each pair of charges. A : Analyze: The force on the 7.00 µc charge by the.00 µc charge is ( C) ( C) F 1 = N m /C ( ( m) cos60 i + sin60 j)= F 1 = 0.5i j Similarly, the force on the 7.00 µc by the 4.00 µc charge is F = N m C C ( C) m ( cos60 i sin60 j) Thus, the total force on the 7.00 µc, expressed as a set of components, is F = F 1 + F = ( i j) N = 0.87 N at 30.0 below the +x axis N = ( 0.503i 0.87j ) N L : Learn: Our calculated answer agrees with our initial estimate. An equivalent approach to this problem would be to find the net electric field due to the two lower charges and apply F=qE to find the force on the upper charge in this electric field. 000 by Harcourt, Inc. All rights reserved.

4 4 Chapter 3 Solutions *3.8 Let the third bead have charge Q and be located distance x from the left end of the rod. This bead will experience a net force given by F = k ( e 3q)Q x i + k ( e q)q d x ( i) The net force will be zero if 3 x = 1 d x, or d x = x 3 This gives an equilibrium position of the third bead of x = 0.634d The equilibrium is stable if the third bead has positive charge. *3.9 (a) F = k ee r = ( N m /C ) ( C) ( m) = N We have F = mv r from which v = ( 8 N) m Fr m = kg = m/s 3.10 The top charge exerts a force on the negative charge to the left, at an angle of tan 1 d /x force k e qq ( d ) which is directed upward and + x to the x-axis. The two positive charges together exert k e qq d 4 + x ( x)i d 4 + x 1/ = ma or for x << d, a k eqq md 3 /8 x (a) The acceleration is equal to a negative constant times the excursion from equilibrium, as in a = ω x, so we have Simple Harmonic Motion with ω = 16k e qq md 3. T = π ω = π md 3 k e qq, where m is the mass of the object with charge Q. v max = ωa = 4a k e qq md 3

5 Chapter 3 Solutions For equilibrium, F e = F g, or qe = mg( j). Thus, E = mg q j. (a) E = mg q j = ( kg)(9.80 m s ) j = ( NC)j C E = mg q ( 9.80 m s ) ( C) kg j = j = ( NC)j 3.1 F y = 0: QEj + mg( j)=0 m = QE g = ( C)(610 N / C) 9.80 m / s = 1.49 grams *3.13 The point is designated in the sketch. The magnitudes of the electric fields, E 1, (due to the C charge) and E (due to the C charge) are E 1 = k eq r = ( N m /C )( C) d (1) E = k eq r = ( N m /C )( C) (d m) () Equate the right sides of (1) and () to get (d m) =.40d or d m = ±1.55d which yields d = 1.8 m or d = 0.39 m The negative value for d is unsatisfactory because that locates a point between the charges where both fields are in the same direction. Thus, d = 1.8 m to the left of the -.50 µc charge If we treat the concentrations as point charges, q E + = k e r = 8.99 N m ( 40.0 C) 109 C = N/C j 1000 m j q E = k e r = 8.99 N m ( 40.0 C) 109 C = N/C j 1000 m j (downward) (downward) E = E + + E = N / C downward 000 by Harcourt, Inc. All rights reserved.

6 6 Chapter 3 Solutions ( ) *3.15 (a) E 1 = k e q = r E = k e q = r = NC ( ) = NC E x = E E 1 cos 60 = cos 60.0 = NC E y = E 1 sin 60.0 = sin 60.0 = NC E = [18.0i 18j] 10 3 NC= [18.0i 18j] kn C F = qe = ( C) ( 18.0i 18j) 10 3 NC= ( 36.0i 436j) 10 3 N = ( 36.0i 436j )mn *3.16 (a) E 1 = k e q 1 r 1 E = k e q r ( j)= ( i)= ( ) j ( j) = NC ( ) = ( NC)i i E = E + E 1 = ( NC)i ( NC)j F = qe = ( C) ( 599i 700j)NC F = ( i j )N = ( 300. i 135. j ) µn 3.17 (a) The electric field has the general appearance shown. It is zero at the center, where (by symmetry) one can see that the three charges individually produce fields that cancel out. You may need to review vector addition in Chapter Three. The magnitude of the field at point P due to each of the charges along the base of the triangle is E = k e q a. The direction of the field in each case is along the line joining the charge in question to point P as shown in the diagram at the right. The x components add to zero, leaving E = k eq a ( sin60.0 )j + k eq ( a sin60.0 )j = 3 k eq a j

