Electromagnetic Fields: Moving Charges and Time-Dependent Flux - Prof. Alfred L. Ritter, Exams of Physics

Download Electromagnetic Fields: Moving Charges and Time-Dependent Flux - Prof. Alfred L. Ritter and more Exams Physics in PDF only on Docsity! Physics 2306 Fall 2007 Third Exam 1) What is the first Maxwell equation? A. There are no magnetic monopoles. B. Moving charges and time-dependent electric flux create magnetic fields. C. Moving charges create electric fields. D. Time-dependent magnetic flux creates conservative electric fields. E. Time-dependent magnetic flux creates nonconservative electric fields. F. Electric charge creates conservative electric fields. G. Electric charge creates nonconservative electric fields. H. There are no magnetic dipoles 2) What is the second Maxwell equation? A. There are no magnetic monopoles. B. Moving charges and time-dependent electric flux create magnetic fields. C. Moving charges create electric fields. D. Time-dependent magnetic flux creates conservative electric fields. E. Time-dependent magnetic flux creates nonconservative electric fields. F. Electric charge creates conservative electric fields. G. Electric charge creates nonconservative electric fields. H. There are no magnetic dipoles 3) What is the third Maxwell equation? A. There are no magnetic monopoles. B. Moving charges and time-dependent electric flux create magnetic fields. C. Moving charges create electric fields. D. Time-dependent magnetic flux creates conservative electric fields. E. Time-dependent magnetic flux creates nonconservative electric fields. F. Electric charge creates conservative electric fields. G. Electric charge creates nonconservative electric fields. H. There are no magnetic dipoles 4) What is the fourth Maxwell equation? A. There are no magnetic monopoles. B. Moving charges and time-dependent electric flux create magnetic fields. C. Moving charges create electric fields. D. Time-dependent magnetic flux creates conservative electric fields. E. Time-dependent magnetic flux creates nonconservative electric fields. F. Electric charge creates conservative electric fields. G. Electric charge creates nonconservative electric fields. H. There are no magnetic dipoles 5 and 6) A positive charge q is moving along the y- axis with a velocity v in the positive y-direction (south to north) as shown in the adjacent figure. At time t = 0, the charge is at the origin. 5) At time t = 0, what is the direction of the magnetic field, generated by the moving charge, at the point (x = 1, y = 1)? A. North B. Into the page C. Out of the page D. South E. East F. West G. Northeast H. Southwest 6) At time t = 0, what is the direction of the magnetic field, generated by the moving charge, at the point (x = −1, y = −1)? A. Northwest B. Out of the page C. Southeast D. Into the page E. East F. West G. North H. South 7) A negative charge − q is moving along the z- axis with a velocity v in the positive z-direction (out of the page) as shown in the adjacent figure. At time t = 0, the charge is at the origin. At time t = 0, what is the direction of the magnetic field, generated by the moving charge, at the point (x = 1, y = 1)? A. North B. Northwest C. Southeast D. South E. East F. West G. Northeast • (x = 1, y =1) • (x = −1, y = −1) N E S W v q • (x = 1, y =1) N (+y-axis) E (+x-axis) vout of page −q W S Use the right hand rule (thumb of right hand in direction of v and fingers curl in the direction of B). Or, for a positive charge, direction of B is v × r where r points from the charge to the point where B is being determined. For a negative charge, direction of B is opposite to v × r. Same explanation for 6 and 7. 13) A square loop of wire is in a homogeneous magnetic field that is pointing into the page as shown in the adjacent figure. The area of the loop is 1.2 m2 and the resistance of the loop is 0.060 ohms. A constant current of 2.0 A flows counterclockwise around the loop because the magnetic field is changing. What is the rate of change of the magnitude of the magnetic field (dB/dt in Tesla/second)? A. 0.50 B. −0.50 C. 1.0 D. −1.0 E. − 0.10 F. 0.10 G. − 0.20 H. 0.20 14) A current I is flowing in a wire as shown in the adjacent figure. The current is increasing with time at a constant rate (dI/dt = constant). The induced current going around the square loop A is A. increasing at a constant rate and going CW around the loop. B. increasing at a constant rate and going CCW around the loop. C. decreasing at a constant rate and going CW around the loop. D. decreasing at a constant rate and going CCW around the loop. E. constant and going CW around the loop. F. constant and going CCW around the loop. G. zero. H. infinite. B × I The changing magnetic field induces an EMF in the loop that is given by Maxwell’s third equation Induced EMF = − dΦB/dt For a homogeneous magnetic field, dΦB/dt = (dB/dt)*area of loop. The induced current in the loop is given by Ohm’s law Iinduced = Induced EMF/R ={(dB/dt)*area of loop}/R You are given Iinduced, the area of the loop, and R; therefore, you can find the magnitude of dB/dt. You can find the sign of dB/dt by Lenz’s law. The induced current is going CCW and therefore the induced magnetic field must be out of the page by the right hand rule. Since this is in the opposite direction to the applied magnetic field, by Lenz’s law, the applied magnetic must be increasing. Therefore, dB/dt is positive. 