A uniform, solid, uniformly charged sphere of radius has a

When the axis of a cathode-ray tube is horizontal in a region in which there is a magnetic field that is directed vertically upward, the electrons emitted from the cathode follow one of the dashed paths to the face of the tube in Figure 26-30. The correct path is (a) 1, (b) 2, (c) 3, (d) 4, (e) 5. SS

We define the direction of the electric field to be the same as the direction of the force on a positive test charge. Why then do we not define the direction of the magnetic field to be the same as the direction of the magnetic force on a moving positive test charge

A flicker bulb is a lightbulb that has a long, thin flexible filament. It is meant to be plugged into an ac outlet that delivers current at a frequency of 60 Hz. There is a small permanent magnet inside the bulb. When the bulb is plugged in the filament oscillates back and forth. At what frequency does it oscillate? Explain your answer

In a cyclotron, the potential difference between the dees oscillates with a period given by Show that the expression to the right of the equal sign has units of seconds if have units of coulombs, teslas, and kilograms, respectively

A nucleus has a charge equal to and a mass that is equal to the mass of seven protons. A nucleus and a proton are both moving perpendicular to a uniform magnetic field The magnitude of the momentum of the proton is equal to the magnitude of the momentum of the nucleus. The path of the proton has a radius of curvature equal to and the path of the nucleus has a radius of curvature equal to The ratio is closest to (a) (b) (c) (d) (e) (f

An electron moving in the direction enters a region that has a uniform magnetic field in the direction. When the electron enters this region, it will (a) be deflected toward the direction, (b) be deflected toward the direction, (c) be deflected toward the direction, (d) be deflected toward the direction, (e) continue undeflected in the direction

In a velocity selector, the speed of the undeflected charged particle is given by the ratio of the magnitude of the electric field to the magnitude of the magnetic field. Show that in fact does have the units of if and are in units of volts per meter and teslas, respectively

How are the properties of magnetic field lines similar to the properties of electric field lines? How are they different

True or false: (a) The magnetic moment of a bar magnet points from its north pole to its south pole. (b) Inside the material of a bar magnet, the magnetic field due to the bar magnet points from the magnets south pole toward its north pole. (c) If a current loop simultaneously has its current doubled and its area cut in half, then the magnitude of its magnetic moment remains the same. (d) The maximum torque on a current loop placed in a magnetic field occurs when the plane of the loop is perpendicular to the direction of the magnetic field.

Show that the von Klitzing constant, gives the SI unit for resistance (the ohm) if and e are in units of joule-seconds and coulombs, respectively.

The theory of relativity states that no law of physics can be described using the absolute velocity of an object, which is in fact impossible to define due to a lack of an absolute reference frame. Instead, the behavior of interacting objects can only be described by the relative velocities between the objects. New physical insights result from this idea. For example, in Figure 26-31 a magnet moving at high speed relative to some observer passes by an electron that is at rest relative to the same observer. Explain why you are sure that a force must be acting on the electron. In what direction will the force point at the instant the north pole of the magnet passes directly underneath the electron? Explain your answer.

Estimate the maximum magnetic force per meter that Earths magnetic field could exert on a current-carrying wire in a 20-A circuit in your house.

CONTEXT-RICH Your friend wants to be a magician and intends to use Earths magnetic field to suspend a current-carrying wire above the stage. He asks you to estimate the minimum current needed to suspend the wire just above Earths surface at the equator (where Earths magnetic field is horizontal). Assume the wire has a linear mass density of Would you advise him to proceed with his plans for this act?

Find the magnetic force on a proton moving in the direction at a speed of in a uniform magnetic field of in the direction.

Apoint particle has a charge equal to and a velocity equal to Find the force on the charge if the magnetic field is (a) (b) (c) and (d)

Auniform magnetic field equal to is in the direction. Find the force exerted by the field on a proton if the velocity of the proton is (a) (b) (c) and (d)

A straight wire segment that is long makes an angle of with a uniform 0.37- magnetic field. Find the magnitude of the force on the wire if the wire has a current of

A straight segment of a current-carrying wire has a current element where and The segment is in a region with a uniform magnetic field given by Find the force on the segment of wire.

What is the force on an electron that has a velocity equal to when it is in a region with a magnetic field given by 0.80 T i

The section of wire shown in Figure 26-32 carries a current equal to from to The segment is in a region that has a magnetic field whose value is Find the total force on the wire and show that the total force is the same as if the wire were in the form of a straight wire directly from to and carrying the same current.