7 Chapter 3 Solutions 7 Goal Solution Three equal positive charges q are at the corners of an equilateral triangle of side a, as shown in Figure P3.17. (a) Assume that the three charges together create an electric field. Find the location of a point (other than ) where the electric field is zero. (Hint: Sketch the field lines in the plane of the charges.) What are the magnitude and direction of the electric field at P due to the two charges at the base? G : The electric field has the general appearance shown by the black arrows in the figure to the right. This drawing indicates that E = 0 at the center of the triangle, since a small positive charge placed at the center of this triangle will be pushed away from each corner equally strongly. This fact could be verified by vector addition as in part below. The electric field at point P should be directed upwards and about twice the magnitude of the electric field due to just one of the lower charges as shown in Figure P3.17. For part, we must ignore the effect of the charge at point P, because a charge cannot exert a force on itself. O : A : The electric field at point P can be found by adding the electric field vectors due to each of the two lower point charges: E = E 1 + E The electric field from a point charge is As shown in the solution figure above, q E 1 = k e to the right and upward at 60 a q E = k e to the left and upward at 60 a q q E = E 1 + E = k e [( cos60 i + sin60 j)+ ( cos60 i + sin60 j) ]= k e a a [ ( sin60 j) ] q = 1.73k e a j L : The net electric field at point P is indeed nearly twice the magnitude due to a single charge and is entirely vertical as expected from the symmetry of the configuration. In addition to the center of the triangle, the electric field lines in the figure to the right indicate three other points near the middle of each leg of the triangle where E = 0, but they are more difficult to find mathematically (a) E = k e q r = ( )( ) (1.1) = 14, 400 N C E x = 0 and E y = (14, 400) sin 6.6 = NC so E = j N C F = Eq = ( j)( ) = j N 000 by Harcourt, Inc. All rights reserved.

8 8 Chapter 3 Solutions 3.19 (a) E = k e q 1 r ~ 1 + k e q 1 r ~ + k e q 3 r 3 ~ 3 = k e q a i + k ( e 3q) E =3.06 k eq a i +5.06k eq a j = 5.91 k eq at 58.8 a a (i cos j sin 45.0 ) + k e 4q a j F = qe= 5.91 k eq a at The magnitude of the field at (x, y) due to charge q at (x 0, y 0 ) is given by E = k e q r where r is the distance from (x 0, y 0 ) to (x, y). Observe the geometry in the diagram at the right. From triangle ABC, r = (x x 0 ) + (y y 0 ), or r = (x x 0 ) + (y y 0 ), sinθ = (y y 0 ), and cosθ = (x x 0 ) r r Thus, E x = Ecosθ = k e q (x x 0 ) r = r k e q(x x 0 ) [(x x 0 ) + (y y 0 ) ] 3/ and E y = Esinθ = k e q (y y 0 ) r = r k e q(y y 0 ) [(x x 0 ) + (y y 0 ) ] 3/ 3.1 The electric field at any point x is E = k e q (x a) k e q (x ( a)) = k eq(4ax) (x a ) When x is much, much greater than a, we find E (4a)(k e q) x 3 3. (a) One of the charges creates at P a field E = k e Q/n R + x at an angle θ to the x-axis as shown. When all the charges produce field, for n > 1, the components perpendicular to the x-axis add to zero. The total field is nk e (Q/n)i R + x cos θ = k e Qxi (R + x ) 3/ A circle of charge corresponds to letting n grow beyond all bounds, but the result does not depend on n. Smearing the charge around the circle does not change its amount or its distance from the field point, so it does not change the field..

9 Chapter 3 Solutions E = k e q r ~ = k e q ( i)+ k e q ( i) + k e q a (a) (3a) ( i) +... = k e qi a = π k e q 6 a i 3.4 E = k eλl d l+d l = = k e Q / l d( l+d) k e Q d( l+d) = ( )( ) (0.90)( ) E = N/C, directed toward the rod. 3.5 E = k e dq x where dq = λ 0 dx dx E = k e λ 0 x 0 x = k e 1 x x 0 = k e λ 0 x 0 The direction is i or left for λ 0 > 0 k 3.6 E = de = e λ 0 x 0 dx( i) x 3 x0 = k e λ 0 x 0 i x 3 dx = k x e λ 0 x 0 i 1 0 x = k e λ 0 x 0 x 0 ( i) k e xq 3.7 E = (x + a ) 3/ = ( )( )x x (x ) 3/ = (x ) 3/ (a) At x = m, E = i N/C = 6.64 i MN/C At x = m, E = i N/C = 4.1 i MN/C (c) At x = m, E = i N/C = 6.40 i MN/C (d) At x = 1.00 m, E = i N/C = i MN/C 000 by Harcourt, Inc. All rights reserved.

