Chapter 27

A particle with a charge of \(-1.24 \times 10^{-8} \mathrm{C}\) is moving with instantancous \(\quad\) velocity \(\quad \vec{v}=\left(4.19 \times 10^{4} \mathrm{m} / \mathrm{s}\right) \hat{\imath}+(-3.85 \times\) \(10^{4} \mathrm{m} / \mathrm{s}\) ) \(\hat{\mathrm{v}}\) . What is the force exerted on this particle by a magnetic field \((a) \vec{B}=(1.40 \mathrm{T}) \hat{\imath}\) and \((b) \vec{B}=(1.40 \mathrm{T}) \hat{k} ?\)

A particle of mass 0.195 g carries a charge of \(-2.50 \times\) \(10^{-8} \mathrm{C}\) . The particle is given an initial horizontal velocity that is due north and has magnitude \(4.00 \times 10^{4} \mathrm{m} / \mathrm{s}\) . What are the magnitude and direction of the minimum magnetic field that will keep the particle moving in the earth's gravitational field in the same horizontal, northward direction?

In a 1.25 - T magnetic field directed vertically upward, a particle having a charge of magnitude 8.50\(\mu \mathrm{C}\) and initially moving northward at 4.75 \(\mathrm{km} / \mathrm{s}\) is deflected toward the east. (a) What is the sign of the charge of this particle? Make a sketch to illustrate how you found your answer. (b) Find the magnetic force on the particle.

A particle with mass \(1.81 \times 10^{-3} \mathrm{kg}\) and a charge of \(1.22 \times\) \(10^{-8} \mathrm{C}\) has, at a given instant, a velocity \(\vec{v}=\left(3.00 \times 10^{4} \mathrm{m} / \mathrm{s}\right) \hat{j}\) What are the magnitude and direction of the particle's accoleration produced by a uniform magnetic field \(\vec{B}=(1.63 \mathrm{T}) \hat{\imath}+\) \((0.980 \mathrm{T}) \hat{\jmath} ?\)

An electron experiences a magnetic force of magnitude \(4.60 \times 10^{-15} \mathrm{N}\) when moving at an angle of \(60.0^{\circ}\) with respect to a magnetic field of magnitude \(3.50 \times 10^{-3} \mathrm{T}\) . Find the speed of the electron.

A particle with charge \(-5.60 \mathrm{nC}\) is moving in a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}=-(1.25 \mathrm{T}) \hat{\boldsymbol{k}} .\) The magnetic force on the particle is measured to be \(\overrightarrow{\boldsymbol{F}}=-\left(3.40 \times 10^{-7} \mathrm{N}\right) \hat{\boldsymbol{i}}+\left(7.40 \times 10^{-7} \mathrm{N}\right) \hat{\boldsymbol{j}}\) (a) Calculate all the components of the velocity of the particle that you can from this information. (b) Are there components of the velocity that are not determined by the measurement of the force? Explain. (c) Calculate the scalar product \(\overrightarrow{\boldsymbol{F}} \cdot \overrightarrow{\boldsymbol{F}}\) . What is the angle between \(\overrightarrow{\boldsymbol{v}}\) and \(\overrightarrow{\boldsymbol{F}} ?\)

A group of particles is traveling in a magnetic field of unknown magnitude and direction. You observe that a proton moving at 1.50 \(\mathrm{km} / \mathrm{s}\) im the \(+x\) -direction experiences a force of \(2.25 \times 10^{-16} \mathrm{N}\) in the \(+y\) -direction, and an electron moving at 4.75 \(\mathrm{km} / \mathrm{s}\) in the \(-\mathrm{z}\) -direction experiences a force of \(8.50 \times 10^{-16} \mathrm{N}\) . (a) What are the magnitude and direction of the magnetic field? (b) What are the magnitude and direction of the magnetic force on an electron moving in the \(-y\) -direction at 3.2 \(\mathrm{km} / \mathrm{s} ?\)

The magnetic flux through one face of a cube is \(+0.120 \mathrm{Wb} .\) (a) What must the total magnetic flux through the other five faces of the cube be? (b) Why didn't you need to know the dimensions of the cube in order to answer part (a) 2 (c) Suppose the magnetic flux is due to a permanent magnet like that shown in Fig. \(27.11 .\) In a sketch, show where the cube in part (a) might be located relative to the magnet.