A straight, stiff, horizontal 25-cm-long wire that has a mass equal to is connected to a source of emf by light, flexible leads. A magnetic field of is horizontal and perpendicular to the wire. Find the current necessary to float the wire, that is, when the wire is released from rest it remains at rest.

ENGINEERING APPLICATION In your physics laboratory class, you have constructed a simple gaussmeter for measuring the horizontal component of magnetic fields. The setup consists of a stiff 50-cm wire that hangs vertically from a conducting pivot so that its free end makes contact with a pool of mercury in a dish below (Figure 26-33). The mercury provides an electrical contact without constraining the movement of the wire. The wire has a mass of and conducts a current downward. (a) What is the equilibrium angular displacement of the wire from vertical if the horizontal component of the magnetic field is and the current is (b) What is the sensitivity of this gaussmeter? That is, what is the ratio of the output to the input (in radians per tesla)?

A 10-cm-long straight wire is parallel with the axis and carries a current of in the direction. The force on this wire due to the presence of a magnetic field is If this wire is rotated so that it is parallel with the axis with the current in the direction, the force on the wire becomes Determine the magnetic field

A 10-cm-long straight wire is parallel with the axis and carries a current of in the direction. The force on this wire due to a uniform magnetic field is If this wire is rotated so that it is parallel with the axis with the current in the direction, the force on the wire becomes Find BS0.20 k .

Acurrent-carrying wire is bent into a closed semicircular loop of radius that lies in the plane (Figure 26-34). The wire is in a uniform magnetic field that is in the direction, as shown. Verify that the force acting on the loop is zero. SSM

A wire bent in some arbitrary shape carries a current The wire is in a region with a uniform magnetic field Show that the total force on the part of the wire from some arbitrary point on the wire (designated as to some other arbitrary point on the wire (designated as is where is the vector from point to point In other words, show that the force on an arbitrary section of the bent wire is the same as the force on a straight section wire carrying the same current and connecting the two endpoints of the arbitrary section.

Aproton moves in a 65-cm-radius circular orbit that is perpendicular to a uniform magnetic field of magnitude (a) What is the orbital period for the motion? (b) What is the speed of the proton? (c) What is the kinetic energy of the proton?

A electron (an electron that has a kinetic energy equal to moves in a circular orbit that is perpendicular to a magnetic field of (a) Find the radius of the orbit. (b) Find the frequency and period of the orbital motion.

A proton, deuteron, and an alpha particle in a region with a uniform magnetic field each follow circular paths that have the same radius. The deuteron has a charge that is equal to the charge a proton has, and an alpha particle has a charge that is equal to twice the charge a proton has. Assume that Compare (a) their speeds, (b) their kinetic energies, and (c) the magnitudes of their angular momenta about the centers of the orbits.

A particle has a charge q, a mass m, a linear momentum of magnitude p, and a kinetic energy K. The particle moves in a circular orbit of radius R perpendicular to a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}\). Show that (a) p=BqR and (b) \(K=\frac{1}{2} B^2 q^2 R^2 / m\).

A beam of particles with velocity enters a region that has a uniform magnetic field in the direction. Show that when the component of the displacement of one of the particles is where is the angle between and the velocity of the particle is in the same direction as it was when the particle entered the field.

A proton that has a speed equal to enters a region with a uniform magnetic field that has a magnitude of and points into the page, as shown in Figure 26-35. The proton enters the region at an angle Find the exit angle and the distance d.

Suppose that in Figure 26-35, the magnetic field has a magnitude of the distance is and is Find the speed at which a particle enters the region and the exit angle if the particle is (a) a proton and (b) a deuteron. Assume that

The galactic magnetic field in some region of interstellar space has a magnitude of A particle of interstellar dust has a mass of and a total charge of How many years does it take for the particle to complete a revolution of the circular orbit caused by its interaction with the magnetic field?

A velocity selector has a magnetic field that has a magnitude equal to and is perpendicular to an electric field that has a magnitude equal to (a) What must the speed of a particle be for that particle to pass through the velocity selector undeflected? What kinetic energy must (b) protons and (c) electrons have in order to pass through the velocity selector undeflected?

