Chapter 28: Magnetic Fields

In a Hall-effect experiment, a current of $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{A}}$sent lengthwise through a conductor$\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{cm}}$wide,$\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{cm}}$ long, and$\mathbf{10}\mathbf{}\mathbf{\text{μm}}$ thick produces a transverse (across the width) Hall potential difference of$\mathbf{10}\mathbf{}\mathbf{\text{μV}}$ when a magnetic field of$\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{\text{T}}$is passed perpendicularly through the thickness of the conductor. From these data, find (a) the drift velocity of the charge carriers and (b) the number density of charge carriers. (c) Show on a diagram the polarity of the Hall potential difference with assumed current and magnetic field directions, assuming also that the charge carriers are electrons.

Question: A wire lying along an xaxis from$\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$to $\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{\text{m}}$carries a current of $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{A}}$ in the positive xdirection. The wire is immersed in a nonuniform magnetic field that is given by localid="1663064054178" $\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{00}\frac{\mathbf{T}}{{\mathbf{m}}^{\mathbf{2}}}\mathbf{)}{\mathbf{x}}^{\mathbf{2}}\hat{\mathbf{i}}\mathbf{-}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{600}\frac{\mathbf{T}}{{\mathbf{m}}^{\mathbf{2}}}\mathbf{)}{\mathbf{x}}^{\mathbf{2}}\hat{\mathbf{j}}$ In unit-vector notation, what is the magnetic force on the wire?

Question: At one instant$\overrightarrow{\mathbf{v}}\mathbf{=}\left(-2.00\hat{i}+4.00\hat{j}-6.00\hat{k}\right)\mathbf{}\mathbf{\text{m/s}}$, is the velocity of a proton in a uniform magnetic field$\overrightarrow{\mathbf{B}}\mathbf{=}\left(2.00\hat{i}-4.00\hat{j}+8.00\hat{k}\right)\mathit{m}\mathit{T}$At that instant, what are (a) the magnetic force acting on the proton, in unit-vector notation, (b) the angle between$\overrightarrow{\mathbf{v}}$ and $\overrightarrow{\mathbf{F}}$, and (c) the angle between$\overrightarrow{\mathbf{v}}$ and $\overrightarrow{\mathbf{B}}$?

Question: An electron has velocity $\overrightarrow{\mathbf{v}}\mathbf{=}\left(32\hat{i}+40\hat{j}\right)\mathbf{\text{km/s}}$ as it enters a uniform magnetic field$\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{60}\hat{\mathbf{i}}\mathbf{ }\mathit{\mu }\mathit{T}$ What are (a) the radius of the helical path taken by the electron and (b) the pitch of that path? (c) To an observer looking into the magnetic field region from the entrance point of the electron, does the electron spiral clockwise or counterclockwise as it moves?

Question: Figure 28-56 shows a homopolar generator, which has a solid conducting disk as rotor and which is rotated by a motor (not shown). Conducting brushes connect this emf device to a circuit through which the device drives current. The device can produce a greater emf than wire loop rotors because they can spin at a much higher angular speed without rupturing. The disk has radius $\mathit{R}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{250}\mathbf{ }\mathbf{\text{m}}$and rotation frequency $\mathit{f}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4000}\mathbf{}\mathbf{\text{Hz}}$, and the device is in a uniform magnetic field of magnitude $\mathit{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{mT}}$that is perpendicular to the disk. As the disk is rotated, conduction electrons along the conducting path (dashed line) are forced to move through the magnetic field. (a) For the indicated rotation, is the magnetic force on those electrons up or down in the figure? (b) Is the magnitude of that force greater at the rim or near the center of the disk? (c)What is the work per unit charge done by that force in moving charge along the radial line, between the rim and the center? (d)What, then, is the emf of the device? (e) If the current is $\mathbf{50}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{A}}$, what is the power at which electrical energy is being produced?

Question: In Fig. 28-57, the two ends of a U-shaped wire of mass m=10.0gand length L=20.0cmare immersed in mercury (which is a conductor).The wire is in a uniform field of magnitude B=0.100T. A switch (unshown) is rapidly closed and then reopened, sending a pulse of current through the wire, which causes the wire to jump upward. If jump height h=3.00m, how much charge was in the pulse? Assume that the duration of the pulse is much less than the time of flight. Consider the definition of impulse (Eq. 9-30) and its relationship with momentum (Eq. 9-31). Also consider the relationship between charge and current (Eq. 26-2).

Question: An electric field of$\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{kV}\mathbf{/}\mathbf{m}$and a perpendicular magnetic field of $\mathbf{0}\mathbf{.}\mathbf{400}\mathbf{}\mathbf{T}$act on a moving electron to produce no net force. What is the electron’s speed?

Figure 28-28 shows the path of an electron in a region of uniform magnetic field. The path consists oftwo straight sections, each between a pair of uniformly charged plates, and two half-circles. Which plate isat the higher electric potential in

(a) the top pair of plates and

(b) the bottom pair?

(c) What is the direction of the magnetic field?

In Fig 28-32, an electron accelerated from rest through potential difference ${\mathbf{V}}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{kV}}$enters the gap between two parallel plates having separation $\mathbf{d}\mathbf{=}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{\text{mm}}$and potential difference ${\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{100}\mathbf{ }\mathbf{\text{V}}$. The lower plate is at the lower potential. Neglect fringing and assume that the electron’s velocity vector is perpendicular to the electric field vector between the plates. In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap?

a) In Checkpoint 5, if the dipole moment is rotated from orientation 2 to orientation 1 by an external agent, is the work done on the dipole by the agent positive, negative, or zero?

(b) Rank the work done on the dipole by the agent for these three rotations, greatest first.$\mathbf{}\mathbf{\text{2}}\mathbf{\to }\mathbf{\text{1,2}}\mathbf{\to }\mathbf{\text{4,2}}\mathbf{\to }\mathbf{\text{3}}$