A circular area with a radius of 6.50 \(\mathrm{cm}\) lies in the \(x y\) -plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field \(B=0.230 \mathrm{T}\) (a) in the \(+z\) -direction; \((b)\) at an angle of \(53.1^{\circ}\) from the \(+z\) -direction; \((\mathrm{c})\) in the \(+y\) -direction?

An open plastic soda bottle with an opening diameter of 2.5 \(\mathrm{cm}\) is placed on a table. A uniform \(1.75-\mathrm{T}\) magnetic field directed upward and oriented \(25^{\circ}\) from vertical encompasses the bottle. What is the total magnetic flux through the plastic of the soda bottle?

A particle with charge \(6.40 \times 10^{-19} \mathrm{C}\) travels in a circular orbit with radius 4.68 \(\mathrm{mm}\) due to the force exerted on it by a magnetic field with magnitude 1.65 \(\mathrm{T}\) and perpendicular to the orbit. (a) What is the magnitude of the linear momentum \(\vec{p}\) of the partcle? (b) What is the magnitude of the angular momentum \(\overrightarrow{\boldsymbol{L}}\) of the particle?

A \(150-\) g ball containing \(4.00 \times 10^{8}\) excess electrons is dropped into a \(125-\mathrm{m}\) vertical shaft. At the bottom of the shaft, the ball suddenly enters a uniform horizontal magnetic field that has magnitude 0.250 \(\mathrm{T}\) and direction from east to west. If air resistance is negligibly small, find the magnitude and direction of the force that this magnetic field exerts on the ball just as it enrers the field.

(a) An 16 nucleus (charge \(+8 e )\) moving horizontally from west to east with a speed of 500 \(\mathrm{km} / \mathrm{s}\) experiences a magnetic force of 0.00320 \(\mathrm{nN}\) vertically downward. Find the magnitude and direction of the weakest magnetic field required to produce this force. Explain how this same force could be caused by a larger magnetic field. (b) An electron moves in a uniform, horizontal, 2.10 - T magnetic field that is toward the west. What must the magnitude and direction of the minimum velocity of the electron be so that the magnetic force on it will be 4.60 \(\mathrm{pN}\) , vertically upward? Explain how the velocity could be greater than this minimum value and the force still have this same magnitude and direction.

A deuteron (the nucleus of an isotope of bydrogen) has a mass of \(3.34 \times 10^{-27} \mathrm{kg}\) and a charge of \(+e .\) The deuteron travels in a circular path with a radius of 6.96 \(\mathrm{mm}\) in a magnetic field with magnitude 2.50 \(\mathrm{T}\) . (a) Find the speed of the deuteron. (b) Find the time required for it to make half a revolution. (c) Through what to potential difference would the deuteron have to be accelerated to acquire this speed?

A beam of protons traveling at 1.20 \(\mathrm{km} / \mathrm{s}\) enters a uniform magnetic field, traveling perpendicular to the field. The beam exits the magnetic field, leaving the field in a direction perpendicular- to its original direction (Fig. 27.48\()\) . The beam travels a distance of 1.18 \(\mathrm{cm}\) while in the field. What is the magnitude of the magnetic field?

An electron in the beam of a TV picture tube is accelerated by a potential difference of 2.00 \(\mathrm{kV}\) . Then it passes through a region of transverse magnetic field, where it moves in a circular arc with radius 0.180 \(\mathrm{m}\) . What is the magnitude of the field?