15) A rectangular loop of wire is leaving a region where the magnetic field is homogeneous, independent of time, and points out of the page (See figure below). Outside the dashed rectangle the magnetic field is zero. The loop is moving at a constant velocity to the right. At the instant of time shown in the figure, the current in the loop is A. zero. B. infinite. C. constant and going CW around the loop. D. constant and going CCW around the loop. E. increasing and going CW around the loop. F. increasing and going CCW around the loop. G. decreasing and going CW around the loop. H. decreasing and going CCW around the loop. 16 and 17) An LR circuit is shown below. The switch is closed at time t = 0. 16) Right after the switch is closed, what is Va − Vb? B • B = zero outside the dashed rectangle v ξ L R S a b The magnetic field created by the current points out of the page over the area of the square loop (The magnetic fields created by the parallel incoming and outgoing currents will cancel inside the loop leaving the magnetic field created by the short vertical section of wire). The magnetic field is proportional to the current by the equation dB = μ0 I dl × r/4πr3 Therefore, if I is increasing at a constant rate, then the magnetic flux in the square loop will be increasing at a constant rate dΦB/dt = constant By Maxwell’s third equation, an induced EMF is generated that is equal to the constant. The induced EMF generates an induced current by Ohm’s law. The induced current goes CW by Lenz’s law. The magnetic flux in the square loop is decreasing at a constant rate as the square loop leaves the region of magnetic field. By Maxwell’s third equation, an induced EMF is generated that is constant. The induced EMF generates an induced current by Ohm’s law. The induced current goes CCW by Lenz’s law. A. zero B. infinite C. ξ/R D. − ξ/R E. ξ F. − ξ G. ξ L/R H. − ξ L/R 17) When the switch has been closed for a time much longer than L/R, what is Va − Vb? A. zero B. infinite C. ξ/R D. − ξ/R E. ξ F. − ξ G. ξ L/R H. − ξ L/R 18) A current I in a long straight wire is flowing downward as shown in the adjacent figure. The current is increasing at a constant rate dI/dt = 5.0 A/sec. A loop of wire is next to the straight wire and it contains a resistance R =3.0 Ω as shown in the figure. If the mutual inductance between the loop and straight wire is 0.40 H, what is Va − Vb (in Volts)? A. Can’t do the problem without knowing the separation between the loop and the straight wire. B. Can’t do the problem without knowing the area of the loop. C. − 2.0 D. 2.0 E. −0.24 F. 0.24 G. − 6.0 H. 6.0 I a • • b R = 3.0 Ω There is no current flow in circuit right after the switch is closed because the inductor acts like an open circuit. Therefore, starting at a and going CW around the circuit Va + ξ + (0)R = Vb The inductor acts like a short circuit after a time much longer than the time constant L/R. Therefore, starting at a and going CCW around the circuit Va + I (0) = Vb ξin the loop = − M dI/dt The magnitude of the current in the loop is Iin the loop = magnitude ξin the loop/R By Lenz’s law, Iin the loop goes CCW around the loop. Therefore, starting from a and going CCW around the loop Va − Iin the loop R = Vb A. 0.11 B. 7.3E-52 C. 2.8E-22 D. 1.4E+51 E. 8.8 F. 0.32 G. 0.054 H. 4.4 24) What is the velocity (in m/s) of the electron at time t = 1.8E-7 seconds? A. − 2.0E+06 j B. 2.0E+06 j C. − 2.0E+06 k D. 2.0E+06 k E. − 1.4E+06 (i + j) F. 1.4E+06 (i + j) G. − 2.0E+06 i H. 2.0E+06 i The magnetic force provides the centripetal force on the electron so that it moves in a circle. The magnitude of the magnetic force is qvB. The centripetal acceleration is given by Newton’s second law v2/r = qvB/m Solve for r. The electron is initially moving in the positive x- direction when it enters the magnetic field. The magnetic force on the electron is FB = q v×B where q is negative. The magnetic field is in the positive z-direction so that the initial magnetic force on the electron is in the positive y-direction. Therefore, the electron goes CCW around the circle. The period of the electron motion (time to go once around the circle) is period = 2πr/v = 2πm/qB= 3.6E-07 seconds Therefore, in 1.8E-07 seconds, the electron has gone half way around the circle and is moving in the negative x-direction. The speed of the electron does not change as it goes around the circle since the magnetic force is always perpendicular to the direction of motion. Constants and equations B = μ0 q v × r/4πr3 dB = μ0 I dl × r/4πr3 ωLC = square root(1/LC) q = Qcos(ωLCt) i = ξ[1 − exp(−Rt/L)]/R area of sphere = 4πR2 volume of sphere = 4πR3/3