Chapter 28

A proton traveling at \(23.0^{\circ}\) with respect to the direction of a magnetic field of strength \(2.60 \mathrm{mT}\) experiences a magnetic force of \(6.50 \times 10^{-17} \mathrm{~N}\). Calculate (a) the proton's speed and (b) its kinetic energy in electron-volts.

A particle of mass \(10 \mathrm{~g}\) and charge \(80 \mu \mathrm{C}\) moves through a uniform magnetic field, in a region where the free-fall acceleration is \(-9.8 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}^{2} .\) The velocity of the particle is a constant \(20 \mathrm{i} \mathrm{km} / \mathrm{s},\) which is perpendicular to the magnetic field. What, then, is the magnetic field?

An electron that has an instantaneous velocity of $$ \vec{v}=\left(2.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{i}}+\left(3.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\right) \hat{\mathrm{j}} $$ is moving through the uniform magnetic field \(\vec{B}=(0.030 \mathrm{~T}) \hat{\mathrm{i}}-\) \((0.15 \mathrm{~T}) \mathrm{j} .\) (a) Find the force on the electron due to the magnetic field. (b) Repeat your calculation for a proton having the same velocity.

An alpha particle travels at a velocity \(\vec{v}\) of magnitude \(550 \mathrm{~m} / \mathrm{s}\) through a uniform magnetic field \(\vec{B}\) of magnitude \(0.045 \mathrm{~T}\). (An alpha particle has a charge of \(+3.2 \times 10^{-19} \mathrm{C}\) and a mass of \(6.6 \times 10^{-27} \mathrm{~kg} .\). The angle between \(\vec{v}\) and \(\vec{B}\) is \(52^{\circ} .\) What is the magnitude of (a) the force \(\vec{F}_{B}\) acting on the particle due to the field and (b) the acceleration of the particle due to \(\vec{F}_{B} ?\) (c) Does the speed of the particle increase, decrease, or remain the same?

An electron moves through a uniform magnetic field given by \(\vec{B}=B_{x} \hat{i}+\left(3.0 B_{x}\right) \hat{j} .\) At a particular instant, the electron has velocity \(\vec{v}=(2.0 \hat{\mathrm{i}}+4.0 \mathrm{j}) \mathrm{m} / \mathrm{s}\) and the magnetic force acting on it is \(\left(6.4 \times 10^{-19} \mathrm{~N}\right) \mathrm{k} .\) Find \(B_{x}\)

A proton moves through a uniform magnetic field given by \(\vec{B}=(10 \hat{\mathrm{i}}-20 \hat{\mathrm{j}}+30 \hat{\mathrm{k}}) \mathrm{mT} .\) At time \(t_{1},\) the proton has a velocity given by \(\vec{v}_{\rightarrow}=v_{x} \hat{i}+v_{y} \hat{j}+(2.0 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{k}}\) and the magnetic force on the proton is \(\vec{F}_{B}=\left(4.0 \times 10^{-17} \mathrm{~N}\right) \hat{\mathrm{i}}+\left(2.0 \times 10^{-17} \mathrm{~N}\right) \mathrm{j} .\) At that instant, what are (a) \(v_{x}\) and (b) \(v_{y} ?\)

An electric field of \(1.50 \mathrm{kV} / \mathrm{m}\) and a perpendicular magnetic field of \(0.400 \mathrm{~T}\) act on a moving electron to produce no net force. What is the electron's speed?