A singly charged ion of 7 \(\mathrm{Li}\) (an isotope of lithium) has a mass of \(1.16 \times 10^{-26} \mathrm{kg} .\) It is accelerated through a potential difference of 220 \(\mathrm{V}\) and then enters a magnetic field with magnitude 0.723 T perpendicular to the path of the ion. What is the radius of the ion's path in the magnetic field?

A proton \(\left(q=1.60 \times 10^{-19} \mathrm{C}, m=1.67 \times 10^{-27} \mathrm{kg}\right)\) moves in a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}=(0.500 \mathrm{T}) \hat{\boldsymbol{i}} .\) At \(t=0\) the proton has velocity components \(v_{x}=1.50 \times 10^{5} \mathrm{m} / \mathrm{s}, v_{y}=0,\) and \(v_{z}=2.00 \times 10^{5} \mathrm{m} / \mathrm{s}(\text { see Example } 27.4) .\) (a) What are the magnitude and direction of the magnetic force acting on the proton? In addition to the magnetic field there is a uniform electric field in the \(+x\) -direction, \(\vec{E}=\left(+2.00 \times 10^{4} \mathrm{V} / \mathrm{m}\right) \hat{\imath}\) (b) Will the proton have a component of acceleration in the direction of the electric field?(c) Describe the path of the proton. Does the electric field affect the radius of the helix? Explain. (d) At \(t=T / 2\) , where \(T\) is the period of the circular motion of the proton, what is the \(x\) -component of the displacement of the proton from its position at \(t=0 ?\)

(a) What is the speed of a beam of electrons when the simultancous influence of an electric field of \(1.56 \times 10^{4} \mathrm{V} / \mathrm{m}\) and a magnetic field of \(4.62 \times 10^{-3} \mathrm{T}\) , with both fields normal to the beam and to each other, produces no deflection of the electrons? (b) In a diagram, show the relative orientation of the vectors \(\overrightarrow{\boldsymbol{v}}, \overrightarrow{\boldsymbol{E}},\) and \(\overrightarrow{\boldsymbol{B}} .\) (c) When the electric field is removed, what is the radius of the electron orbit? What is the period of the orbit?

A \(150-\mathrm{V}\) battery is connected across two parallel metal plates of area 28.5 \(\mathrm{cm}^{2}\) and separation \(8.20 \mathrm{mm} .\) A beam of alpha particles (charge \(+2 e,\) mass \(6.64 \times 10^{-27} \mathrm{kg} )\) is accelerated from rest through a potential difference of 1.75 \(\mathrm{kV}\) and enters the region between the plates perpendicular to the electric field. What magnitude and direction of magnetic field are needed so that the alpha particles emerge undeflected from between the plates?

Crussed \(\vec{E}\) and \(\vec{B}\) Fields. A particle with initial velocity \(\overrightarrow{\boldsymbol{v}}_{0}=\left(5.85 \times 10^{3} \mathrm{m} / \mathrm{s}\right) \hat{\boldsymbol{j}}\) enters a region of uniform electric and magnetic fields. The magnetic field in the region is \(\overrightarrow{\boldsymbol{B}}=\) \(-(1.35 \mathrm{T}) \hat{\boldsymbol{k}} .\) Calculate the magnitude and direction of the electric field in the region if the particle is to pass through undeflected, for a particle of charge \((\mathrm{a})+0.640 \mathrm{nC}\) and \((\mathrm{b})-0.320 \mathrm{nC}\) . You can ignore the weight of the particle.

Determining the Mass of an Isotope. The electric field between the plates of the velocity selector in a Bainbridge mass spectrometer (see Fig. 27.22) is 1.12 \(\times 10^{5} \mathrm{V} / \mathrm{m}\) , and the magnetic field in both regions is 0.540 T. A stream of singly charged selenium ions moves in a circular path with a radius of 31.0 \(\mathrm{cm}\) in the magnetic field. Determine the mass of one selenium ion and the mass number of this selenium isotope. (The mass number is equal to the mass of the isotope in atomic mass units, rounded to the nearest integer. One atomic mass unit \(=1 \mathbf{u}=1.66 \times 10^{-27} \mathrm{kg} .\) .