10 10 Chapter 3 Solutions 3.8 E = k e Qx (x + a ) 3/ For a maximum, de dx = Qk 1 e (x + a ) 3/ 3x (x + a ) 5/ = 0 x + a 3x = 0 or x = a Substituting into the expression for E gives E = k e Qa ( 3 a ) 3/ = k e Q 3 = 3 a k e Q 3 3a = Q 6 3π e 0 a 3.9 E = π k e σ 1 x x + R E = π( )( ) 1 x x + ( 0.350) = x x (a) At x = m, E = NC= 383 MN C At x = m, E = NC= 34 MN C (c) At x = m, E = NC= 80.7 MN C (d) At x =.00 m, E = NC= 6.68 MN C 3.30 (a) From Example 3.9: E = π k e σ 1 σ = Q πr = Cm x x + R E = ( N C)(0.900) = NC = 93.6 MN/C appx: E = πk e σ = 104 MN/C (about 11% high) E = ( N/C) cm = ( N C)( ) = MN/C cm appx: E = k e Q r = ( ) (0.30) = MN/C (about 0.6% high)

11 Chapter 3 Solutions The electric field at a distance x is E x = πk e σ 1 x x + R This is equivalent to E x = πk e σ R x For large x, R x << 1 and 1+ R so Substitute σ = Q/π R, E x = x 1+ R x E x = πk e σ 1 k e Q( 1 x ) [ 1+R (x )] 1 1+R (x ) = πk eσ 1+R (x ) 1 1+R (x ) [ ] = k e Q x + R [ ] But for x > > R, 1 x + R 1 x, so E x k e Q for a disk at large distances x 3.3 The sheet must have negative charge to repel the negative charge on the Styrofoam. The magnitude of the upward electric force must equal the magnitude of the downward gravitational force for the Styrofoam to "float" (i.e., F e = F g ). Thus, σ qe = mg, or q = mg which gives σ = e 0 e 0 mg q 3.33 Due to symmetry E y = de y = 0, and E x = de sin θ = k e where dq = λ ds = λr dθ, so that, E x = k eλ r π 0 sinθ dθ dq sin θ r π = k eλ r ( cosθ) 0 = k eλ r where λ = q L Solving, and r = L π. Thus, E x = k e qπ L = ( N m /C )( C)π (0.140 m) E = E x = N/C Since the rod has a negative charge, E = ( i) N/C = 1.6 i MN/C 000 by Harcourt, Inc. All rights reserved.

12 1 Chapter 3 Solutions 3.34 (a) We define x = 0 at the point where we are to find the field. One ring, with thickness dx, has charge Qdx/h and produces, at the chosen point, a field de = k e x Qdx (x + R ) 3/ i h The total field is E = d + h k de = e Qxdx h(x + R ) 3/ i = k e Qi d h all charge d + h x = d (x + R ) 3/ xdx E = k e Qi h (x + R ) 1/ d + h ( 1/) x = d = k e Qi h 1 (d + R ) 1/ 1 (d + h) + R 1/ Think of the cylinder as a stack of disks, each with thickness dx, charge Q dx/h, and chargeper-area σ = Qdx/ πr h. One disk produces a field de = π k e Qdx πr h 1 x (x + R ) 1/ i So, E = de all charge = d + h x = d k e Qdx R 1 h x (x + R ) 1/ i E = k e Qi R h E = k e Qi R h d + h dx 1 d d + h x = d d + h d (d + h) + R (x + R ) 1/ xdx = k e Qi R x h 1/ +(d + R ) 1/ d+ h d 1 (x + R ) 1/ d + h 1/ d E= 1/ k e Qi R h +(d + R ) 1/ (d + h) + R h 3.35 (a) k The electric field at point P due to each element of length dx, is de = e dq (x + y ) along the line joining the element of length to point P. By symmetry, and is directed E x = de x = 0 and since dq = λ dx, E = E y = de y = de cos θ where cos θ = y (x + y ) 1 Therefore, dx (x + y ) 3 = k e λ sinθ 0 y For a bar of infinite length, θ 90 and E y = k e λ y