A proton travels through uniform magnetic and electric fields. The magnetic field is \(\vec{B}=-2.50 \hat{\mathrm{i}} \mathrm{mT} .\) At one instant the velocity of the proton is \(\vec{v}=2000 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s} .\) At that instant and in unit-vector notation, what is the net force acting on the proton if the electric field is (a) \(4.00 \mathrm{k} \mathrm{V} / \mathrm{m}\) (b) \(-4.00 \hat{\mathrm{k}} \mathrm{V} / \mathrm{m},\) and (c) \(4.00 \hat{\mathrm{i}} \mathrm{V} / \mathrm{m} ?\)

An ion source is producing \({ }^{6} \mathrm{Li}\) ions, which have charge \(+e\) and mass \(9.99 \times 10^{-27} \mathrm{~kg} .\) The ions are accelerated by a potential difference of \(10 \mathrm{kV}\) and pass horizontally into a region in which there is a uniform vertical magnetic field of magnitude \(B=1.2 \mathrm{~T}\). Calculate the strength of the electric field, to be set up over the same region, that will allow the \({ }^{6} \mathrm{Li}\) ions to pass through without any deflection.

A strip of copper \(150 \mu \mathrm{m}\) thick and \(4.5 \mathrm{~mm}\) wide is placed in a uniform magnetic field \(\vec{B}\) of magnitude \(0.65 \mathrm{~T},\) with \(\vec{B}\) perpendicular to the strip. A current \(i=23 \mathrm{~A}\) is then sent through the strip such that a Hall potential difference \(V\) appears across the width of the strip. Calculate \(V\). (The number of charge carriers per unit volume for copper is \(8.47 \times 10^{28}\) electrons \(\left./ \mathrm{m}^{3} .\right)\)

An alpha particle can be produced in certain radioactive decays of nuclei and consists of two protons and two neutrons. The particle has a charge of \(q=+2 e\) and a mass of \(4.00 \mathrm{u},\) where \(\mathrm{u}\) is the atomic mass unit, with \(1 \mathrm{u}=1.661 \times 10^{-27} \mathrm{~kg} .\) Suppose an alpha particle travels in a circular path of radius \(4.50 \mathrm{~cm}\) in a uniform magnetic field with \(B=1.20 \mathrm{~T}\). Calculate (a) its speed, (b) its period of revolution, (c) its kinetic energy, and (d) the potential difference through which it would have to be accelerated to achieve this energy.

An electron of kinetic energy \(1.20 \mathrm{keV}\) circles in a plane perpendicular to a uniform magnetic field. The orbit radius is \(25.0 \mathrm{~cm}\). Find (a) the electron's speed, (b) the magnetic field magnitude, (c) the circling frequency, and (d) the period of the motion.

What uniform magnetic field, applied perpendicular to a beam of electrons moving at \(1.30 \times 10^{6} \mathrm{~m} / \mathrm{s},\) is required to make the electrons travel in a circular arc of radius \(0.350 \mathrm{~m} ?\)

An electron is accelerated from rest by a potential difference of \(350 \mathrm{~V}\). It then enters a uniform magnetic field of magnitude \(200 \mathrm{mT}\) with its velocity perpendicular to the field. Calculate (a) the speed of the electron and (b) the radius of its path in the magnetic field.

Find the frequency of revolution of an electron with an energy of \(100 \mathrm{eV}\) in a uniform magnetic field of magnitude \(35.0 \mu \mathrm{T} .\) (b) Calculate the radius of the path of this electron if its velocity is perpendicular to the magnetic field.

An electron follows a helical path in a uniform magnetic field of magnitude \(0.300 \mathrm{~T}\). The pitch of the path is \(6.00 \mu \mathrm{m},\) and the magnitude of the magnetic force on the electron is \(2.00 \times 10^{-15} \mathrm{~N}\). What is the electron's speed?

A source injects an electron of speed \(v=1.5 \times 10^{7} \mathrm{~m} / \mathrm{s}\) into a uniform magnetic field of magnitude \(B=1.0 \times 10^{-3} \mathrm{~T}\). The velocity of the electron makes an angle \(\theta=10^{\circ}\) with the direction of the magnetic field. Find the distance \(d\) from the point of injection at which the electron next crosses the field line that passes through the injection point.C

A positron with kinetic energy \(2.00 \mathrm{keV}\) is projected into a uniform magnetic field \(\vec{B}\) of magnitude \(0.100 \mathrm{~T},\) with its velocity vector making an angle of \(89.0^{\circ}\) with \(\vec{B}\). Find (a) the period, (b) the pitch \(p\), and (c) the radius \(r\) of its helical path.

