Chapter 27

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

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.

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.

An electron moves at \(2.50 \times 10^{6} \mathrm{m} / \mathrm{s}\) through a region in which there is a magnetic field of unspecified direction and magnitude \(7.40 \times 10^{-2} \mathrm{T}\) (a) What are the largest and smallest possible magnitudes of the acceleration of the electron due to the magnetic field? (b) If the actual acceleration of the electron is one-fourth of the largest magnitude in part (a), what is the angle between the electron velocity and the magnetic field?

A particle with charge 7.80\(\mu \mathrm{C}\) is moving with velocity \(\vec{\boldsymbol{v}}=-\left(3.80 \times 10^{3} \mathrm{m} / \mathrm{s}\right) \hat{\boldsymbol{J}}\) . The magnetic force on the particle is measured to be \(\vec{\boldsymbol{F}}=+\left(7.60 \times 10^{-3} \mathrm{N}\right) \hat{\boldsymbol{\imath}}-\left(5.20 \times 10^{-3} \mathrm{N}\right) \hat{\boldsymbol{k}}\) (a) Calculate all the components of the magnetic field you can from this information. (b) Are there components of the magnetic field that are not determined by the measurement of the force? Explain. (c) Calculate the scalar product \(\vec{B} \cdot \vec{F} .\) What is the angle between \(\vec{B}\) and \(\vec{\boldsymbol{F}} ?\)

A particle with charge \(-5.60 \mathrm{nC}\) is moving in a uniform magnetic field \(\vec{\boldsymbol{B}}=-(1.25 \mathrm{T}) \hat{\boldsymbol{k}}\) . The magnetic force on the particle is measured to be \(\vec{\boldsymbol{F}}=-\left(3.40 \times 10^{-7} \mathrm{N}\right) \hat{\imath}+(7.40 \times\) \(10^{-7} \mathrm{N} ) \hat{J}\) . (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 \(\vec{v} \cdot\) What is the angle between \(\vec{v}\) and \(\vec{F} ?\)

A group of particles is traveling in a magnetic field of unknown magnitude and direction. You observe that a proton moving at 1.50 \(\mathrm{km} / \mathrm{s}\) in 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 \(-z\) -direction experiences a force of \(8.50 \times\) \(10^{-16} \mathrm{N}\) in the \(+y\) -direction. (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 3.20 \(\mathrm{km} / \mathrm{s}\) ?

A flat, square surface with side length 3.40 \(\mathrm{cm}\) is in the \(x y\) -plane at \(z=0 .\) Calculate the magnitude of the flux through this surface produced by a magnetic field \(\vec{\boldsymbol{B}}=(0.200 \mathrm{T}) \hat{\boldsymbol{\imath}}+\) \((0.300 \mathrm{T}) \hat{\boldsymbol{J}}-(0.500 \mathrm{T}) \hat{\boldsymbol{k}} .\)

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; \((\mathrm{b})\) at an angle of \(53.1^{\circ}\) from the \(+z\) -direction; \((\mathrm{c})\) in the \(+y\) -direction?

A horizontal rectangular surface has dimensions 2.80 \(\mathrm{cm}\) by 3.20 \(\mathrm{cm}\) and is in a uniform magnetic field that is directed at an angle of \(30.0^{\circ}\) above the horizontal. What must the magnitude of the magnetic field be in order to produce a flux of \(4.20 \times 10^{-4} \mathrm{Wb}\) through the surface?

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 enters the field.

An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of \(6.64 \times 10^{-27}\) kg) traveling horizontally at 35.6 \(\mathrm{km} / \mathrm{s}\) enters a uniform, vertical, 1.10 -T magnetic field. (a) What is the diameter of the path followed by this alpha particle? (b) What effect does the magnetic field have on the speed of the particle? (c) What are the magnitude and direction of the acceleration of the alpha particle while it is in the magnetic field? (d) Explain why the speed of the particle does not change even though an unbalanced external force acts on it.

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 particle? (b) What is the magnitude of the angular momentum \(\vec{L}\) of the particle?

(a) An \(\mathrm{}^{16} \mathrm{O}\) 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 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 hydrogen) 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 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. E27.24). 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 dif- ference 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 \(\vec{\boldsymbol{B}}=(0.500 \mathrm{T}) \hat{\boldsymbol{l}}\) . At \(t=0\) the proton has velocity components \(v_{x}=1.50 \times 10^{5} \mathrm{m} / \mathrm{s}, v_{y}=0\) and \(v_{z}=2.00 \times 10^{5} \mathrm{m} / \mathrm{s}\) (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 simultaneous 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 \(\vec{\boldsymbol{v}}, \vec{\boldsymbol{E}},\) and \(\vec{\boldsymbol{B}}\) . (c) When the electric field is removed, what is the radius of the electron orbit? What is the period of the orbit?

