# 2 Coulomb s Law and Electric Field 23.13, 23.17, 23.23, 23.25, 23.26, 23.27, 23.62, 23.77, 23.78

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1 College of Engineering and Technology Department of Basic and Applied Sciences PHYSICS I Sheet Suggested Problems 1 Vectors 2 Coulomb s Law and Electric Field 23.13, 23.17, 23.23, 23.25, 23.26, 23.27, 23.62, 23.77, Electric Flux and Gauss s Law 24.15, 24.19, 24.23, 24.30, 24.47, 24.48, 24.60, 24.61, Electric Potential Energy and Electric Potential 25.3, 25.12, 25.13, 25.16, 25.18, 25.23, 25.27, 25.31, Capacitance 26.23, 26.28, 26.29, 26.48, 26.53, 26.57, 26.61, 26.65, 26.67, DC Circuits 28.25, 28.27, Magnetic Force and Magnetic Torque 29.8, 29.11, 29.12, 29.22, 29.31, 29.39, 29.44, Sources of Magnetic Field 9 Faraday s Law of Induction 31.46, 31.53, Additional Exercises 1 Vectors Question 1.1. If a person moves from the origin to point A with x = 4, y = 3, z = 5: a) Write down the displacement vector in cartesian form. b) What is the unit vector in the direction of the displacement? Question 1.2. For the following vectors: Find the unit vector Â in cartesian form. Find the polar forms of the vectors. a) A = 3î 5ĵ [Ans: Â = î ĵ, A = 5.831, ] 1

2 b) A = 7î 15ĵ [Ans: Â = 0.423î 0.906ĵ, A = , ] c) A = 3î+4ĵ [Ans: Â = 0.6î+0.8ĵ, A = 5, ] Question 1.3. Find the cartesian forms of the following vectors: a) A = 5, 30 [Ans: A = 4.33î+2.5ĵ] b) A = 10, 257 [Ans: A = 2.25î 9.74ĵ] c) A = 34, 55 with the y-axis in the fourth-quadrant [Ans: A = 27.85î 19.50ĵ] d) A = 55, 25 with the x-axis in the third-quadrant [Ans: A = 49.85î 23.24ĵ] Question 1.4. A person moves from point A( 2,1,7) to point B(4, 3, 5). a) Find the displacement vector in cartesian form. [Ans: 6î 4ĵ 12ˆk] b) What is the magnitude of the displacement vector? [Ans: 14] c) Find the unit vector in the direction of the displacement. [Ans: 0.428î 0.286ĵ 0.857ˆk] Question 1.5. For the following force vectors acting on a body: F 1 = 5î 10ĵ(N) F2 = 15î+5ĵ(N) F3 = 4, 45 (N), what is the resultant force vector. [Ans: 7.17î 2.17ĵ(N)] Question 1.6. The vectors A and B are given as: A = 3î+4ĵ B = 4 î 3ĵ a) Find the magnitudes of both A and B. [Ans: A = 5, B = 5] b) Put the following vectors in polar forms: C = A 5 B [Ans: 29.83, ] D = 2 A+ B [Ans: 5.385, ] Question 1.7. [Scalar Product] Find the angle between the following vectors: a) A = 12î 15ĵ +25ˆk, B = 12î+20ĵ +30ˆk [Ans: 75.2 ] b) A = 10î 25ĵ 50ˆk, B = 12î+20ĵ [Ans: ] c) A = 10î 50ˆk, B = 12î [Ans: ] 2