An electron follows a helical path in a uniform magnetic field given by \(\vec{B}=(20 \hat{\mathrm{i}}-50 \hat{\mathrm{j}}-30 \hat{\mathrm{k}}) \mathrm{mT} .\) At time \(t=0,\) the elec- tron's velocity is given by \(\vec{v}=(20 \hat{\mathrm{i}}-30 \mathrm{j}+50 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s} .\) (a) What is the angle \(\phi\) between \(\vec{v}\) and \(\vec{B} ?\) The electron's velocity changes with time. Do (b) its speed and (c) the angle \(\phi\) change with time? (d) What is the radius of the helical path?

A proton circulates in a cyclotron, beginning approximately at rest at the center. Whenever it passes through the gap between dees, the electric potential difference between the dees is \(200 \mathrm{~V}\). (a) By how much does its kinetic energy increase with each passage through the gap? (b) What is its kinetic energy as it completes 100 passes through the gap? Let \(r_{100}\) be the radius of the proton's circular path as it completes those 100 passes and enters a dee, and let \(r_{101}\) be its next radius, as it enters a dee the next time. (c) By what percentage does the radius increase when it changes from \(r_{100}\) to \(r_{101} ?\) That is, what is $$ \text { percentage increase }=\frac{r_{101}-r_{100}}{r_{100}} 100 \% ? $$

A cyclotron with dee radius \(53.0 \mathrm{~cm}\) is operated at an oscillator frequency of \(12.0 \mathrm{MHz}\) to accelerate protons. (a) What magnitude \(B\) of magnetic field is required to achieve resonance? (b) At that field magnitude, what is the kinetic energy of a proton emerging from the cyclotron? Suppose, instead, that \(B=1.57 \mathrm{~T}\). (c) What oscillator frequency is required to achieve resonance now? (d) At that frequency, what is the kinetic energy of an emerging proton?

In a certain cyclotron a proton moves in a circle of radius \(0.500 \mathrm{~m} .\) The magnitude of the magnetic field is \(1.20 \mathrm{~T}\). (a) What is the oscillator frequency? (b) What is the kinetic energy of the proton, in electron-volts?

A horizontal power line carries a current of \(5000 \mathrm{~A}\) from south to north. Earth's magnetic field \((60.0 \mu \mathrm{T})\) is directed toward the north and inclined downward at \(70.0^{\circ}\) to the horizontal. Find the (a) magnitude and (b) direction of the magnetic force on \(100 \mathrm{~m}\) of the line due to Earth's field.

A wire \(1.80 \mathrm{~m}\) long carries a current of \(13.0 \mathrm{~A}\) and makes an angle of \(35.0^{\circ}\) with a uniform magnetic field of magnitude \(B=1.50 \mathrm{~T}\). Calculate the magnetic force on the wire.

\(\bullet 43\) A single-turn current loop, carrying a current of \(4.00 \mathrm{~A},\) is in the shape of a right triangle with sides \(50.0,120,\) and \(130 \mathrm{~cm} .\) The loop is in a uniform magnetic field of magnitude \(75.0 \mathrm{mT}\) whose direction is parallel to the current in the \(130 \mathrm{~cm}\) side of the loop. What is the magnitude of the magnetic force on (a) the \(130 \mathrm{~cm}\) side, (b) the \(50.0 \mathrm{~cm}\) side, and \((\mathrm{c})\) the \(120 \mathrm{~cm}\) side \(?\) (d) What is the magnitude of the net force on the loop?