13 Chapter 3 Solutions 13 *3.36 (a) The whole surface area of the cylinder is A = π r + π rl = π rr+ ( L). Q = σa = ( Cm )π( m) [ m m] = C For the curved lateral surface only, A = πrl. Q = σa = ( Cm )π( m) ( m) = C (c) Q = ρv = ρπr L = ( Cm 3 )π( m) ( m)= C *3.37 (a) Every object has the same volume, V = 8( m) 3 = m 3. For each, Q = ρv = ( Cm 3 )( m 3 )= C We must count the 9.00 cm squares painted with charge: (i) 6 4 = 4 squares Q = σa = ( Cm )4.0( m )= C (ii) 34 squares exposed Q = σa = ( Cm )34.0( m )= C (iii) 34 squares Q = σa = ( Cm )34.0( m )= C (iv) 3 squares Q = σa = ( Cm )3.0( m )= C (c) (i) total edge length: = 4 ( m) Q = λ = ( Cm)4 ( m)= C (ii) Q = λ = ( Cm)44 ( m)= C (iii) Q = λ = ( Cm)64 ( m)= C 000 by Harcourt, Inc. All rights reserved.

14 14 Chapter 3 Solutions (iv) Q = λ = ( Cm)40 ( m)= C

15 Chapter 3 Solutions (a) q 1 = 6 q 18 = 1 3 q 1 is negative, q is positive 3.41 F = qe = ma a = qe m v = v i + at v = qet m electron: v e = ( )(50)( ) = m/s in a direction opposite to the field proton: v p = ( )(50)( ) = m/s in the same direction as the field 3.4 (a) a = qe m = ( )( ) ( ) v = v i + a(x x i ) = ms so a = i ms 0=v i + ( )(0.0700) v i = i ms (c) v = v i + at 0= ( )t t = s 000 by Harcourt, Inc. All rights reserved.

16 16 Chapter 3 Solutions 3.43 (a) a = qe m = = m/s v = v i + at = ( )t t = s (c) x x i = 1 ( v i + v)t x = 1 ( ) ( )= 11.7 m (d) K = 1 mv = 1 ( kg)( m/s) = J 3.44 The required electric field will be in the direction of motion. We know that Work = K So, Fd = 1 mv i (since the final velocity = 0) 1 This becomes Eed = 1 mv i mv i or E = ed J E = ( C)(0.100 m) = N/C (in direction of electron's motion) 3.45 The required electric field will be in the direction of motion. Work done = K so, Fd = 1 mv i (since the final velocity = 0) which becomes eed = K and E = K ed

17 Chapter 3 Solutions 17 Goal Solution The electrons in a particle beam each have a kinetic energy K. What are the magnitude and direction of the electric field that stops these electrons in a distance of d? G : We should expect that a larger electric field would be required to stop electrons with greater kinetic energy. Likewise, E must be greater for a shorter stopping distance, d. The electric field should be in the same direction as the motion of the negatively charged electrons in order to exert an opposing force that will slow them down. O : The electrons will experience an electrostatic force F = qe. Therefore, the work done by the electric field can be equated with the initial kinetic energy since energy should be conserved. A : The work done on the charge is W = F d = qe d and K i + W = K f = 0 Assuming v is in the + x direction, K + ( e)e di = 0 ee ( di)= K E is therefore in the direction of the electron beam: E = K ed i L : As expected, the electric field is proportional to K, and inversely proportional to d. The direction of the electric field is important; if it were otherwise the electron would speed up instead of slowing down! If the particles were protons instead of electrons, the electric field would need to be directed opposite to v in order for the particles to slow down The acceleration is given by v = v i + a(x x i )or v = 0 + a( h) Solving, a = v h Now F = ma: mgj + qe = mv j h Therefore qe = mv h + mg j (a) Gravity alone would give the bead downward impact velocity 9.80 ( m/s )( 5.00 m) = 9.90 m / s To change this to 1.0 m/s down, a downward electric field must exert a downward electric force. q = m E v h g = kg N/C N s kg m (1.0 m/s) (5.00 m) 9.80 m/s = 3.43 µc 000 by Harcourt, Inc. All rights reserved.