3 2 Coulomb s Law and Electric Field Question 2.1. Three charges, each of magnitude 3nC, are at separate corners of a square of side 5 cm. The two charges at opposite corners are positive, and the other charge is negative. Find the magnitude of the force exerted by these charges on a fourth charge q = 3nC at the remaining corner. [Ans: 29.6 µn] Question 2.2. A charge Q 1 = 100 k C is located at x = 10, y = 0, z = 0. A charge Q 2 = C is located at x = 10, y = 8, z = 40. Let the coordinates of a point B be x = 10, y = 8, k z = 0. Let A be the origin. a) CalculatetheelectricfieldvectoratthepointA: E A. [Ans: E A = 6î 4ĵ 20ˆk(N/C)] b) If a charge of Q 3 = 2nC is placed at the origin A, what is the electric force vector F on Q 3? [Ans: F Q3 = 12î 8ĵ 40ˆk(nN)] z Q =37044/k 2 (10,8,40) Q = 100/k 1 A (10,0,0) B (10,8,0) y x Question 2.3. A point charge of 5µC is located at x = 3cm and a second point charge of 8µC is located at x = 4cm. Where should a third charge of 6µC be placed so that the electric field at x = 0 is zero? [Ans: (2.38cm,0)] Question 2.4. Four charges of equal magnitude are arranged at the corners of a square of side L as shown in the figure. a) Find the magnitude and direction of the ( force exerted ) on the charge in the lower left corner by the other charges. [Ans: k q2 1 1 L 2 2 (î+ĵ)] 2 b) Show that the electric field at the midpoint of one of the sides of the square is directed along that side toward the negative charge and has a magnitude E given by: ( E = k 8q ) 5 1 L

4 3 q (0,L) A(L/2,L) q (L,L) (0,0) q q (L,0) Question 2.5. A point charge of 2.5µC is located at the origin. A second point charge of 6µC is at x = 1m, y = 0.5m. a) Find the x and y coordinates of the position where the electric field is zero. [Ans: ( 1.82 m, 0.91 m)] b) Find the x and y coordinates where the force on an: electron, proton, is zero. [Ans: ( 1.82 m, 0.91 m)] 3 Electric Flux and Gauss s Law Question 3.1. Consider a uniform electric field E = 2î 3ĵ + 5ˆk(kN/C). What is the flux of this field through an area of 5cm 2 with a normal vector of 7î 12ĵ + 17ˆk? [Ans: N.m 2 /C] Question 3.2. A uniform electric field E 4 1 = 9 10 N/Cî, is directed out of one face of a parallelopiped, and another uniform electric field, E2 4 = N/Cî, is directed into the opposite face, as shown in the figure. Assuming that there are no other electric-field lines crossing the surfaces of the parallelopiped, determine the net charge contained within, and the average volume charge density. [Ans: Q inside = nC, ρ = 1.77µC/m 3 ] Top view 6cm Side view E 1 E cm 10 cm 4

5 Question 3.3. The electric field vector at the surface of the earth is 200 N/C toward the center. The field vector is reduced to 20N/C at a height of 1400m. What is the average volumetric charge density in the air enclosed between the earth surface and the height given? [Ans: C/m 3 ] Question 3.4. An infinite charged plane in the z x plane with a surface charge density of 70nC/m 2, a second infinite plane carrying a surface charge density of 50nC/m 2, intersects the z x plane at the z-axis and makes an angle of 40 with the z x plane as shown in the figure. Find the electric field at the point A (x = 6m,y = 2m), and the point B (x = 6m,y = 8m). Three dimensional view Side view B y σ 2 y σ 2 A x θ σ1 x θ z σ 1 [Ans: E A = î ĵ(N/C), E B = î ĵ(N/C)] Question 3.5. An infinite plane of charge with surface charge density σ 1 = 3µC/m 2 is parallel tothexz planeaty = 0.6m. Asecondinfiniteplaneofchargewithsurfacechargedensityσ 2 = 2µC/m 2 isparallel totheyz planeatx = 1m. Asphere ofradius1mwithitscenter inthexy plane at the intersection of the two charged planes (x = 1m,y = 0.6m) has a surface charge density σ 3 = 3µC/m 2. Find the magnitude and direction of the electric field on the x axis at (a) x = 0.4m and (b) x = 2.5m. [Ans: E 5 x=0.4m = (N/C)î (N/C)ĵ, 5 E x=2.5m = (N/C)î (N/C)ĵ] Question 3.6. An infinitely long, thick, non-conducting cylindrical shell of inner radius a and outer radius b has a uniform charge density ρ. Find the electric field everywhere. [Ans: E r b = ρ(b2 a 2 ) 2ǫ 0 r, E a r b = ρ(r2 a 2 ), E 2ǫ 0 r r a = 0] 4 Electric Potential Energy and Electric Potential Question 4.1. Three point charges 5µC, 8µC, and 6µC are placed at the vertices of a rectangle as shown in the figure. a) Calculate the electric potential at points a with x = 5m, y = 9m and b located at the center of the intersection of the diagonals of the rectangle. [Ans: V A = 7.6kV, V B = 25.2kV] b) What is the work done needed to move a charge of 4µC from a to b? [Ans: W ex A B = J] 5