Figure \(28-43\) shows a wire ring of radius \(a=1.8 \mathrm{~cm}\) that is perpendicular to the general direction of a radially symmetric, diverging magnetic field. The magnetic field at the ring is everywhere of the same magnitude \(B=3.4 \mathrm{mT},\) and its direction at the ring everywhere makes an angle \(\theta=20^{\circ}\) with a normal to the plane of the ring. The twisted lead wires have no effect on the problem. Find the magnitude of the force the field exerts on the ring if the ring carries a current \(i=4.6 \mathrm{~mA}\)

A wire \(50.0 \mathrm{~cm}\) long carries a \(0.500 \mathrm{~A}\) current in the positive direction of an \(x\) axis through a magnetic field \(\vec{B}=\) \((3.00 \mathrm{mT}) \hat{\mathrm{j}}+(10.0 \mathrm{mT}) \hat{\mathrm{k}} .\) In unit- vector notation, what is the mag- netic force on the wire?

A \(1.0 \mathrm{~kg}\) copper rod rests on two horizontal rails \(1.0 \mathrm{~m}\) apart and carries a current of \(50 \mathrm{~A}\) from one rail to the other. The coefficient of static friction between rod and rails is \(0.60 .\) What are the (a) magnitude and (b) angle (relative to the vertical) of the smallest magnetic field that puts the rod on the verge of sliding?

A long, rigid conductor, lying along an \(x\) axis, carries a current of \(5.0 \mathrm{~A}\) in the negative \(x\) direction. A magnetic field \(\vec{B}\) is present, given by \(\vec{B}=3.0 \hat{\mathrm{i}}+8.0 x^{2} \mathrm{j},\) with \(x\) in meters and \(\vec{B}\) in milliteslas. Find, in unit-vector notation, the force on the \(2.0 \mathrm{~m}\) segment of the conductor that lies between \(x=1.0 \mathrm{~m}\) and \(x=3.0 \mathrm{~m}\).

An electron moves in a circle of radius \(r=5.29 \times 10^{-11} \mathrm{~m}\) with speed \(2.19 \times 10^{6} \mathrm{~m} / \mathrm{s} .\) Treat the circular path as a current loop with a constant current equal to the ratio of the electron's charge magnitude to the period of the motion. If the circle lies in a uniform magnetic field of magnitude \(B=7.10 \mathrm{mT}\), what is the maximum possible magnitude of the torque produced on the loop by the field?

A magnetic dipole with a dipole moment of magnitude \(0.020 \mathrm{~J} / \mathrm{T}\) is released from rest in a uniform magnetic field of magnitude \(52 \mathrm{mT}\). The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is \(0.80 \mathrm{~mJ}\). (a) What is the initial angle between the dipole moment and the magnetic field? (b) What is the angle when the dipole is next (momentarily) at rest?

Two concentric, circular wire loops, of radii \(r_{1}=20.0 \mathrm{~cm}\) and \(r_{2}=30.0 \mathrm{~cm},\) are located in an \(x y\) plane; each carries a clockwise current of 7.00 A (Fig. \(28-48\) ). (a) Find the magnitude of the net magnetic dipole moment of the system. (b) Repeat for reversed current in the inner loop.

A circular wire loop of radius \(15.0 \mathrm{~cm}\) carries a current of \(2.60 \mathrm{~A}\). It is placed so that the normal to its plane makes an angle of \(41.0^{\circ}\) with a uniform magnetic field of magnitude \(12.0 \mathrm{~T}\). (a) Calculate the magnitude of the magnetic dipole moment of the loop. (b) What is the magnitude of the torque acting on the loop?

A circular coil of 160 turns has a radius of \(1.90 \mathrm{~cm} .\) (a) Calculate the current that results in a magnetic dipole moment of magnitude \(2.30 \mathrm{~A} \cdot \mathrm{m}^{2}\). (b) Find the maximum magnitude of the torque that the coil, carrying this current, can experience in a uniform \(35.0 \mathrm{mT}\) magnetic field.

The magnetic dipole moment of Earth has magnitude \(8.00 \times 10^{22} \mathrm{~J} / \mathrm{T}\). Assume that this is produced by charges flowing in Earth's molten outer core. If the radius of their circular path is \(3500 \mathrm{~km},\) calculate the current they produce.