18 18 Chapter 3 Solutions 3.47 (a) t = x v = = s = 111 ns a y = qe m = ( )( ) ( ) = m/s y y i = v yi t + 1 a y t y = 1 ( )( ) = m = 5.67 mm (c) v x = m/s v y = v y i + a y = ( )( ) = m/s 3.48 a y = qe m = ( )(390) ( ) = m/s (a) t = v i sinθ a y from projectile motion equations t = ( )sin = s) = 1.0 ns (c) h = v i sin θ = ( ) sin 30.0 a y ( = 1.3 mm ) R = v i sin θ = ( ) sin 60.0 a y ( = 4.4 mm ) 3.49 v i = m/s (a) a y = ee m = ( )(70) ( ) = ms R = v i sin θ a y = m so that ( ) sin θ = sin θ = θ = θ = 53.1 t = R R = v ix v i cosθ If θ = 36.9, t = 167 ns If θ = 53.1, t = 1 ns

19 Chapter 3 Solutions 19 *3.50 (a) The field, E 1, due to the C charge is in the x direction. E 1 = k e q r = ( N m /C )( C) (.50 m) i = 5.75i N/C Likewise, E and E 3, due to the C charge and the C charge are E = k e q r = ( N m /C )( C) (.00 m) i = 11. N/C E 3 = ( N m /C )( C) (1.0 m) i = 18.7 N/C E R = E 1 + E + E 3 = 4. N/C in +x direction. E 1 = k e q r E = k e q r E 3 = k e q r = ( 8.46 N / C) ( 0.43i j) = ( 11. N / C) ( +j) = ( 5.81 N / C) ( 0.371i j) E x = E 1x + E 3x = 4.1i N/C E y = E 1y + E y + E 3y = 8.43j N/C E R = 9.4 N/C θ = 63.4 above x axis 3.51 The proton moves with acceleration a p = qe ( m = C) ( 640 N / C) = ms kg ( while the e has acceleration a e = C) ( 640 N/C ) = ms = 1836 a p kg (a) We want to find the distance traveled by the proton (i.e., d = 1 a p t ), knowing: 4.00 cm = 1 a pt + 1 a et = 1837( 1 a pt ) Thus, d = 1 a pt = 4.00 cm 1837 = 1.8 µm 000 by Harcourt, Inc. All rights reserved.

20 0 Chapter 3 Solutions The distance from the positive plate to where the meeting occurs equals the distance the sodium ion travels (i.e., d Na = 1 a Na t ). This is found from: 4.00 cm = 1 a Na t + 1 a Cl t : 4.00 cm = 1 ee.99 u t + 1 ee u This may be written as t 4.00 cm = 1 a Na t + 1 ( 0.649a Na)t = a Na t so d Na = 1 a Na t = 4.00 cm 1.65 =.43 cm 3.5 From the free-body diagram shown, F y = 0 and So T cos 15.0 = N T = N From F x = 0, we have qe = T sin 15.0 or q = T sin 15.0 E = ( N) sin N/C = C = 5.5 µc 3.53 (a) Let us sum force components to find F x = qe x T sin θ = 0, and F y = qe y + T cos θ mg = 0 Combining these two equations, we get q = mg (E x cot θ + E y ) = ( )(9.80) (3.00 cot ) 10 5 = C = 10.9 nc From the two equations for F x and F y we also find Free Body Diagram for Goal Solution T = qex sin 37.0 = N = 5.44 mn

21 Chapter 3 Solutions 1 Goal Solution A charged cork ball of mass 1.00 g is suspended on a light string in the presence of a uniform electric field, as shown in Fig. P3.53. When E = ( 3.00i j) 10 5 N / C, the ball is in equilibrium at θ = Find (a) the charge on the ball and the tension in the string. G : O : (a) Since the electric force must be in the same direction as E, the ball must be positively charged. If we examine the free body diagram that shows the three forces acting on the ball, the sum of which must be zero, we can see that the tension is about half the magnitude of the weight. The tension can be found from applying Newton's second law to this statics problem (electrostatics, in this case!). Since the force vectors are in two dimensions, we must apply ΣF = ma to both the x and y directions. A : Applying Newton's Second Law in the x and y directions, and noting that ΣF = T + qe + F g = 0, ΣF x = qe x T sin 37.0 = 0 (1) ΣF y = qe y + T cos 37.0 mg = 0 () We are given E x = N / C and E y = N / C; substituting T from (1) into (): mg q = E E y + x tan 37.0 = ( kg)(9.80 m / s ) = C tan N/C Using this result for q in Equation (1), we find that the tension is T = qe x sin 37.0 = N L : The tension is slightly more than half the weight of the ball ( F g = N) so our result seems reasonable based on our initial prediction (a) Applying the first condition of equilibrium to the ball gives: ΣF x = qe x T sinθ = 0 or T = qe x sinθ = qa sinθ and ΣF y = qe y + T cosθ mg = 0 or qb + T cosθ = mg Substituting from the first equation into the second gives: qacotθ ( + B)= mg, or q = mg Acotθ + B Substituting the charge into the equation obtained from ΣF x yields T = mg Acotθ + B A sinθ = mga Acosθ + Bsinθ 000 by Harcourt, Inc. All rights reserved.