A beam of protons is moving in the direction with a speed of through a region in which the electric field is perpendicular to the magnetic field. The beam is not deflected in this region. (a) If the magnetic field has a magnitude of and points in the direction, find the magnitude and direction of the electric field. (b) Would electrons that have the same velocity as the protons be deflected by these fields? If so, in what direction would they be deflected? If not, why not?

The plates of a Thomson apparatus are long and are separated by The end of the plates is from the tube screen. The kinetic energy of the electrons is If a potential difference of is applied across the deflection plates, by how much will the point where the beam of electrons strikes the screen be displaced?

Chlorine has two stable isotopes, and Chlorine gas which consists of singly ionized ions is to be separated into its isotopic components using a mass spectrometer. The magnetic field strength in the spectrometer is What is the minimum value of the potential difference through which these ions must be accelerated so that the separation between them, after they complete their semicircular path, is

In a mass spectrometer, a singly ionized ion has a mass equal to and is accelerated through a 2.50-kV potential difference. It then enters a region where it is deflected by a magnetic field of (a) Find the radius of curvature of the ions orbit. (b) What is the difference in the orbital radii of the 26Mg and 24Mg ions? Assume that their mass ratio is 26:24. SSM

A beam of singly ionized and ions passes through a velocity selector and enters a region of uniform magnetic field with a velocity that is perpendicular to the direction of the field. If the diameter of the orbit of the ions is what is the diameter of the orbit for ions? Assume their mass ratio is

Using Example 26-6, determine the time required for a ion and a ion to complete the semicircular path.

Before entering a mass spectrometer, ions pass through a velocity selector consisting of parallel plates that are separated by and have a potential difference of The magnetic field strength is in the region between the plates. The magnetic field strength in the mass spectrometer is Find (a) the speed of the ions entering the mass spectrometer and (b) the difference in the diameters of the orbits of singly ionized and The mass of a ion is

A cyclotron for accelerating protons has a magnetic field strength of and a radius of (a) What is the cyclotrons frequency? (b) Find the kinetic energy of the protons when they emerge. (c) How will your answers change if deuterons are used instead of protons?

A certain cyclotron that has a magnetic field whose magnitude is is designed to accelerate protons to a kinetic energy of (a) What is the cyclotron frequency for this cyclotron? (b) What must the minimum radius of the magnet be to achieve this energy? (c) If the alternating potential difference applied to the dees has a maximum value of how many revolutions must the protons make before emerging with kinetic energies of

Show that for a given cyclotron the cyclotron frequency for accelerating deuterons is the same as the frequency for accelerating alpha particles and is half the frequency for accelerating protons in the same magnetic field. The deuteron has a charge that is equal to the charge a proton has, and an alpha particle has a charge that is equal to twice the charge a proton has. Assume that

Show that the radius of the orbit of a charged particle in a cyclotron is proportional to the square root of the number of orbits completed.

Asmall circular coil consisting of 20 turns of wire lies in a region with a uniform magnetic field whose magnitude is The arrangement is such that the normal to the plane of the coil makes an angle of with the direction of the magnetic field. The radius of the coil is and the wire carries a current of (a) What is the magnitude of the magnetic moment of the coil? (b) What is the magnitude of the torque exerted on the coil?

What is the maximum torque on a 400-turn circular coil of radius that carries a current of and is in a region with a uniform magnetic field of

A current-carrying wire is in the shape of a square of edge length The square lies in the plane. The wire carries a current of What is the magnitude of the torque on the wire if it is in a region with a uniform magnetic field of magnitude 0.30 T that points in the (a) _z direction and (b) _x direction? SSM

Acurrent-carrying wire is in the shape of an equilateral triangle of edge length The triangle lies in the plane. The wire carries a current of What is the magnitude of the torque on the wire if it is in a region with a uniform magnetic field of magnitude that points in the (a) direction and (b) direction?

A rigid wire is in the shape of a square of edge length The square has mass and the wire carries current The square lies on a flat horizontal surface in a region where there is a magnetic field of magnitude that is parallel to two edges of the square. What is the minimum value of so that one edge of the square will lift off the surface?

A rectangular current-carrying 50-turn coil, as shown in Figure 26-36, is pivoted about the axis. (a) If the wires in the plane make an angle with the axis, what angle does the magnetic moment of the coil make with the unit vector (b) Write an expression for in terms of the unit vectors and where is a unit vector in the direction of the magnetic moment. (c) What is the magnetic moment of the coil? (d) Find the torque on the coil when there is a uniform magnetic field in the region occupied by the coil. (e) Find the potential energy of the coil in this field. (The potential energy is zero when u _ 0.)

