MIT OpenCourseWare http://ocw.mit.edu Electromagnetic Field Theory: A Problem Solving Approach For any use or distribution of this textbook, please cite as follows: Markus Zahn, Electromagnetic Field Theory: A Problem Solving Approach. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.110 The Electric Field2-7-4 Constant Voltage SphereIf the sphere is kept at constant voltage V0, the imagecharge q'= -qRID at distance b = R2/D from the spherecenter still keeps the sphere at zero potential. To raise thepotential of the sphere to V0, another image charge,Qo= 41reoRVo (26)must be placed at the sphere center, as in Figure 2-29b. Theforce on the sphere is thenfq= qR Qo (27)4veo D(D-b)2 2D (27)PROBLEMSSection 2.11. Faraday's "ice-pail" experiment is repeated with thefollowing sequence of steps:(i) A ball with total charge Q is brought inside aninsulated metal ice-pail without touching.(ii) The outside of the pail is momentarily connected tothe ground and then disconnected so that once againthe pail is insulated.(iii) Without touching the pail, the charged ball is removed.(a) Sketch the charge distribution on the inside and outsideof the pail during each step.(b) What is the net charge on the pail after the charged ballis removed?2. A sphere initially carrying a total charge Q is brought intomomentary contact with an uncharged identical sphere.(a) How much charge is on each sphere?(b) This process is repeated for N identical initiallyuncharged spheres. How much charge is on each of thespheres including the original charged sphere?(c) What is the total charge in the system after the Ncontacts?Section 2.23. The charge of an electron was first measured by Robert A.Millikan in 1909 by measuring the electric field necessary tolevitate a small charged oil drop against its weight. The oildroplets were sprayed and became charged by frictionalelectrification.Problems 111+. + Total charge q+ R ++ ++ +StEoA spherical droplet of radius R and effective mass densityp. carries a total charge q in a gravity field g. What electricfield Eoi, will suspend the charged droplet? Millikan found bythis method that all droplets carried integer multiples ofnegative charge e -1.6 x 10-coul.4. Two small conducting balls, each of mass m, are at the endof insulating strings of length I joined at a point. Charges aregplaced on the balls so that they are a distance d apart. Acharge QI is placed on ball 1. What is the charge Q2 on ball 2?5. A point charge -Qi of mass m travels in a circular orbit ofradius R about a charge of opposite sign Q2.Q2(a) What is the equilibrium angular speed of the charge-Qi?(b) This problem describes Bohr's one electron model ofthe atom if the charge -Q1 is that of an electron and Q2 = Zeis the nuclear charge, where Z is the number of protons.According to the postulates of quantum mechanics theangular momentum L of the electron must be quantized,L = mvR = nh/2i, n = 1, 2, 3, -where h = 6.63 x 10- 3 4 joule-sec is Planck's constant. What arethe allowed values of R?112 The Electric Field(c) For the hydrogen atom (Z = 1) what is the radius of thesmallest allowed orbit and what is the electron's orbital veloc-ity?6. An electroscope measures charge by the angular deflectionof two identical conducting balls suspended by an essentiallyweightless insulating string of length 1. Each ball has mass Min the gravity field g and when charged can be considered apoint charge.IQ/2 Q/2A total charge Q is deposited on the two balls of the elec-troscope. The angle 0 from the normal obeys a relation of theformtan 0 sin2 0 = constWhat is the constant?7. Two point charges qi and q2 in vacuum with respectivemasses mi and m2 attract (or repel) each other via thecoulomb force.mi, q1 m2, q2* 0<- r--ri(a) Write a single differential equation for the distancebetween the charges r = r2-rl. What is the effective mass ofthe charges? (Hint: Write Newton's law for each charge andtake a mass-weighted difference.)(b) If the two charges are released from rest at t = 0 when adistance ro from one another, what is their relative velocityv = dr/dt as a function of r? Hint:dv dv dr dv d 1 )dt dr dt dr dr 2^··_IIBB8ss~Problems 113(c) What is their position as a function of time? Separatelyconsider the cases when the charges have the same oropposite polarity. Hint:Let = /rU2 du sin2 2 audu _ -In• u+ a)S2 2(d) If the charges are of opposite polarity, at what time willthey collide? (Hint: If you get a negative value of time,check your signs of square roots in (b).)(e) If the charges are taken out of the vacuum and placedin a viscous medium, the velocity rather than the accelerationis proportional to the forcef1V1 = f , 92V2 =f2where 1and 32 are the friction coefficients for each charge.Repeat parts (a)-(d) for this viscous dominated motion.8. A charge q of mass m with initial velocity v= voi, isinjected at x =0 into a region of uniform electric field E =Eoi,. A screen is placed at the position x = L. At what height hdoes the charge hit the screen? Neglect gravity.hf9. A pendulum with a weightless string of length I has on itsend a small sphere with charge q and mass m. A distance DQ I Qq~i2q114 The ElectricFieldaway on either side of the pendulum mass are two fixedspheres each carrying a charge Q. The three spheres are ofsufficiently small size that they can be considered as pointcharges and masses.(a) Assuming the pendulum displacement f to be small(6<< D), show that Newton's law can be approximately writtenasdtWhat is 0w?Hint:1 1 2fsin 06~1' (D f)2 D D(b) At t = 0 the pendulum is released from rest with f = 6o.What is the subsequent pendulum motion?(c) For what values of qQ is the motion unbounded withtime?Y 10. Charges Q, Q, and q lie on the corners of an equilateraltriangle with sides of length a.(a) What is the force on the charge q?(b) What must q be for E to be zero half-way up the altitudeat P?----'a --11. Find the electric field along the z axis due to four equalmagnitude point charges q placed on the vertices of a squarewith sides of length a in the xy plane centered at the origin.a i-Problems 115when:(a) the charges have the same polarity, q = q2= q3 = q4-4q;(b) the charges alternate in polarity, ql = q3 q, q2 = q4-q;(c) the charges are q, = q2 q, qs3 = q4-.Section 2.312. Find the
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