6 c) What is the work needed to assemble the three charges q 1, q 2 and q 3? [Ans: U = J] q = 8µC 2 y=3m a(5,9) b q = 5µC 1 q = 6µ C 3 x = 4m Question 4.2. Two spherical shells of charge are as shown in the figure. Shell 1 has its center in the origin, with R 1 = 5m, Q 1 = 100/k, and shell 2 has its center C at x = 2m, with R 2 = 2m, Q 2 = 128 2/k. Find: a) The electric field at point A with x = 6m, y = 8m. [Ans: E A = 1.6î+1.8ĵ(N/C)] b) The electric field at point B with x = 0m, y = 2m. [Ans: E B = 16( î+ĵ)(n/c)] c) The electric potential at point A. [Ans: V A = 26V] d) The electric potential at point B. [Ans: V B = 84V] e) The work needed to move a charge of 5nC from A to B. [Ans: 290nJ] A Q 1 R 1 B Q 2 C R 2 Question 4.3. A spherical conductor of radius R 1 is charged to 20kV. When it is connected by a long, fine wire to a second conducting sphere far away, its potential drops to 12kV. What is the radius of the second sphere? [Ans: R 2 = 2 3 R 1] 5 Capacitors Question 5.1. A parallel-plate capacitor has square plates of size 10 cm and a separation of d = 4mm. A dielectric slab of constant κ = 2 has the same area as the plates. 6

7 a) What is the capacitance without the dielectric? [Ans: 22.1 pf] b) What is the capacitance if the dielectric slab fills the space between the plates? [Ans: 44.2pF] c) What is the capacitance if a dielectric slab of thickness 3mm is inserted into the 4mm gap? [Ans: 35.36pF] Question 5.2. In the following circuit of capacitors, where C 1 = 3µF, C 2 = 1.5µF and C 3 = 2µF: a) Find the equivalent capacitance between AC. [Ans: µf] b) If a battery of 12V is connected between A and C, find the total charge on the capacitors C 1, C 2 and C 3. [Ans: Q 1 = µC, Q 2 = µC, Q 3 = 16.61µC] A C 3 C 2 C 1 C Question 5.3 (Final 2/1/2006). Question 5.4. Find the capacitance of the parallel-plate capacitor shown in the figure. 7

8 A d d/2 d/2 κ 1 κ 2 κ 3 [Ans: ( κ1 κ 2 + κ ) 3 C 0 ] κ 1 +κ 2 2 x/2 x/2 6 EMF and DC Circuits Question 6.1. In the circuit shown in the figure, find: a) The currents in the loops when S is open. [Ans: 2.25A, 0A] b) The currents in the loops when S is closed.[ans: 2.29A, 0.109A] (The questions below are to be answered when S is closed). c) The potential difference V AB = V B V A between A and B.[Ans: 5.7V] d) The power delivered by the 24V emf. [Ans: 54.96W] e) The power dissipated in the 5Ω resistor. [Ans: 26.22W] A 2 Ω 4 Ω S 3 Ω 24V i 1 12V i 2 12 Ω 12 Ω 6 Ω 5 Ω B Question 6.2. In the circuit shown, find a) the readings of the ammeters, b) the power delivered by the 3V emf, c) the power dissipated in the 6 Ω resistor. 8

9 1.5 Ω A 1 3V 3Ω A 2 A3 5Ω 6Ω 6V A 4 [Ans: A 1 = A, A 2 = A, A 3 = A, A 4 = A, P 3V = 1.61W, P 6Ω = 2.413W] 7 Magnetic Force and Magnetic Torque Question 7.1. Find the magnetic force F on a charge q = 5nC with velocity v = 3î 4ĵ + 7ˆk(m/s), in a magnetic field B = 7î 10ĵ +13ˆk(T). [Ans: F = 90î+50ĵ 10ˆk(nN)] Question 7.2. A proton moving in a circular path perpendicular to a constant magnetic field takes 1.0 µs to complete one revolution. Determine the magnitude of the field. [Ans: B = T] Question 7.3. A proton moves in a circular orbit of radius 65cm perpendicular to a uniform magnetic field of magnitude 0.75T. a) What is the period for this motion? b) Find the speed of the proton. c) Find the kinetic energy of the proton. Question 7.4 (Question 28.27, page 878). A proton with speed v enters a region of uniform magnetic field B = 0.6T, which is into the page as shown in the figure. Suppose that the distance d = 0.4cm and θ = 24. Find the speed v and the angle φ. [Ans: v = m/s, φ = 24 ] 9