A current loop, carrying a current of \(5.0 \mathrm{~A},\) is in the shape of a right triangle with sides \(30,40,\) and \(50 \mathrm{~cm} .\) The loop is in a uniform magnetic field of magnitude \(80 \mathrm{mT}\) whose direction is parallel to the current in the \(50 \mathrm{~cm}\) side of the loop. Find the magnitude of (a) the magnetic dipole moment of the loop and (b) the torque on the loop.

A circular loop of wire having a radius of \(8.0 \mathrm{~cm}\) carries a current of 0.20 A. A vector of unit length and parallel to the dipole moment \(\vec{\mu}\) of the loop is given by \(0.60 \hat{i}-0.80 \hat{j}\). (This unit vector gives the orientation of the magnetic dipole moment vector.) If the loop is located in a uniform magnetic field given by \(\vec{B}=\) \((0.25 \mathrm{~T}) \hat{\mathrm{i}}+(0.30 \mathrm{~T}) \hat{\mathrm{k}},\) find \((\mathrm{a})\) the torque on the loop (in unit- vector notation) and (b) the orientation energy of the loop.

A wire of length \(25.0 \mathrm{~cm}\) carrying a current of \(4.51 \mathrm{~mA}\) is to be formed into a circular coil and placed in a uniform magnetic field \(\vec{B}\) of magnitude \(5.71 \mathrm{mT}\). If the torque on the coil from the field is maximized, what are (a) the angle between \(\vec{B}\) and the coil's magnetic dipole moment and (b) the number of turns in the coil? (c) What is the magnitude of that maximum torque?

A proton of charge \(+e\) and mass \(m\) enters a uniform magnetic field \(\vec{B}=B \hat{i}\) with an initial velocity \(\vec{v}=v_{0 x} \hat{i}+v_{0 y} \hat{j} .\) Find an expression in unit-vector notation for its velocity \(\vec{v}\) at any later time \(t\).

A stationary circular wall clock has a face with a radius of \(15 \mathrm{~cm} .\) Six turns of wire are wound around its perimeter; the wire carries a current of \(2.0 \mathrm{~A}\) in the clockwise direction. The clock is located where there is a constant, uniform external magnetic field of magnitude \(70 \mathrm{mT}\) (but the clock still keeps perfect time). At exactly 1: 00 P.M., the hour hand of the clock points in the direction of the external magnetic field. (a) After how many minutes will the minute hand point in the direction of the torque on the winding due to the magnetic field? (b) Find the torque magnitude.

A wire lying along a \(y\) axis from \(y=0\) to \(y=0.250 \mathrm{~m}\) carries a current of \(2.00 \mathrm{~mA}\) in the negative direction of the axis. The wire fully lies in a nonuniform magnetic field that is given by \(\vec{B}=(0.300 \mathrm{~T} / \mathrm{m}) y \hat{\mathrm{i}}+(0.400 \mathrm{~T} / \mathrm{m}) y \hat{\mathrm{j}} .\) In unit-vector notation, what is the magnetic force on the wire?

Atom 1 of mass \(35 \mathrm{u}\) and atom 2 of mass \(37 \mathrm{u}\) are both singly ionized with a charge of \(+e .\) After being introduced into a mass spectrometer (Fig. \(28-12\) ) and accelerated from rest through a potential difference \(V=7.3 \mathrm{kV},\) each ion follows a circular path in a uniform magnetic field of magnitude \(B=0.50 \mathrm{~T}\). What is the distance \(\Delta x\) between the points where the ions strike the detector?

An electron with kinetic energy \(2.5 \mathrm{keV}\) moving along the positive direction of an \(x\) axis enters a region in which a uniform electric field of magnitude \(10 \mathrm{kV} / \mathrm{m}\) is in the negative direction of the \(y\) axis. A uniform magnetic field \(\vec{B}\) is to be set up to keep the electron moving along the \(x\) axis, and the direction of \(\vec{B}\) is to be chosen to minimize the required magnitude of \(\vec{B}\). In unit-vector notation, what \(\vec{B}\) should be set up?