For the coil in Problem 52 the magnetic field is now Find the torque exerted on the coil when is equal to (a) (b) (c) and (d)

A small bar magnet has a length equal to and its magnetic moment is aligned with a uniform magnetic field of magnitude The bar magnet is then rotated through an angle of about an axis perpendicular to its length. The observed torque on the bar magnet has a magnitude of (a) Find the magnetic moment of the magnet. (b) Find the potential energy of the magnet.

A wire loop consists of two semicircles connected by straight segments (Figure 26-37). The inner and outer radii are \(0.30 \mathrm{~m}\) and \(0.50 \mathrm{~m}\), respectively. A current of \(1.5 \mathrm{~A}\) is in this wire and the current in the outer semicircle is in the clockwise direction. What is the magnetic moment of this current loop?

A wire of length is wound into a circular coil that has turns. Show that when the wire carries a current the magnetic moment of the coil has a magnitude given by IL2>(4pN).

Aparticle that has a charge and a mass moves with angular velocity in a circular path of radius (a) Show that the average current created by this moving particle is and that the magnetic moment of its orbit has a magnitude of (b) Show that the angular momentum of this particle has the magnitude of and that the magnetic moment and angular momentum vectors are related by where is the angular momentum about the center of the circle.

A uniformly charged nonconducting cylindrical shell (Figure 26-38) has length inner and outer radii and respectively, a charge density and an angular velocity about its axis. Derive an expression for the magnetic moment of the cylinder.

A uniform nonconducting thin rod of mass and length has a uniform charge per unit length and rotates with angular speed about an axis through one end and perpendicular to the rod. (a) Consider a small segment of the rod of length and charge at a distance from the pivot (Figure 26-39). Show that the average current created by this moving segment is and show that the magnetic moment of this segment is (b) Use this to show that the magnitude of the magnetic moment of the rod is (c) Show that the magnetic moment and angular momentum are related by where is the total charge on the rod.

A nonuniform, nonconducting thin disk of mass radius and total charge has a charge per unit area that varies as and a mass per unit area that is given by The disk rotates with angular speed about its central axis. (a) Show that the magnetic moment of the disk has a magnitude which can be alternatively rewritten as (b) Show that the magnetic moment and angular momentum are related by

A spherical shell of radius carries a constant surface charge density The shell rotates about its diameter with angular speed Find the magnitude of the magnetic moment of the rotating spherical shell.

A uniform, solid, uniformly charged sphere of radius has a volume charge density The sphere rotates about an axis through its center with angular speed Find the magnitude of the magnetic moment of the rotating sphere

Auniform, thin, uniformly charged disk of mass radius and uniform surface charge density rotates with angular speed about an axis through its center and perpendicular to the disk (Figure 26-40). The disk is in a region with a uniform magnetic field that makes an angle with the rotation axis. Calculate (a) the magnitude of the torque exerted on the disk by the magnetic field and (b) the precession frequency of the disk in the magnetic field.

Ametal strip that is wide and thick carries a current of in a region with a uniform magnetic field of as shown in Figure 26-41. The Hall voltage is measured to be (a) Calculate the drift speed of the free electrons in the strip. (b) Find the number density of the free electrons in the strip. (c) Is point a or point b at the higher potential? Explain your answer.

The number density of free electrons in copper is electrons per cubic centimeter. If the metal strip in Figure 26-41 is copper and the current is find (a) the drift speed and (b) the potential difference Assume that the magnetic field strength is

ENGINEERING APPLICATION A copper strip has free electrons per cubic centimeter, is wide, is thick, and is used to measure the magnitudes of unknown magnetic fields that are perpendicular to it. Find the magnitude when the current is and the Hall voltage is (a) (b) and (c)

BIOLOGICAL APPLICATION Because blood contains ions, moving blood in the presence of a magnetic field develops a Hall voltage across the diameter of an artery. Alarge artery that has a diameter of can have blood flowing through it with a maximum speed of If a section of the artery is in a magnetic field of what is the maximum potential difference across the diameter of the artery?

The Hall coefficient is a property of conducting material (just as resistivity is). It is defined as where is the component of the current density in the material, is the component of the magnetic field, and is the component of the resulting Hall electric field. Show that the Hall coefficient is equal to where is the charge of the charge carriers (-e if they are electrons). (The Hall coefficients of monovalent metals, such as copper, silver, and sodium, are therefore negative.)