22 Chapter 3 Solutions Goal Solution A charged cork ball of mass m is suspended on a light string in the presence of a uniform electric field, as shown in Figure P3.53. When E = ( Ai + Bj) N / C, where A and B are positive numbers, the ball is in equilibrium at the angle θ. Find (a) the charge on the ball and the tension in the string. G : This is the general version of the preceding problem. The known quantities are A, B, m, g, and θ. The unknowns are q and T. O : The approach to this problem should be the same as for the last problem, but without numbers to substitute for the variables. Likewise, we can use the free body diagram given in the solution to problem 53. A : Again, Newton's second law: T sin θ + qa = 0 (1) and + T cosθ + qb mg = 0 () (a) Substituting T = qa qacosθ, into Eq. (), sinθ Isolating q on the left, q = Substituting this value into Eq. (1), T = sinθ + qb = mg mg Acot θ + B mga Acos θ + Bsinθ L : If we had solved this general problem first, we would only need to substitute the appropriate values in the equations for q and T to find the numerical results needed for problem 53. If you find this problem more difficult than problem 53, the little list at the Gather step is useful. It shows what symbols to think of as known data, and what to consider unknown. The list is a guide for deciding what to solve for in the Analysis step, and for recognizing when we have an answer F = k e q 1 q r tan θ = θ = 14.0 F 1 = ( )( ) (0.150) = 40.0 N F 3 = ( )( ) (0.600) =.50 N F = ( )( ) (0.619) =.35 N F x = F 3 F cos 14.0 = cos 14.0 = 4.78 N F y = F 1 F sin 14.0 = sin 14.0 = 40.6 N F net = F x + F y = ( 4.78) + ( 40.6) = 40.9 N tan φ = F y = 40.6 F x 4.78 φ = 63

23 Chapter 3 Solutions From Fig. A: dcos = cm, or d = cm cos From Fig. B: θ = sin 1 d 50.0 cm = 15.0 cm sin cm cos 30.0 = 0.3 F q mg = tanθ Figure A or F q = mg tan 0.3 (1) k From Fig. C: F q = F cos 30.0 = e q m cos 30.0 () k Equating equations (1) and (), e q m cos 30.0 =mg tan 0.3 Figure B q = mg ( m ) tan 0.3 k e cos 30.0 ( 9.80 m s ) m tan 0.3 q = kg 8.99 ( 10 9 N m C )cos 30.0 q = C = C= 0.05 µc Figure C 3.57 Charge Q/ resides on each block, which repel as point charges: F = k e(q/)(q/) L = k(l L i ) Q = L kl ( L i ) = ( m) k e ( 100 N / m) ( m) N m /C = 6.7 µc 3.58 Charge Q/ resides on each block, which repel as point charges: F = k e Q Solving for Q, Q = L kl L i k e ( Q ) L = kl ( L i ) 000 by Harcourt, Inc. All rights reserved.