A straight \(2.00-\mathrm{m}, 150-\mathrm{g}\) wire carries a current in a region where the earth's magnetic field is horizontal with a magnitude of 0.55 gauss. (a) What is the minimum value of the current in this wire so that its weight is completely supported by the magnetic force due to earth's field, assuming that no other forces except gravity act on it? Does it scem likely that such a wire could support this size of current? (b) Show how the wire would have to be oriented relative to the earth's magnetic field to be supported in this way.

An electromagnet produces a magnetic field of 0.550 T in a cylindrical region of radius 2.50 \(\mathrm{cm}\) between its poles. A straight wire carrying a current of 10.8 \(\mathrm{A}\) passes through the center of this region and is perpendicular to both the axis of the cylindrical region and the magnetic field. What magnitude of force is exerted on the wire?

A straight, vertical wire carries a current of 1.20 \(\mathrm{A}\) downward in a region between the poles of a large superconducting electromagnet, where the magnetic field has magnitude \(B=\) 0.588 \(\mathrm{T}\) and is horizontal. What are the magnitude and direction of the magnetic force on a \(1.00-\mathrm{cm}\) section of the wire that is in this uniform magnetic field, if the magnetic field direction is (a) east; (b) south; (c) \(30.0^{\circ}\) south of west?

A horizontal rod 0.200 \(\mathrm{m}\) long is mounted on a balance and carries a current. At the location of the rod a uniform horizontal magnetic field has magnitude 0.067 T and direction perpendicular to the rod. The magnetic force on the rod is measured by the balance and is found to be 0.13 \(\mathrm{N}\) . What is the current?

\(\mathbf{A}\) thin 50.0 -cm-long metal bar with mass 750 g rests on, but is not attached to, two metallic supports in a uniform 0.450 - T magnetic field, as shown in Fig. 27.51 . A battery and a \(25.0-\Omega\) resistor in series are connected to the supports. (a) What is the highest voltage the battery can have without breaking the circuit at the supports? (b) The battery voltage has the maximum value calculated in part (a). If the resistor suddenly gets partially short- circuited, decreasing its resistance to 2.0\(\Omega\) , find the initial acceleration of the bar.

The plane of a \(5.0 \mathrm{cm} \times 8.0 \mathrm{cm}\) rectangular loop of wire is parallel to a \(0.19-\mathrm{T}\) magnetic field. The loop carries a current of 6.2 \(\mathrm{A}\) . (a) What torque acts on the loop? (b) What is the magnetic moment of the loop? (c) What is the maximum torque that can be obtained with the same total length of wire carrying the same current in this magnetic field?

Magnetic Moment of the Hydrogen Atom. In the Bohr model of the hydrogen atom (see Section \(38.5 ),\) in the lowest energy state the electron orbits the proton at a speed of \(2.2 \times\) \(10^{6} \mathrm{m} / \mathrm{s}\) in a circular orbit of radius \(5.3 \times 10^{-11} \mathrm{m}\) (a) What is the orbital period of the electron? (b) If the orbiting electron is considered to be a current loop, what is the current \(I ?\) (c) What is the magnetic moment of the atom due to the motion of the electron?

A coil with magnetic moment 1.45 \(\mathrm{A} \cdot \mathrm{m}^{2}\) is oriented initially with its magnetic moment antiparallel to a uniform \(0.835-\mathrm{T}\) magnetic field. What is the change in potential energy of the coil when it is rotated \(180^{\circ}\) so that its magnetic moment is parallel to the field?