Crossed \(\vec{E}\) and \(\vec{B}\) Fields. A particle with initial velocity \(\vec{\boldsymbol{v}}_{0}=\left(5.85 \times 10^{3} \mathrm{m} / \mathrm{s}\right) \hat{\boldsymbol{J}}\) enters a region of uniform electric and magnetic fields. The magnetic field in the region is \(\vec{\boldsymbol{B}}=\) \(-(1.35 \mathrm{T}) \hat{k}\) . 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 (a) \(+0.640 \mathrm{nC}\) and \((\mathrm{b})-0.320 \mathrm{nC.}\) You can ignore the weight of the particle.

A singly ionized (one electron removed) \(^{40} \mathrm{K}\) atom passes through a velocity selector consisting of uniform perpendicular electric and magnetic fields. The selector is adjusted to allow ions having a speed of 4.50 \(\mathrm{km} / \mathrm{s}\) to pass through undeflected when the magnetic field is 0.0250 T. The ions next enter a second uniform magnetic field \(\left(B^{\prime}\right)\) oriented at right angles to their velocity. 40 contains 19 protons and 21 neutrons and has a mass of \(6.64 \times 10^{-26} \mathrm{kg} .\) (a) What is the magnitude of the electric field in the velocity selector? (b) What must be the magnitude of \(B^{\prime}\) so that the ions will be bent into a semicircle of radius 12.5 \(\mathrm{cm} ?\)

Singly ionized (one electron removed) atoms are accelerated and then passed through a velocity selector consisting of perpendicular electric and magnetic fields. The electric field is 155 \(\mathrm{V} / \mathrm{m}\) and the magnetic field is 0.0315 T. The ions next enter a uniform magnetic field of magnitude 0.0175 \(\mathrm{T}\) that is oriented perpendicular to their velocity. (a) How fast are the ions moving when they emerge from the velocity selector? (b) If the radius of the path of the ions in the second magnetic field is \(17.5 \mathrm{cm},\) what is their mass?

A straight, \(2.5-\mathrm{m}\) wire carries a typical household current of 1.5 \(\mathrm{A}\) (in one direction) at a location where the earth's magnetic field is 0.55 gauss from south to north. Find the magnitude and direction of the force that our planet's magnetic field exerts on this wire if is oriented so that the current in it is running (a) from west to east, (b) vertically upward, (c) from north to south. (d) Is the magnetic force ever large enough to cause significant effects undernormal household conditions?

A straight, 2.00 -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 seem 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 \(\mathrm{T}\) in a cylindrical region of radius 2.50 \(\mathrm{cm}\) between its poles. A straight wire carrying a current of 10.8 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 A downward in a region between the poles of a large superconductingelectromagnet, 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 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. E27.51), the resistance \(R _ { \text { f of the } }\) 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 mechani- cal power, the current supplied to it is 4.82 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?

When a particle of charge \(q > 0\) moves with a velocity of \(\vec { \boldsymbol { v } } _ { 1 }\) at \(45.0 ^ { \circ }\) from the \(\pm x\) -axis in the \(x y\) -plane, a uniform magnetic field exerts a force \(F _ { 1 }\) along the \(- z\) -axis (Fig. \(P 27.55 ) .\) When the same particle moves with a velocity \(\vec { \boldsymbol { v } } _ { 2 }\) with the same magnitude as \(\vec { \boldsymbol { v } } _ { 1 }\) but along the \(+ z\) -zaxis, a force \(\vec { \boldsymbol { F } } _ { 2 }\) of magnitude \(F _ { 2 }\) 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 \(\vec { F } _ { 1 }\) 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 0.650 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?

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?

You wish to hit a target from several meters away with a charged coin having a mass of 4.25\(\mathrm { g }\) and a charge of \(+ 2500 \mu \mathrm { C }\) . The coin is given an initial velocity of 12.8\(\mathrm { m } / \mathrm { s }\) , and a downward, uniform electric field with field strength 27.5\(\mathrm { N } / \mathrm { C }\) exists through-out the region. If you aim directly at the target and fire the coin horizontally, what magnitude and direction of uniform magnetic field are needed in the region for the coin to hit the target?

A cyclotron is to accelerate protons to an energy of 5.4 MeV. The superconducting electromagnet of the cyclotron produces a 2.9 -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)? 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?