24 4 Chapter 3 Solutions *3.59 According to the result of Example 3.7, the lefthand rod creates this field at a distance d from its righthand end: k e Q E = d(a + d) df = k eqq a dx d(d + a) F = k eq a b x = b a dx x(x + a) = k eq a 1 a ln a + x x b b a F = +k eq 4a ln a + b b b + ln b a = k eq 4a ln b (b a)(b + a) = k e Q 4a ln b b 4a *3.60 The charge moves with acceleration of magnitude a given by F = ma = q E (a) a = q E m = C (1.00 N/C) kg = m/s Then v = v i + at = 0 + at gives t = v a = m/s = 171 µs m/s t = v a = vm qe (c) From t = vm qe = ( m/s)( kg) ( C)(1.00 N/C) = s, as E increases, t gets shorter in inverse proportion Q = λdl = λ 0 cos θ Rdθ = λ 0 R sin θ = λ 0 R [1 ( 1)] = λ 0 R Q = 1.0 µc = (λ 0 )(0.600) m = 1.0 µc so λ 0 = 10.0 µc/m ( λdl) µc df y = 4π e 0 R µc cosθ = 4π e 0 R λ 0 cos θrdθ 90.0 F y = N m C ( C)( C/m) (0.600 m) cos θdθ F y = π/ ( N) ( + 1 cosθ )dθ π/ F y = ( N) 1 π + 1 sin θ ( 4 π/)= N Downward. Since the leftward and rightward forces due to the two halves of the semicircle cancel out, F x = 0.

25 Chapter 3 Solutions At equilibrium, the distance between the charges is r = ( m)sin10.0 = m Now consider the forces on the sphere with charge +q, and use ΣF y = 0: ΣF y = 0: T cos 10.0 = mg, or T = mg cos 10.0 (1) ΣF x = 0: F net = F F 1 = T sin10.0 () F net is the net electrical force on the charged sphere. Eliminate T from () by use of (1). F net = mgsin 10.0 cos 10.0 = mg tan 10.0 = ( kg) ( 9.80 m / s )tan 10.0 = N F net is the resultant of two forces, F 1 and F. F 1 is the attractive force on +q exerted by q, and F is the force exerted on +q by the external electric field. F net = F F 1 or F = F net + F 1 ( C) ( C) ( m) = N F 1 = N m /C Thus, F = F net + F 1 yields F = N N =.1 10 N and F = qe, or E = F q =.1 10 N C = N/C = 443 kn/c 3.63 (a) From the Q charge we have F e T sinθ = 0 and mg T cosθ = 0 Combining these we find F e mg = T sinθ T cosθ = tanθ From the Q charge we have F e T 1 sinθ 1 = 0 and mg T 1 cosθ 1 =0 Combining these we find F e mg = T 1 sinθ 1 T 1 cosθ 1 = tanθ 1 or θ = θ 1 F e = k e QQ r = k eq r If we assume θ is small then. Substitute expressions for F e and tan θ into either equation found in part (a) and solve for r. F e = tanθ then mg and solving for r we find 000 by Harcourt, Inc. All rights reserved.

26 position is x = 1 a( 3 + cot 81.7 ) = 0.939a θ cosθ sin θ( 3 + cotθ) 6 Chapter 3 Solutions 3.64 At an equilibrium position, the net force on the charge Q is zero. The equilibrium position can be located by determining the angle θ corresponding to equilibrium. In terms of lengths s, 1 a 3, and r, shown in Figure P3.64, the charge at the origin exerts an attractive force k e Qq (s + 1 a 3). The other two charges exert equal repulsive forces of magnitude k e Qq r. The horizontal components of the two repulsive forces add, balancing the attractive force, F net = k e Qq cos θ 1 r (s + 1 a 3) = 0 From Figure P3.64, r = 1 a sin θ s = 1 a cot θ The equilibrium condition, in terms of θ, is F net = 4 a k e Qq cos θ sin θ Thus the equilibrium value of θ is cos θ sin θ( 3 + cot θ) = 1. 1 ( 3 + cot θ) = 0 One method for solving for θ is to tabulate the left side. To three significant figures the value of θ corresponding to equilibrium is The distance from the origin to the equilibrium (a) The distance from each corner to the center of the square is ( L ) + ( L ) = L The distance from each positive charge to Q is then z + L. Each positive charge exerts a force directed along the line joining q and Q, of magnitude The line of force makes an angle with the z-axis whose cosine is k e Qq z + L z z + L The four charges together exert forces whose x and y components 4k add to zero, while the z-components add to F = e Qqz z + L 3k