A dc motor with its rotor and field coils connected in series has an internal resistance of 3.2\(\Omega\) . When the motor is running at full load on a \(120-\mathrm{V}\) line, the emf in the rotor is 105 \(\mathrm{V}\) . (a) What is the current drawn by the motor from the line? (b) What is the power delivered to the motor? (c) What is the mechanical power developed by the motor?

In a shunt-wound dc motor with the field coils and rotor connected in parallel (Fig. \(27.56 ),\) the resistance \(R_{t}\) of the field coils is \(106 \Omega,\) and the resistance \(R_{r}\) of the rotor is 5.9\(\Omega\) . When a potential difference of 120 \(\mathrm{V}\) is applied to the brushes and the motor is running at full speed delivering mechanical power, the current supplied to it is 4.82 \(\mathrm{A}\) (a) What is the current in the field coils? (b) What is the current in the rotor? (c) What is the induced emf developed by the motor? (d) How much mechanical power is developed by this motor?

A shunt-wound de motor with the field coils and rotor connected in parallel (Fig. 27.56 ) operates from a \(120-\mathrm{V}\) dc power linc. The resistance of the ficld windings, \(R_{f},\) is 218\(\Omega\) . The resistance of the rotor, \(R_{r}\) is 5.9 . When the motor is running, the rotor develops an emf \(\mathcal{E}\) . The motor draws a current of 4.82 A from the line. Friction losses amount to 45.0 W. Compute (a) the field current; \((b)\) the rotor current; \((c)\) the emf \(\mathcal{E} ;\) (d) the rate of development of thermal energy in the field windings; (e) the rate of development of thermal energy in the rotor; \((f)\) the power input to the motor; (g) the efficiency of the motor.

When a particle of charge \(q>0\) moves with a velocity of \(\overrightarrow{\boldsymbol{v}}_{1}\) at \(45.0^{\circ}\) from the \(+x\) -axis in the \(x y\) -plane, a uniform magnetic field exerts a force \(\overrightarrow{\boldsymbol{F}}_{1}\) along the \(-z\) -axis (Fig. \(27.58 ) .\) When the same particle moves with a velocity \(\overrightarrow{\boldsymbol{v}}_{2}\) with the same magnitude as \(\overrightarrow{\boldsymbol{v}}_{1}\) but along the \(+z\) -axis, a force \(\overrightarrow{\boldsymbol{F}}_{2}\) of magnitude \(\boldsymbol{F}_{2}\) is exerted on it along the \(+x\) -axis. (a) What are the magnitude (in terms of \(q\) , \(v_{1},\) and \(F_{2} )\) and direction of the magnetic field? (b) What is the magnitude of \(\overrightarrow{\boldsymbol{F}}_{1}\) in terms of \(F_{2} ?\)

A particle with charge \(9.45 \times 10^{-8} \mathrm{C}\) is moving in a region where there is a uniform magnetic field of 0.450 \(\mathrm{T}\) in the \(+x\) - direction. At a particular instant of time the velocity of the particle has components \(v_{x}=-1.68 \times 10^{4} \mathrm{m} / \mathrm{s}, v_{y}=-3.11 \times 10^{4} \mathrm{m} / \mathrm{s}\) and \(v_{z}=5.85 \times 10^{4} \mathrm{m} / \mathrm{s}\) . What are the components of the force on the particle at this time?

A cycloron is to accelerate protons to an energy of 5.4 MeV. The superconducting electromagnet of the cyclotron produces a \(3.5-\) T magnetic field perpendicular to the proton orbits. (a) When the protons have achieved a kinetic energy of 2.7 \(\mathrm{MeV}\) , what is the radius of their circular orbit and what is their angular speed? (b) Repeat part (a) when the protons have achieved their final kinetic energy of 5.4 \(\mathrm{MeV}\) .

