(II) A Hall probe used to measure magnetic field

(a) What is the force per meter of length on a straight wire carrying a 9.40-A current when perpendicular to a 0.90-T uniform magnetic field? (b) What if the angle between the wire and field is 35.0?

Calculate the magnitude of the magnetic force on a 240-m length of wire stretched between two towers and carrying a 150-A current. The Earths magnetic field of 5.0 X 10-5 T makes an angle of 68 with the wire.

A 1.6-m length of wire carrying 4.5 A of current toward the south is oriented horizontally. At that point on the Earths surface, the dip angle of the Earths magnetic field makes an angle of 41 to the wire. Estimate the magnitude of the magnetic force on the wire due to the Earths magnetic field of 5.5 X 10 5 T at this point.

The magnetic force per meter on a wire is measured to be only 25 percent of its maximum possible value. Sketch the relationship of the wire and the field if the force had been a maximum, and sketch the relationship as it actually is, calculating the angle between the wire and the magnetic field.

The force on a wire is a maximum of 7.50 X 10-2 N when placed between the pole faces of a magnet. The current flows horizontally to the right and the magnetic field is vertical. The wire is observed to jump toward the observer when the current is turned on. (a) What type of magnetic pole is the top pole face? (b) If the pole faces have a diameter of 10.0 cm, estimate the current in the wire if the field is 0.220 T. (c) If the wire is tipped so that it makes an angle of 10.0 with the horizontal, what force will it now feel?

Suppose a straight 1.00-mm-diameter copper wire could just float horizontally in air because of the force due to the Earths magnetic field B, which is horizontal, perpendicular to the wire, and of magnitude 5.0 X 10_5T. What current would the wire carry? Does the answer seem feasible? Explain briefly.

A stiff wire 50.0 cm long is bent at a right angle in the middle. One section lies along the z axis and the other is along the line y = 2x in the xy plane. A current of 20.0 A flows in the wiredown the z axis and out the line in the xy plane. The wire passes through a uniform magnetic field given by B = (0.318i)T. Determine the magnitude and direction of the total force on the wire.

A long wire stretches along the x axis and carries a 3.0-A current to the right (+x). The wire is in a uniform magnetic field B = (0.20i - 0.36j + 0.25k) T. Determine the components of the force on the wire per cm of length.

A current-carrying circular loop of wire (radius r, current I) is partially immersed in a magnetic field of constant magnitude B0 directed out of the page as shown in Fig. 27-39. Determine the net force on the loop due to the field in terms of 0O. (Note that 0O points to the dashed line, above which B = 0.)

A 2.0-m-long wire carries a current of 8.2 A and is immersed within a uniform magnetic field B. When this wire lies along the +x axis, a magnetic force F = ( -2 .5 j) N acts on the wire, and when it lies on the +y axis, the force is F = (2.5i - 5.0k) N. Find B.

A curved wire, connecting two points a and b, lies in a plane perpendicular to a uniform magnetic field B and carries a current I. Show that the resultant magnetic force on the wire, no matter what its shape, is the same as that on a straight wire connecting the two points carrying the same current I. See Fig. 27-40. FIGURE 27-40 Problem 11.

A circular loop of wire, of radius r, carries current I. It is placed in a magnetic field whose straight lines seem to diverge from a point a distance d below the loop on its axis. (That is, the field makes an angle 0 with the loop at all points, Fig. 27-41, where tan0 = r /d .) Determine the force on the loop. FIGURE 27-41 Problem 12.

Determine the magnitude and direction of the force on an electron traveling 8.75 X 105 m /s horizontally to the east in a vertically upward magnetic field of strength 0.45 T.

(I) An electron is projected vertically upward with a speed of 1.70 X 106m /s into a uniform magnetic field of 0.480 T that is directed horizontally away from the observer. Describe the electrons path in this field.

(I) Alpha particles o f charge q = +2e and mass m = 6.6 X 10-27 kg are emitted from a radioactive source at a speed of 1.6 X 107m/s. What magnetic field strength would be required to bend them into a circular path of radius r = 0.18 m?

(I) Find the direction of the force on a negative charge for each diagram shown in Fig. 27-42, where v (green) is the velocity of the charge and B (blue) is the direction of the magnetic field. ( means the vector points inward. O means it points outward, toward you.)

(I) Determine the direction of B for each case in Fig. 27-43, where F represents the maximum magnetic force on a positively charged particle moving with velocity v. FIGURE 27-43 Problem 17.

(II) What is the velocity of a beam of electrons that goes undeflected when passing through perpendicular electric and magnetic fields of magnitude 8.8 X 103V/m and 7.5 X 10-3 T, respectively? What is the radius of the electron orbit if the electric field is turned off?

(II) A doubly charged helium atom whose mass is 6.6 X 10-27kg is accelerated by a voltage of 2700 V. (a) What will be its radius of curvature if it moves in a plane perpendicular to a uniform 0.340-T field? (b) What is its period of revolution?

(II) A proton (mass rap), a deuteron (ra = 2rap,Q = e), and an alpha particle (ra = 4rap, Q = 2e) are accelerated by the same potential difference V and then enter a uniform magnetic field B, where they move in circular paths perpendicular to B. Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton.