27 Chapter 3 Solutions 7 For z << L, the magnitude of this force is F z 4k e Qqz L 3 = 4 3 k e Qq L 3 z = ma z Therefore, the object s vertical acceleration is of the form a z = ω z with ω = 4 3 k e Qq ml 3 = k eqq 18 ml 3 Since the acceleration of the object is always oppositely directed to its excursion from equilibrium and in magnitude proportional to it, the object will execute simple harmonic motion with a period given by T = π ω = π ml ( 18) 14 k e Qq = 3 3 π ml ( 8) 14 k e Qq 3.66 (a) The total non-contact force on the cork ball is: F = qe + mg = m g + qe m, which is constant and directed downward. Therefore, it behaves like a simple pendulum in the presence of a modified uniform gravitational field with a period given by: T = π L g + qe m = π m ( N/C) 9.80 m / s C kg = s Yes. Without gravity in part (a), we get T = π L qe m T = π m ( C) N/C kg = s (a.8% difference) (a) Due to symmetry the field contribution from each negative charge is equal and opposite to each other. Therefore, their contribution to the net field is zero. The field contribution of the +q charge is y E = k e q r = k e q ( 3 a 4) = 4k q e 3a x in the negative y direction, i.e., E = 4k e q 3a j 000 by Harcourt, Inc. All rights reserved.

28 8 Chapter 3 Solutions If F e = 0, then E at P must equal zero. In order for the field to cancel at P, the 4q must be above + q on the y-axis. k Then, E = 0 = e q ( 1.00 m) + k e(4q) y, which reduces to y = 4.00 m. Thus, y =±.00 m. Only the positive answer is acceptable since the 4q must be located above + q. Therefore, the 4q must be placed.00 meters above point P along the + y axis The bowl exerts a normal force on each bead, directed along the radius line or at 60.0 above the horizontal. Consider the free-body diagram of the bead on the left: ΣF y = nsin 60.0 mg = 0, or n = mg sin 60.0 Also, ΣF x = F e + ncos 60.0 =0, n or k e q R = ncos 60.0 = mg tan 60.0 = mg 3 F e 60.0 Thus, q = R mg k e 3 1 mg 3.69 (a) There are 7 terms which contribute: 3 are s away (along sides) 3 are s away (face diagonals) and sin θ = 1 = cos θ 1 is 3 s away (body diagonal) and sinφ = 1 3 The component in each direction is the same by symmetry. F = k eq s (i + j + k) = k e q (1.90)(i + j + k) s F = F x + F y + F z = 3.9 k e q s away from the origin

29 Chapter 3 Solutions (a) Zero contribution from the same face due to symmetry, opposite face contributes 4 k e q r sin φ where r = s + s + s = 1. 5 s = 1. s E = 4 k e qs 4 k r 3 = e q (1.) 3 s =.18 k e q s sin φ = s/r The direction is the k direction. k *3.71 de = e dq x m xi m j x + ( m) = k eλ ( xi m j)dx x m [ ] 3 de m y x E = de all charge = k e λ m x=0 ( xi m j)dx [ x + ( m) ] 3 x dq E = k e λ +i x + ( m) m + ( m)j x ( m) x + ( m) m E = N m C C i m [ m + j( 6.4 0) m ] E = ( i j ) 10 3 NC= ( i j ) kn C 3.7 By symmetry E x = 0. Using the distances as labeled, q E y = k e (a + y ) sin θ + q (a + y ) q sin θ y But sin θ = y y, so E = E (a +y y = k e q 1 ) (a + y ) 3 y Expand (a +y ) 3 as (a + y ) 3 = y 3 (3 )a y Therefore, for a << y, we can ignore terms in powers higher than, and we have E = k e q 1 y 3 a y 4 1 y or E = k e 3qa y 4 j 000 by Harcourt, Inc. All rights reserved.

30 30 Chapter 3 Solutions 3.73 The field on the axis of the ring is calculated in Example 3.8, E = E x = The force experienced by a charge q placed along the axis of the ring is x F = k e Qq (x + a ) 3/ and when x << a, this becomes F = k e Qq a 3 x This expression for the force is in the form of Hooke's law, with an effective spring constant of k = k e Qq a 3 k e xq (x + a ) 3/ Since ω = π f = km, we have f = 1 π k e Qq ma The electrostatic forces exerted on the two charges result in a net torque τ = Fa sin θ = Eqa sin θ. For small θ, sin θ θ and using p = qa, we have The torque produces an angular acceleration given by Combining these two expressions for torque, we have This equation can be written in the form τ = Epθ. τ = Iα = I d θ dt d θ dt + Ep I θ =0 d θ dt = ω θ where ω = Ep I This is the same form as Equation and the frequency of oscillation is found by comparison with Equation 13.19, or f = 1 π pe I = 1 π qae I