The magnetic poles of a small cyclotron produce a magnetic field with magnitude 0.85 \(\mathrm{T}\) . The poles have a radius of \(0.40 \mathrm{m},\) which is the maximum radius of the orbits of the accelerated particles. (a) What is the maximum energy to which protons \(\left(q=1.60 \times 10^{-19} \mathrm{C}, m=1.67 \times 10^{-27} \mathrm{kg}\right)\) can be accelerated by this cyclotron? Give your answer in electron volts and in joules. (b) What is the time for one revolution of a proton orbiting at this maximum radius? (c) What would the magnetic-field magnitude have to be for the maximum energy to which a proton can be accelerated to be twice that calculated in part (a)? (d) For \(B=0.85 \mathrm{T}\) . what is the maximum energy to which alpha particles \(\left(q=3.20 \times 10^{-19} \mathrm{C}, m=6.65 \times 10^{-27} \mathrm{kg}\right)\) can be accelerated by this cyclotron? How does this compare to the maximum energy for protons?

The force on a charged particle moving in a magnetic field can be computed as the vector sunt of the forces due to each separate component of the magnetic field. As an example, a particle with charge \(q\) is moving with speed \(v\) in the \(-y\) -direction. It is moving in a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}=\overrightarrow{\boldsymbol{B}}_{\boldsymbol{x}} \hat{\boldsymbol{i}}+\boldsymbol{B}_{\boldsymbol{y}} \hat{\boldsymbol{j}}+\boldsymbol{B}_{\boldsymbol{z}} \hat{\boldsymbol{k}}\) (a) What are the components of the force \(\overrightarrow{\boldsymbol{F}}\) exerted on the particle by the magnetic field? (b) If \(q>0,\) what must the signs of the components of \(\overrightarrow{\boldsymbol{B}}\) be if the components of \(\overrightarrow{\boldsymbol{F}}\) are all nonnegative? (c) If \(q<0\) and \(B_{x}=B_{y}=B_{z}>0,\) find the direction of \(\vec{F}\) and find the magnitude of \(\vec{F}\) in terms of \(|q|, v,\) and \(B_{x}\) .

In the electron gun of a TV picture tube the electrons (charge \(-e,\) mass \(m )\) are accelerated by a voltage \(V .\) After leaving the electron gun, the electron beam travels a distance \(D\) to the screen; in this region there is a transverse magnetic field of magnitude \(B\) and no electric field. (a) Sketch the path of the electron beam in the tube. (b) Show that the approximate deflection of the beam due to this magnetic field is $$ d=\frac{B D^{2}}{2} \sqrt{\frac{e}{2 m V}} $$ (Hint: Place the origin at the center of the electron beam's arc and compare an undefiected beam's path to the deflected beam's path.) (c) Evaluate this expression for \(V=750 \mathrm{V}, D=50 \mathrm{cm},\) and \(B=5.0 \times 10^{-5} \mathrm{T}\) (comparable to the earth's field). Is this deflection significant?

A particle with negative charge \(q\) and mass \(m=2.58 \times\) 10 \(^{-15} \mathrm{kg}\) is traveling through a region containing a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}=-(0.120 \mathrm{T}) \hat{\boldsymbol{k}} .\) At a particular instant of time the velocity of the particle is \(\overrightarrow{\boldsymbol{v}}=\left(1.05 \times 10^{6} \mathrm{m} / \mathrm{s}\right)(-3 \hat{\imath}+4 \hat{\jmath}+\) 12\(\hat{k} )\) and the force \(\vec{F}\) on the particle has a magnitude of 1.25 \(\mathrm{N}\) . (a) Determine the charge \(q\) . (b) Determine the acceleration \(\overrightarrow{\boldsymbol{d}}\) of the particle. (c) Explain why the path of the particle is a helix, and determine the radius of curvature \(R\) of the circular component of the helical path. (d) Determine the cyclotron frequency of the particle. (e) Although helical motion is not periodic in the full sense of the word, the \(x\) - and \(y\) -coordinates do vary in a periodic way. If the coordinates of the particle at \(t=0\) are \((x, y, z)=(R, 0,0),\) determine its coordinates at a time \(t=2 T,\) where \(T\) is the period of the motion in the \(x y\) -plane.