(II) For a particle of mass ra and charge q moving in a circular path in a magnetic field B, (a) show that its kinetic energy is proportional to r2, the square of the radius of curvature of its path, and (b) show that its angular momentum is L = qBr2, about the center of the circle.

(II) An electron moves with velocity v = (7.0i 6.0j) X 104m/s in a magnetic field B = (0.80i + 0.60j)T. Determine the magnitude and direction of the force on the electron.

(II) A 6.0-MeV (kinetic energy) proton enters a 0.20-T field, in a plane perpendicular to the field. What is the radius of its path? See Section 23-8.

(II) An electron experiences the greatest force as it travels 2.8 X 106m/s in a magnetic field when it is moving northward. The force is vertically upward and of magnitude 8.2 X 1 0 13 N. What is the magnitude and direction of the magnetic field?

(II) A proton moves through a region of space where there is a magnetic field B = (0.45i + 0.38j)T and an electric field E = (3.0i - 4.2j) X 103V/m. At a given instant, the protons velocity is v = (6.0i + 3.0j - 5.0k) X 103m/s. Determine the components of the total force on the proton.

(II) An electron experiences a force F = (3.8i - 2.7j) X10-13N when passing through a magnetic field B = (0.85 T)k. Determine the electrons velocity.

(II) A particle of charge q moves in a circular path of radius r in a uniform magnetic field B. If the magnitude of the magnetic field is doubled, and the kinetic energy of the particle remains constant, what happens to the angular

(II) An electron enters a uniform magnetic field B = 0.28 T at a 45 angle to B. Determine the radius r and pitch p (distance between loops) of the electrons helical path assuming Fig. 27-44. FIGURE 27-44 Problem 28.

(II) A particle with charge q and momentum p, initially moving along the x axis, enters a region where a uniform magnetic field B = 2?0k extends over a width x = as shown in Fig. 27-45. The particle is deflected a distance d in the +y direction as it traverses the field. Determine (a) whether q is positive or negative, and (b) the magnitude of its momentum p. FIGURE 27-45 0 Problem 29.

(II) The path of protons emerging from an accelerator must be bent by 90 by a bending magnet so as not to strike a barrier in their path a distance d from their exit hole in the accelerator. Show that the field B in the bending magnet, which we assume is uniform and can extend over an area d X d, must have magnitude B > (2mK/e2d2)2, where ra is the mass of a proton and K is its kinetic energy.

(Ill) Suppose the Earths magnetic field at the equator has magnitude 0.50 X 10_4T and a northerly direction at all points. Estimate the speed a singly ionized uranium ion (ra = 238 u, q = e) would need to circle the Earth 5.0 km above the equator. Can you ignore gravity? [Ignore relativity.]

(Ill) A 3.40-g bullet moves with a speed of 155 m/s perpendicular to the Earths magnetic field of 5.00 X 10 5 T. If the bullet possesses a net charge of 18.5 X 10_9C, by what distance will it be deflected from its path due to the Earths magnetic field after it has traveled 1.00 km?

(Ill) A proton moving with speed v = 1.3 X 105m/s in a field-free region abruptly enters an essentially uniform magnetic field B = 0.850 T (B v). If the proton enters the magnetic field region at a 45 angle as shown in Fig. 27-46, (a) at what angle does it leave, and (b) at what distance x does it exit the field? FIGURE 27-46

(Ill) A particle with charge +q and mass m travels in a uniform magnetic field B = B0k. At time t = 0, the particles speed is v0 and its velocity vector lies in the xy plane directed at an angle of 30 with respect to the y axis as shown in Fig. 27-47. At a later time t = ta, the particle will cross the x axis at x = a. In terms of q, m, v0, and B0, determine (a) a, and (b) ta.

(I) How much work is required to rotate the current loop (Fig. 27-22) in a uniform magnetic field B from (a) 6 = 0 (jx||B) to 6 = 180, (b) Q = 90 to 6 = -90?

(I) A 13.0-cm-diameter circular loop of wire is placed with the plane of the loop parallel to the uniform magnetic field between the pole pieces of a large magnet. When 4.20 A flows in the coil, the torque on it is 0.185 m N. What is the magnetic field strength?

(II) A circular coil 18.0 cm in diameter and containing twelve loops lies flat on the ground. The Earths magnetic field at this location has magnitude 5.50 X 105 T and points into the Earth at an angle of 66.0 below a line pointing due north. If a 7.10-A clockwise current passes through the coil, determine (a) the torque on the coil, and (b) which edge of the coil rises up, north, east, south, or west.

(II) Show that the magnetic dipole moment |x of an electron orbiting the proton nucleus of a hydrogen atom is related to the orbital momentum L of the electron by a = - LT.

(II) A 15-loop circular coil 22 cm in diameter lies in the xy plane. The current in each loop of the coil is 7.6 A clockwise, and an external magnetic field B = (0.55i + 0.60j - 0.65k) T passes through the coil. Determine (a) the magnetic moment of the coil, jl; (b) the torque on the coil due to the external magnetic field; (c) the potential energy U of the coil in the field (take the same zero for U as we did in our discussion of Fig. 27-22).