A particle with charge \(q\) is moving with speed \(v\) in the \(- y\) -direction. It is moving in a uniform magnetic field \(\vec { B } =\) \(B _ { x } \hat { i } + B _ { y } \hat { J } + B _ { z } \hat { k } .\) (a) What are the components of the force \(\vec { \boldsymbol { F } }\) exerted on the particle by the magnetic field? (b) If \(q > 0 ,\) what must the signs of the components of \(\vec { B }\) be if the components of \(\vec { \boldsymbol { F } }\) are all nonnegative? (c) If \(q < 0\) and \(B _ { x } = B _ { y } = B _ { z } > 0 ,\) find the direction of \(\vec { \boldsymbol { F } }\) and find the magnitude of \(\vec { \boldsymbol { F } }\) in terms of \(| q | , v ,\) and \(B _ { x }\)

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 \(\vec { \boldsymbol { B } } = - ( 0.120 \mathrm { T } ) \hat { \boldsymbol { k } }\) . At a particular instant of time the velocity of the particle is \(\vec { \boldsymbol { v } } = \left( 1.05 \times 10 ^ { 6 } \mathrm { m } / \mathrm { s } \right) ( - 3 \hat { \imath } + 4 \hat { \jmath } + 12 \hat { \boldsymbol { k } } )\) and the force \(\vec { \boldsymbol { F } }\) on the particle has a magnitude of 2.45\(\mathrm { N }\) . (a) Determine the charge \(q .\) (b) Determine the acceleration \(\vec { a }\) 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. (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 \(\vec { \boldsymbol { B } }\) . The magnetic force on the particle is \(\vec { \boldsymbol { F } } = F _ { 0 } ( 3 \hat { \boldsymbol { \imath } } + 4 \hat { \boldsymbol { J } } ) ,\) where \(F _ { 0 }\) 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 \(\vec { B } .\)

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. P27.69). There is a uniform, vertical magnetic field \(\vec { B }\) at all points (produced by an arrangement of magnets not shown in the figure). To keep the wire from sliding 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 has 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 \(\vec { \boldsymbol { B } }\) 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. Overreliance 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 12\(\mathrm { C }\) 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 \(1.99 \times 10 ^ { - 26 } \mathrm { kg } \left( ^ { 12 } \mathrm { C } \right)\) 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 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 \(\vec { B }\) fills the region between the rails (Fig. \(P 27.74 )\) (a) Find the magnitude and direction of the net force on the con- ducting 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 \(B = 0.80 \mathrm { T } , \quad I = 2.0 \times 10 ^ { 3 } \mathrm { A } , \quad m = 25 \mathrm { kg }\) 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 7.40 A in the \(+ z\) -direction. The magnetic field is uniform and has components \(B _ { x } = - 0.242\) 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 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 \(2.99 \times 10 ^ { - 26 } \mathrm { kg } ( 8 \mathrm { O } ) .\) If the magnetic field is \(0.050 \mathrm { T } ,\) what must be the accelerating potential \(V\) so that these two isotopes will be separated by 4.00\(\mathrm { cm }\) at the detector?

Quark Model of the Neutron. The neutron is a particle with zero charge. Nonetheless, it has a nonzero magnetic moment with \(z\) -component \(9.66 \times\) \(10 ^ { - 27 } \mathrm { A } \cdot \mathrm { m } ^ { 2 } .\) This be explained by the internal structure of the neutron. A substantial body of evidence indicates that a neutron is composed of three fundamental particles called of three fundamental particles called quarks: an "up" (u) quark, of charge \(+ 2 e / 3 ,\) and two "down" \(( d )\) quarks, each of charge \(- e / 3 .\) The combination of the three quarks produces a net charge of \(2 e / 3 - e / 3 - e / 3 = 0\) . If the quarks are in motion, they can produce a nonzero magnetic moment. As a very simple model, suppose the \(u\) quark moves in a counterclockwise circular path and the \(d\) quarks move in a clock- wise circular path, all of radius \(r\) and all with the same speed \(v\) (Fig. P27.84). (a) Determine the current due to the circulation of the \(u\) quark. (b) Determine the magnitude of the magnetic moment due to the circulating \(u\) quark. (c) Determine the magnitude of the magnetic moment of the three-quark system. (Be careful to use the correct magnetic moment directions.) (d) With what speed \(v\) must the quarks move if this model is to reproduce the magnetic moment of the neutron? Use \(r = 1.20 \times 10 ^ { - 15 } \mathrm { m }\) (the radius of the neutron) for the radius of the orbits.

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 \(\vec { B }\) is not uniform? Figure P27.85 shows a square loop of wire that lies 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 Eq. (27.20). (c) Find the magnitude and direction of the net magnetic force on the loop.

A circular loop of wire with area \(A\) lies in the \(x y\) -plane. As viewed along the \(z\) -axis looking in the \(- z\) -direction toward the origin, a current \(I\) is circulating clockwise around the loop. The torque produced by an external magnetic field \(\vec { B }\) is given by \(\vec { \tau } = D ( 4 \hat { \imath } - 3 \hat { \jmath } ) ,\) where \(D\) is a positive constant, and for this orientation of the loop the magnetic potential energy \(U = - \vec { \mu } \cdot B\) is negative. The magnitude of the magnetic field is \(B _ { 0 } = 13 D / I A\) . (a) Determine the vector magnetic moment of the current loop. (b) Determine the components \(B _ { x } , B _ { y }\) and \(B _ { z }\) of \(\vec { B } .\)