10 B in φ d θ Question 7.5. The wire segment in the figure carries a current of 1.8A from a to b in the presence of a magnetic field of B = 1.2ˆkT. Find the total force on the wire. [Ans: F = î(N)] z b 4cm a 3cm y x Question 7.6. A 10cm length of wire carries a current of 2.0A in the positive x-direction. The force on this wire due to the presence of a magnetic field B is F = 3ĵ + 2ˆk(N). If this wire is now rotated so that the current flows in the positive y-direction, the force on the wire is F = 3î 2ˆk(N). Determine the magnetic field vector B. 8 Magnetic Field from an Electric Current Question 8.1. Three long, parallel, straight wires pass through the corners of an equilateral triangle of sides 10cm as shown in the figure. If each current is 15A, find: a) The force per unit length on the upper wire. [Ans: F = 780ĵ(µN)] b) The magnetic field B at the upper wire due to the two lower wires. [Ans: B = 52î(µT)] 10cm 10cm 10cm 10

11 Question 8.2. Two long parallel conductors each carry a 2A current. The figure shows an end-view of the conductors, with both currents coming out of the page. a) What is the force per unit length acting on the two wires? Is it attractive or repulsive? b) Determine the magnitude and direction of the magnetic field at point B. 2m 2m 2m A Question 8.3. A solenoid 2.7m long has a radius of 0.85cm and 600 turns. It carries a current I of 2.5A. What is the approximate magnetic field B on the axis of the solenoid? [Ans: 0.7 mt] Question 8.4 (Ampère s Law). A wire of radius 0.5cm carries a current of 100A that is uniformly distributed over its cross-sectional area. Find B a) 0.1cm from the center of the wire. [Ans: 0.8mT] b) at the surface of the wire. [Ans: 4mT] c) at a point outside the wire 0.2cm from the surface of the wire. [Ans: 2.86mT] Question 8.5 (Ampère s Law). A tightly wound toroid of inner radius 1cm and outer radius 2cm has 1000 turns of wire and carries a current of 1.5A. a) What is the magnetic field at a distance of 1.1cm from the center? [Ans: 0.027T] b) What is the field 1.5cm from the center? [Ans: 0.02T] 9 Magnetic Induction (Faraday s Law) Question 9.1. A magnetic field B = 3tî 5sin(5πt)ĵ + 17t3ˆk(T) is applied at a coil with area A = 3m 2, and with a normal vector n = 4î+5ĵ 7ˆk. Find the induced emf at t = 4s. [Ans: E ind = 1.925kV] Question 9.2. The rectangular coil in the figure has 80 turns, is 25cm wide and 30cm long, and is located in a magnetic field B = 1.4T directed out of the page as shown, with only half of the coil in the region of the magnetic field. The resistance of the coil is 24Ω. Find the magnitude and direction of the induced current if the coil is moved with a speed of 2m/s (a) to the right, (b) up, (c) to the left, and (d) down. [Ans: right 0A, up 2.33A (clockwise), left 0 A, down 2.33 A (counter-clockwise)] 11

12 B out Question 9.3. Suppose the coil in Question 9.2 is rotated about its vertical centerline at constant angular velocity of 2 rad/s. Find the induced current as a function of time. [Ans: I(t) = 0.35 sin(2t)] Question 9.4. [Final 13/6/2004] A Constants B Theory Coulomb constant k N.m 2 /C 2 Fundamental charge e C Mass of electron m e Kg Mass of proton m p Kg Permittivity of free space ǫ C 2 /N.m 2 Permeability of free space µ 0 4π 10 7 N/A 2 Electric field due to an infinite plane of charge. Electric field due to a spherical shell of charge. 12

13 Electric field due to an infinite line of charge. Electrostatic potential energy of a charge configuration. Capacitance of a parallel plate capacitor. Microscopic model of current: I = nqva (p.773). Circular motion of a charge in a magnetic field. Motional emf. Comparison between electric force and magnetic force. Units of all quantities studied in the course. All equations studied in the course. 13

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