Aluminum has a density of and a molar mass of The Hall coefficient of aluminum is (See Problem 68 for the definition of What is the number of conduction electrons per aluminum atom?

A long wire parallel to the axis carries a current of in the direction. The wire occupies a region that has a uniform magnetic field Find the magnetic force per unit length on the wire.

An alpha particle travels in a circular path of radius in a region with a magnetic field whose magnitude is Find (a) the period, (b) the speed, and (c) the kinetic energy (in electron volts) of the alpha particle. (The mass of an alpha particle is

The pole strength of a bar magnet is defined by where is the magnetic moment of the magnet and is the position of the north-pole end of the magnet relative to the south-pole end. Show that the torque exerted on a bar magnet in a uniform magnetic field is the same as if a force is exerted on the north pole of the magnet and a force is exerted on the south pole.

A particle of mass and charge enters a region where there is a uniform magnetic field parallel with the axis. The initial velocity of the particle is so the particle moves in a helix. (a) Show that the radius of the helix is (b) Show that the particle takes a time to complete each turn of the helix. (c) What is the component of the displacement of the particle during the time given in Part (b)?

A metal crossbar of mass rides on a parallel pair of long horizontal conducting rails separated by a distance and connected to a device that supplies constant current to the circuit, as shown in Figure 26-42. The circuit is in a region with a uniform magnetic field whose direction is vertically downward. There is no friction and the bar starts from rest at (a) In which direction will the bar start to move? (b) Show that at time the bar has a speed of

Assume that the rails Problem 74 are frictionless but tilted upward so that they make an angle with the horizontal, and with the current source attached to the low end of the rails. The magnetic field is still directed vertically downward. (a) What minimum value of is needed to keep the bar from sliding down the rails? (b) What is the acceleration of the bar if is twice the value found in Part (a)? SSM

A long, narrow bar magnet that has magnetic moment parallel to its long axis is suspended at its center as a frictionless compass needle. When placed in region with a horizontal magnetic field the needle lines up with the field. If it is displaced by a small angle show that the needle will oscillate about its equilibrium position with frequency where is the moment of inertia of the needle about the point of suspension.

A straight 20-m-long conducting wire is parallel to the axis and is moving in the direction with a speed of in a region that has a magnetic field given by (a)Amagnetic force acting on the conduction electrons leaves one end negatively charged due to an electron surplus and the other end positively charged due to an electron deficit. This charge separation process continues until the electric field due to the accumulated positive and negative charges exerts forces on the remaining conduction electrons that exactly balance the magnetic forces acting on them. Find the magnitude and direction of this electric field in the steady-state situation. (b) Which end of the wire is positively charged and which end is negatively charged? (c) Suppose the moving wire is long. What is the potential difference between its two ends due to this electric field?

A circular loop of wire that has a mass and a constant current is in a region with a uniform magnetic field. It is initially in equilibrium and its magnetic moment is aligned with the magnetic field. The loop is given a small angular displacement about an axis through its center and perpendicular to the magnetic field and then released. What is the period of the subsequent motion? (Assume that the only torque exerted on the loop is due to the magnetic field and that there are no other forces acting on the loop.)

A small bar magnet has a magnetic moment that makes an angle with the axis. The magnet is in a region that has a nonuniform magnetic field given by Using and show that there is a net magnetic force on the magnet that is given by

A proton, a deuteron, and an alpha particle all have the same kinetic energy. They are moving in a region with a uniform magnetic field that is perpendicular to each of their velocities. Let and be the radii of their circular orbits, respectively. The deuteron has a charge that is equal to the charge a proton has, and an alpha particle has a charge that is equal to twice the charge a proton has. Find the ratios and Assume that

ENGINEERING APPLICATION, CONTEXT-RICH Your forensic chemistry group, working closely with local law enforcement agencies, has acquired a mass spectrometer similar to that discussed in the text. It employs a uniform magnetic field that has a magnitude of To calibrate the mass spectrometer, you decide to measure the masses of various carbon isotopes by measuring the position of impact of the various singly ionized carbon ions that have entered the spectrometer with a kinetic energy of A wire chamber with position sensitivity of is part of the apparatus. What will be the limit on its mass resolution (in kg) for ions in this mass range, that is, those whose mass is on the order of that of a carbon atom?