A magnetic field exerts a torque \(\tau\) on a round current-carrying loop of wire. What will be the torque on this loop (in terms of \(\tau )\) if its diameter is tripled?

A particle of charge \(q>0\) is moving at speed \(v\) in the \(+z\) -direction through a region of uniform magnetic field \(\overrightarrow{\boldsymbol{B}} .\) The magnetic force on the particle is \(\overrightarrow{\boldsymbol{F}}=F_{0}(3 \hat{\imath}+4 \hat{\jmath}),\) where \(\boldsymbol{F}_{0}\) is a positive constant. (a) Determine the components \(B_{x}, B_{y},\) and \(B_{z},\) or at least as many of the three components as is possible from the information given. (b) If it is given in addition that the magnetic field has magnitude \(6 F_{0} / q v,\) determine as much as you can about the remaining components of \(\overrightarrow{\boldsymbol{B}}\) .

A straight piece of conducting wire with mass \(M\) and length \(L\) is placed on a friction- less incline tilted at an angle \(\theta\) from the horizontal (Fig. 27.61 ) There is a uniform, vertical magnetic field \(\overrightarrow{\boldsymbol{B}}\) at all points (produced by an arrangement of magnets not shown in the figure). To keep the wire from shiding down the incline, a voltage source is attached to the ends of the wire. When just the right amount of current flows through the wire, the wire remains at rest. Determine the magnitude and direction of the current in the wire that will cause the wire to remain at rest. Copy the figure and draw the direction of the current on your copy. In addition, show in a free-body diagram all the forces that act on the wire.

A plastic circular loop of radius \(R\) and a positive charge \(q\) is distributed uniformly around the circumference of the loop. The loop is then rotated around its central axis, perpendicular to the plane of the loop, with angular speed \(\omega\) . If the loop is in a region where there is a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}\) directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop.

Determining Diet. One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon- 12 . Overneliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use a mass spectrometer to separate the \(^{10} \mathrm{C}\) and \(^{13} \mathrm{C}\) isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.50 \(\mathrm{km} / \mathrm{s}\) , and you want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 \(\mathrm{cm}\) for the 12 \(\mathrm{C}\) . The measured masses of these isotopes are \(1.99 \times 10^{-25} \mathrm{kg}\left(^{12} \mathrm{C}\right)\) and \(2.16 \times 10^{-26} \mathrm{kg}\left(^{13} \mathrm{C}\right) .\) (a) What strength of magnetic field is required? (b) What is the diameter of the \(^{13} \mathrm{C}\) semicircle? (c) What is the separation of the \(^{12}\mathrm{C}\) and \(^{13}\mathrm{C}\) ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?

An Electromagnetic Rail Gun. A conducting bar with mass \(m\) and length \(L\) slides over horizontal rails that are connected to a voltage source. The voltage source maintains a constant current \(I\) in the rails and bar, and a constant, uniform, vertical magnetic field \(\overrightarrow{\boldsymbol{B}}\) fills the region between the rails (Fig. 27.63\()\) . (a) Find the magnitude and direction of the net force on the conducting bar Ignore friction, air resistance, and electrical resistance. (b) If the bar has mass \(m\) , find the distance \(d\) that the bar must move along the rails from rest to attain speed \(v\) . (c) It has been suggested that rail guns based on this principle could accelerate payloads into earth orbit or beyond. Find the distance the bar must travel along the rails if it is to reach the escape speed for the earth \((11.2 \mathrm{km} / \mathrm{s}) .\) Let \(B=0.50 \mathrm{T}, \quad I=2.0 \times 10^{3} \mathrm{A}\) \(m=25 \mathrm{kg},\) and \(L=50 \mathrm{cm} .\) For simplicity assume the net force on the object is equal to the magnetic force, as in parts \((a)\) and \((b)\) , even though gravity plays an important role in an actual launch in space.

