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 2 7 4 The Electric Field Constant Voltage Sphere If the sphere is kept at constant voltage V0 the image charge q qRID at distance b R 2 D from the sphere center still keeps the sphere at zero potential To raise the potential of the sphere to V0 another image charge 26 Qo 41reoRVo must be placed at the sphere center as in Figure 2 29b The force on the sphere is then fq 4veo qR D D b 2 Qo 2D 27 27 PROBLEMS Section 2 1 1 Faraday s ice pail experiment is repeated with the following sequence of steps i A ball with total charge Q is brought inside an insulated metal ice pail without touching ii The outside of the pail is momentarily connected to the ground and then disconnected so that once again the pail is insulated iii Without touching the pail the charged ball is removed a Sketch the charge distribution on the inside and outside of the pail during each step b What is the net charge on the pail after the charged ball is removed 2 A sphere initially carrying a total charge Q is brought into momentary contact with an uncharged identical sphere a How much charge is on each sphere b This process is repeated for N identical initially uncharged spheres How much charge is on each of the spheres including the original charged sphere c What is the total charge in the system after the N contacts Section 2 2 3 The charge of an electron was first measured by Robert A Millikan in 1909 by measuring the electric field necessary to levitate a small charged oil drop against its weight The oil droplets were sprayed and became charged by frictional electrification Problems Total charge q R 111 StEo A spherical droplet of radius R and effective mass density p carries a total charge q in a gravity field g What electric field Eoi will suspend the charged droplet Millikan found by this method that all droplets carried integer multiples of negative charge e 1 6 x 10 coul 4 Two small conducting balls each of mass m are at the end of insulating strings of length I joined at a point Charges are g placed on the balls so that they are a distance d apart A charge 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 of radius 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 of the atom if the charge Q1 is that of an electron and Q2 Ze is the nuclear charge where Z is the number of protons According to the postulates of quantum mechanics the angular momentum L of the electron must be quantized L mvR nh 2i 34 n 1 2 3 joule sec is Planck s constant What are where h 6 63 x 10 the allowed values of R 112 The Electric Field c For the hydrogen atom Z 1 what is the radius of the smallest allowed orbit and what is the electron s orbital velocity 6 An electroscope measures charge by the angular deflection of two identical conducting balls suspended by an essentially weightless insulating string of length 1 Each ball has mass M in the gravity field g and when charged can be considered a point charge I Q 2 Q 2 A total charge Q is deposited on the two balls of the electroscope The angle 0 from the normal obeys a relation of the form tan 0 sin 2 0 const What is the constant 7 Two point charges qi and q2 in vacuum with respective masses mi and m 2 attract or repel each other via the coulomb force B mi q1 B 8 ri r m2 q2 0 s s a Write a single differential equation for the distance between the charges r r 2 rl What is the effective mass of the charges Hint Write Newton s law for each charge and take a mass weighted difference b If the two charges are released from rest at t 0 when a distance ro from one another what is their relative velocity v dr dt as a function of r Hint dv dv dr dv dt dr dt dr d 1 dr 2 II Problems 113 c What is their position as a function of time Separately consider the cases when the charges have the same or opposite polarity Hint r Let U2 du sin du a 2 2 u S2 In 2 u a d If the charges are of opposite polarity at what time will they 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 placed in a viscous medium the velocity rather than the acceleration is proportional to the force f 1V1 f 9 2 V2 f 2 where 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 is injected at x 0 into a region of uniform electric field E Eoi A screen is placed at the position x L At what height h does the charge hit the screen Neglect gravity q i2 hf q 9 A pendulum with a weightless string of length I has on its end a small sphere with charge q and mass m A distance D Q I Q 114 The ElectricField away on either side of the pendulum mass are two fixed spheres each carrying a charge Q The three spheres are of sufficiently small size that they can be considered as point charges and masses a Assuming the pendulum displacement f to be small 6 D show that Newton s law can be approximately written as dt What is 0w Hint sin 06 1 1 D f 2 1 D 2f 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 with time Y 10 Charges Q Q and q lie on the corners of an equilateral triangle 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 altitude at P a 11 Find the electric field along the z axis due to four equal magnitude point …
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