Problems set # 7 Physics 169 March 31, 2015

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1 Prof. Anchordoqui Problems set # 7 Physics 69 March 3, 05. (i) Determine the initial direction of the deflection of charged particles as they enter the magnetic fields as shown in Fig.. (ii) At the Equator near Earths surface, the magnetic field is approximately 50.0 µt northward and the electric field is about 00 N/C downward in fair weather. Find the gravitational, electric, and magnetic forces on an electron with an instantaneous velocity of m/s directed to the east in this environment. (iii) Consider an electron near the Earths equator. In which direction does it tend to deflect if its velocity is directed: (a) downward, (b) northward, (c) westward, or (d) south-eastward?. A particle of charge e is moving with an initial velocity v when it enters midway between two plates where there exists a uniform magnetic field pointing into the page, as shown in Fig.. You may ignore effects of the gravitational force. (i) Is the trajectory of the particle deflected upward or downward? (ii) What is the magnitude of the velocity of the particle if it just strikes the end of the plate? 3. The entire x y plane to the right of the origin O is filled with a uniform magnetic field of magnitude B pointing out of the page, as shown in Fig. 3. Two charged particles travel along the negative x axis in the positive x direction, each with velocity v, and enter the magnetic field at the origin O. The two particles have the same mass m, but have different charges, q and q. When propagate thorugh the magnetic field, their trajectories both curve in the same direction (see sketch in Fig. 3), but describe semi-circles with different radii. The radius of the semi-circle traced out by particle is exactly twice as big as the radius of the semi-circle traced out by particle. (i) Are the charges of these particles positive or negative? Explain your reasoning. (ii) What is the ratio q /q? 4. Shown in Fig. 4 are the essentials of a commercial mass spectrometer. This device is used to measure the composition of gas samples, by measuring the abundance of species of different masses. An ion of mass m and charge q = +e is produced in source S, a chamber in which a gas discharge is taking place. The initially stationary ion leaves S, is accelerated by a potential difference V > 0, and then enters a selector chamber, S, in which there is an adjustable magnetic field B, pointing out of the page and a deflecting electric field E, pointing from positive to negative plate. Only particles of a uniform velocity v leave the selector. The emerging particles at S, enter a second magnetic field B, also pointing out of the page. The particle then moves in a semicircle, striking an electronic sensor at a distance x from the entry slit. Express your answers to the questions below in terms of E E, e, x, m, B B, and V. (i) What magnetic field B in the selector chamber is needed to insure that the particle travels straight through? (ii) Find an expression for the mass of the particle after it has hit the electronic sensor at a distance x from the entry slit. 5. Electrons in a beam are accelerated from rest through a potential difference V. The beam enters an experimental chamber through a small hole. As shown in Fig. 5, the electron velocity vectors lie within a narrow cone of half angle φ oriented along the beam axis. We wish to use a uniform magnetic field directed parallel to the axis to focus the beam, so that all of the electrons can pass through a small exit port on the opposite side of the chamber after they travel the length d of the chamber. What is the required magnitude of the magnetic field? [Hint: Because every

2 Section 9.. Magnetic Fields and Forces Determine the initial direction of the deflection of charged particles as they enter the magnetic fields as shown in Figure P9.. (a) + (c) B in B right (b) (d) + Figure P9. Figure : Problem. B up B at 45. Consider an electron near the Earth s equator. In which direction does it tend to deflect if its velocity is directed electron passes through the same potential difference and the angle φ is small, they all require the same time interval to travel the axial distance d. 6. A circular ring of radius R lying in the xy plane carries a steady current I, as shown in the Fig. 6. What is the magnetic field at a point P on the axis of the loop, at a distance z from the center? 7. Find the magnetic field at point P due to the current distribution shown in Fig A thin uniform ring of radius R and mass M carrying a charge +Q rotates about its axis with constant angular speed ω. Find the ratio of the magnitudes of its magnetic dipole moment to its angular momentum. (This is called the gyromagnetic ratio.) 9. A wire ring lying in the xy-plane with its center at the origin carries a counterclockwise I. There is a uniform magnetic field B = Bî in the +x-direction. The magnetic moment vector µ is perpendicular to the plane of the loop and has magnitude µ = IA and the direction is given by right-hand-rule with respect to the direction of the current. What is the torque on the loop? 0. A nonconducting sphere has mass 80.0 g and radius 0.0 cm. A flat compact coil of wire with 5 turns is wrapped tightly around it, with each turn concentric with the sphere. As shown in Fig. 8, the sphere is placed on an inclined plane that slopes downward to the left, making an angle θ with the horizontal, so that the coil is parallel to the inclined plane. A uniform magnetic field of T vertically upward exists in the region of the sphere. What current in the coil will enable the sphere to rest in equilibrium on the inclined plane? Show that the result does not depend on the value of θ (a) downward eastward? 3. An electron m lar to a magn in the negati magnetic field 4. A proton trave of 37.0 with in the y dir magnetic forc 5. A proton mov B at m the z direct of the field. 6. An electron i then enters a (a) the maxi magnetic forc 7. A proton mo field of.70 T velocity and th