A wire 25.0 \(\mathrm{cm}\) long lies along the \(z\) -axis and carries a current of 9.00 \(\mathrm{A}\) in the \(+z\) -direction. The magnetic field is uniform and has components \(B_{x}=-0.242 \mathrm{T}, B_{y}=-0.985 \mathrm{T}\) , and \(B_{z}=\) \(-0.336 \mathrm{T} .\) (a) Find the components of the magnetic force on the wire. (b) What is the magnitude of the net magnetic force on the wire?

Paleoclimate. Climatologists can determine the past temperature of the earth by comparing the ratio of the isotope oxygen-18 to the isotope oxygen- 16 in air trapped in ancient ice sheets, such as those in Greenland. In one method for separating these isotopes, a sample containing both of them is first singly ionized (one electron is removed) and then accelerated from rest through a potential difference \(V\) . This beam then enters a magnetic field \(B\) at right angles to the field and is bent into a quarter circle. A particle detector at the end of the path measures the amount of each isotope, (a) Show that the separation \(\Delta r\) of the two isotopes at the detector is given by $$ \Delta r=\frac{\sqrt{2 e V}}{e B}\left(\sqrt{m_{18}}-\sqrt{m_{16}}\right) $$ where \(m_{16}\) and \(m_{18}\) are the masses of the two oxygen isotopes, (b) The measured masses of the two isotopes are \(2.66 \times 10^{-26} \mathrm{kg}\) \(\left(^{16} \mathrm{O}\right)\) and \(2.99 \times 10^{-25} \mathrm{kg}\) \(\left(^{18} \mathrm{O}\right)\). If the magnetic field is 0.050 T, what must be the accelerating potential \(V\) so that these two isotopes will be separated by 4.00 \(\mathrm{cm}\) at the detector?

Force on a Current Loop in a Nonuniform Magnetic Field. It was shown in Section 27.7 that the net force on a current loop in a uniform magnetic field is zero. But what if \(\overrightarrow{\boldsymbol{B}}\) is not uniform? Figure 27.70 shows a square loop of wire that ties in the \(x y\) -plane. The loop has corners at \((0,0),(0, L),(L, 0)\) , and \((L, L)\) and carries a constant current \(I\) in the clockwise direction. The magnetic field has no \(x\) -component but has both \(y-\) and \(z\) -components: \(\vec{B}=\) \(\left(B_{0} z / L\right) \hat{j}+\left(B_{0} y / L\right) \hat{k},\) where \(B_{0}\) is a positive constant. (a) Sketch the magnetic field lines in the \(y z\) -plane. (b) Find the magnitude and direction of the magnetic force exerted on each of the sides of the loop by integrating \(\mathrm{Eq} .(27.20) .\) (c) Find the magnitude and direction of the net magnetic force on the loop.

An insulated wire with mass \(m=5.40 \times 10^{-5} \mathrm{kg}\) is bent into the shape of an inverted U such that the horizontal part has a length \(l=15.0 \mathrm{cm} .\) The bent ends of the wire are partially immersed in two pools of mercury, with 2.5 \(\mathrm{cm}\) of each end below the mercury's surface. The entire structure is in a region containing a uniform \(0.00650-\mathrm{T}\) magnetic field directed into the page (Fig. 27.71\()\) . An electrical connection from the mercury pools is made through the ends of the wires. The mercury pools are connected to a \(1.50-\mathrm{V}\) battery and a switch \(\mathrm{S}\) . When switch \(\mathrm{S}\) is closed, the wire jumps 35.0 \(\mathrm{cm}\) into the air, measured from its initial position. (a) Determine the speed \(v\) of the wire as it leaves the mercury. (b) Assuming that the current \(I\) through the wire was constant from the time the switch was closed until the wire left the mercury, determine \(I\) (c) Ignoring the resistance of the mercury and the circuit wires, determine the resistance of the moving wire.