Physicist S. A. Goudsmit devised a method for measuring the mass of heavy ions by timing their period of revolution in a known magnetic field. A singly charged ion of iodine makes 7.00 rev in a \(45.0 \mathrm{mT}\) field in \(1.29 \mathrm{~ms} .\) Calculate its mass in atomic mass units.

At time \(t=0,\) an electron with kinetic energy \(12 \mathrm{keV}\) moves through \(x=0\) in the positive direction of an \(x\) axis that is parallel to the horizontal component of Earth's magnetic field \(\vec{B}\). The field's vertical component is downward and has magnitude \(55.0 \mu \mathrm{T}_{\therefore}\) (a) What is the magnitude of the electron's acceleration due to \(\vec{B} ?\) (b) What is the electron's distance from the \(x\) axis when the electron reaches coordinate \(x=20 \mathrm{~cm} ?\)

A particle with charge \(2.0 \mathrm{C}\) moves through a uniform magnetic field. At one instant the velocity of the particle is \((2.0 \hat{i}+4.0 \hat{j}+6.0 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}\) and the magnetic force on the particle is \((4.0 \hat{\mathrm{i}}-20 \mathrm{j}+12 \mathrm{k}) \mathrm{N}_{\rightarrow}\) The \(x\) and \(y\) components of the magnetic field are equal. What is \(\vec{B} ?\)

A proton, a deuteron \((q=+e, m=2.0 \mathrm{u}),\) and an alpha particle \((q=+2 e, m=4.0 \mathrm{u})\) all having the same kinetic energy enter a region of uniform magnetic field \(\vec{B},\) moving perpendicular to \(\vec{B}\) What is the ratio of (a) the radius \(r_{d}\) of the deuteron path to the radius \(r_{p}\) of the proton path and (b) the radius \(r_{a}\) of the alpha particle path to \(r_{p} ?\)

Bainbridge's mass spectrometer, shown in Fig. 28-54, separates ions having the same velocity. The ions, after entering through slits, \(\mathrm{S}_{1}\) and \(\mathrm{S}_{2}\), pass through a velocity selector composed of an electric field produced by the charged plates \(P\) and \(P^{\prime},\) and a magnetic field \(\vec{B}\) perpendicular to the electric field and the ion path. The ions that then pass undeviated through the crossed \(\vec{E}\) and \(\vec{B}\) fields enter into a region where a second magnetic field \(\vec{B}^{\prime}\) exists, where they are made to follow circular paths. A photographic plate (or a modern detector) registers their arrival. Show that, for the ions, \(q / m=E / r B B^{\prime}\), where \(r\) is the radius of the circular orbit.

A proton, a deuteron \((q=+e, m=2.0 \mathrm{u}),\) and an alpha particle \((q=+2 e, m=4.0 \mathrm{u})\) are accelerated through the same potential difference and then enter the same region of uniform magnetic field \(\vec{B},\) moving perpendicular to \(\vec{B}\). What is the ratio of (a) the proton's kinetic energy \(K_{p}\) to the alpha particle's kinetic energy \(K_{a}\) and \((\mathrm{b})\) the deuteron's kinetic energy \(K_{d}\) to \(K_{a} ?\) If the radius of the proton's circular path is \(10 \mathrm{~cm},\) what is the radius of (c) the deuteron's path and (d) the alpha particle's path?

An electron is moving at \(7.20 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a magnetic field of strength \(83.0 \mathrm{mT}\). What is the (a) maximum and (b) minimum magnitude of the force acting on the electron due to the field? (c) At one point the electron has an acceleration of magnitude \(4.90 \times 10^{14} \mathrm{~m} / \mathrm{s}^{2} .\) What is the angle between the electron's velocity and the magnetic field?