3 magnitude B pointing out of the page, as shown. Two charged particles travel along the negative x axis in the positive x direction, each with velocity v!, and enter the magnetic field at the origin O. The two particles have the same mass m, but have different charges, q and q. When in the magnetic field, their trajectories both curve in the same direction (see sketch), but describe semi-circles with different radii. The radius of the semi-circle traced out by particle is exactly twice as big as the radius of the semi-circle traced out by particle. Problem 3: Particle Orbits in a Uniform Magnetic Field The entire x-y plane to the right of the origin O is filled with a uniform magnetic field of magnitude B pointing out of the page, as shown. Two charged particles travel along the (a) negative Are the x axis charges in the of positive these particles x direction, Figure positive : Problem each or with negative?. velocity Explain v!, and your enter the magnetic reasoning. field at the origin O. The two particles have the same mass m, but have different charges, q and q. When!! in the! magnetic field, their trajectories both curve in the same Solution: direction (see Because sketch), FB but = qv describe! B, the semi-circles charges of these with particles different are radii. POSITIVE. The radius of the semi-circle traced out by particle is exactly twice as big as the radius of the semi-circle (b) traced What out is by the particle ratio q. / q? Solution: We first find an expression for the radius R of the semi-circle traced out by a particle with charge q in terms of q, v! v!, B, and m. The magnitude of the force on the charged particle is qvb and the magnitude of the acceleration for the circular orbit is v / R. Therefore applying Newton s Second Law yields mv qvb =. R We can solve this for the radius of the circular orbit mv R = qb Therefore (a) Are the the charges charged of these ratio particles Figure positive 3: Problem or negative? 3. Explain your reasoning. q! mv "! mv " R = # $ # $ =.! q % RB & % R B & R!! Solution: Because FB = qv! B, the charges of these particles are POSITIVE. (b) What is the ratio q / q?

4 lectric field E, pointing from positive to negative plate. Only particles of a uniform elocity v! leave the selector. The emerging particles at S, enter a second magnetic B!, also pointing out of the page. The particle then moves in a semicircle, striking an lectronic sensor at a distance x from the entry slit. Express your answers to the uestions below in terms of E! E!, e, x, m, B! B!, and! V. 58. Review Problem. A wire having a linear mass density of.00 g/cm is placed on a horizontal surface that has a coefficient of kinetic friction of The wire carries a current of.50 A toward the east and slides horizontally to the north. What are the magnitude and direction of the smallest magnetic field that enables the wire to move in this fashion? 59. Electrons in a beam are accelerated from rest through a potential difference V. The beam enters an experimental chamber through a small hole. As shown in Figure P9.59, the electron velocity vectors lie within a narrow cone of half angle oriented along the beam axis. We wish to use a uniform magnetic field directed parallel to the axis to focus the beam, so that all of the electrons can pass through a small a) What magnetic field B! exit Figure port 4: Problem on the 4. opposite side of the in the selector chamber is needed to insure tha chamber after they travel the length d of the chamber. particle travels What straight is the through? required magnitude of the magnetic field? Hint: Because every electron passes through the same potential difference and the angle is small, they all olution: We first find require an expression the same time interval for the to speed travel the of axial the distance particle d. after it 0.0 is accelerate cm. A flat co he potential difference!v, in terms of m, e, and!v. The change in tightly kinetic around energi sphere. As shown! K = (/ ) mv. The change in potential energy is! U = " e! V From conservation o d nergy,! K = "! U, we have that (/ )mv = e! V. φ o the speed is V Entrance port v = e! V m Figure P9.59 Figure 5: Problem 5. nside the selector the force on the charge is given by Exit port 60. Review Problem. A proton is at rest at the plane vertical boundary of a region! containing!! a uniform! vertical magnetic field B. An alpha Fe particle = e( E + moving v! Bhorizontally ). makes a head-on elastic collision with the proton. Immediately after the collision, both particles enter the magnetic field, moving perpendicular to the direction of the field. The long. The springs wire and the circ a magnetic field i springs stretch an tude of the magn 6. A hand-held ele Model the motor ing electric curre produced by an e sider only one in will consider mot because the mag described in Sec mates of the ma current in it, its ar that they are relat the input power t and the useful ou 63. A nonconductin an inclined plane an angle with th the inclined plan vertically upward current in the co rium on the incli depend on the va

5 m 5: Magnetic Field of a Ring of Current ular ring of radius R lying in the xy plane carries a steady current I, as sho ure below. agnetic Fields Figure 6: Problem 6. s the magnetic field at a point P on the axis of the loop, at a distance z from? netic field at point P due to the following current distribution. on: We shall use the Biot-Savart law to find the magnetic field on the symmetr Figure 7: Problem 7. e to the straight wire segments are zero at P because d s r and urce point:

6 mber. field? same ey all nce d. Exit port that they are related according to Equation 9.. Note that the input power to the motor is electric, given by I V, and the useful output power is mechanical,. 63. A nonconducting sphere has mass 80.0 g and radius 0.0 cm. A flat compact coil of wire with 5 turns is wrapped tightly around it, with each turn concentric with the sphere. As shown in Figure P9.63, the sphere is placed on an inclined plane that slopes downward to the left, making an angle with the horizontal, so that the coil is parallel to the inclined plane. A uniform magnetic field of T vertically upward exists in the region of the sphere. What current in the coil will enable the sphere to rest in equilibrium on the inclined plane? Show that the result does not depend on the value of. B ertical l magmakes iately field,. The of the ticle is hat of θ Figure P9.63 Figure 8: Problem 0. e top ft and. The 00 cm 64. A metal rod having a mass per unit length carries a current I. The rod hangs from two vertical wires in a uniform vertical magnetic field as shown in Figure P9.64. The wires make an angle with the vertical when in equilibrium. Determine the magnitude of the magnetic field. θ

P ROBL E M S. 10. A current-carrying conductor experiences no magnetic force when placed in a certain manner in a uniform magnetic field. Explain. 11.

P ROBL E M S. 10. A current-carrying conductor experiences no magnetic force when placed in a certain manner in a uniform magnetic field. Explain. 11. 918 C HAPTER 29 Magnetic Fields 10. A current-carrying conductor experiences no magnetic force when placed in a certain manner in a uniform magnetic field. Explain. 11. s it possible to orient a current