(Ill) Suppose a nonconducting rod of length d carries a uniformly distributed charge Q. It is rotated with angular velocity <a about an axis perpendicular to the rod at one end, Fig. 27-48. Show that the magnetic dipole moment of this rod is \Q(od2. [Hint: Consider the motion of each infinitesimal length of the rod.] FIGURE 27-48 Problem 40

(I) If the current to a motor drops by 12%, by what factor does the output torque change?

(I) A galvanometer needle deflects full scale for a 63.0-/iA current. What current will give full-scale deflection if the magnetic field weakens to 0.800 of its original value?

(I) If the restoring spring of a galvanometer weakens by 15% over the years, what current will give full-scale deflection if it originally required 46 /iA?

(I) What is the value of q/m for a particle that moves in a circle of radius 8.0 mm in a 0.46-T magnetic field if a crossed 260-V/m electric field will make the path straight?

(II) An oil drop whose mass is determined to be 3.3 X 10-15 kg is held at rest between two large plates separated by 1.0 cm as in Fig. 27-31. If the potential difference between the plates is 340 V, how many excess electrons does this drop have?

(II) A Hall probe, consisting of a rectangular slab of current-carrying material, is calibrated by placing it in a known magnetic field of magnitude 0.10 T. When the field is oriented normal to the slabs rectangular face, a Hall emf of 12 mV is measured across the slabs width. The probe is then placed in a magnetic field of unknown magnitude B, and a Hall emf of 63 mV is measured. Determine B assuming that the angle 6 between the unknown field and the plane of the slabs rectangular face is (a) 6 = 90, and (b) 6 = 60.

(II) A Hall probe used to measure magnetic field strengths consists of a rectangular slab of material (free-electron density n) with width d and thickness t, carrying a current I along its length L The slab is immersed in a magnetic field of magnitude B oriented perpendicular to its rectangular face (of area Id), so that a Hall emf is produced across its width d. The probes magnetic sensitivity, defined as Kh = %H/IB , indicates the magnitude of the Hall emf achieved for a given applied magnetic field and current. A slab with a large KH is a good candidate for use as a Hall probe, (a) Show that = 1 lent. Thus, a good Hall probe has small values for both n and t. (b) As possible candidates for the material used in a Hall probe, consider (i) a typical metal {n 1 X 1029/m 3) and (ii) a (doped) semiconductor (n 3 X 1022/m 3). Given that a semiconductor slab can be manufactured with a thickness of 0.15 mm, how thin (nm) should a metal slab be to yield a value equal to that of the semiconductor slab? Compare this metal slab thickness with the 0.3-nm size of a typical metal atom, (c) For the typical semiconductor slab described in part (b), what is the expected value for when I = 100 mA and B = 0.1 T?

(II) A rectangular sample of a metal is 3.0 cm wide and 680 [xm thick. When it carries a 42-A current and is placed in a 0.80-T magnetic field it produces a 6.5-/jlV Hall emf. Determine: (a) the Hall field in the conductor; (b) the drift speed of the conduction electrons; (c) the density of free electrons in the metal.

(II) In a probe that uses the Hall effect to measure magnetic fields, a 12.0-A current passes through a 1.50-cm-wide 1.30-mm-thick strip of sodium metal. If the Hall emf is 1.86 /xV, what is the magnitude of the magnetic field (take it perpendicular to the flat face of the strip)? Assume one free electron per atom of Na, and take its specific gravity to be 0.971.

(II) The Hall effect can be used to measure blood flow rate because the blood contains ions that constitute an electric current, (a) Does the sign of the ions influence the emf? (b) Determine the flow velocity in an artery 3.3 mm in diameter if the measured emf is 0.13 mV and B is 0.070 T. (In actual practice, an alternating magnetic field is used.)

(I) In a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2, and 22.8 cm. The largest radius corresponds to an atomic mass of 76 u. What are the atomic masses of the other isotopes?

(II) One form of mass spectrometer accelerates ions by a voltage V before they enter a magnetic field B. The ions are assumed to start from rest. Show that the mass of an ion is m = qB2R 2/2V, where R is the radius of the ions path in the magnetic field and q is their charge.

(II) Suppose the electric field between the electric plates in the mass spectrometer of Fig. 27-33 is 2.48 X 104 V /m and the magnetic fields are B = B' = 0.58 T. The source contains carbon isotopes o f mass numbers 12,13, and 14 from a long dead piece of a tree. (To estimate atomic masses, multiply by 1.66 X 10-27 kg.) How far apart are the lines formed by the singly charged ions of each type on the photographic film? What if the ions were doubly charged?

(II) A mass spectrometer is being used to monitor air pollutants. It is difficult, however, to separate molecules with nearly equal mass such as CO (28.0106 u) and N2 (28.0134 u). How large a radius of curvature must a spectrometer have if these two molecules are to be separated at the film or detectors by 0.65 mm?

(II) An unknown particle moves in a straight line through crossed electric and magnetic fields with E = 1.5 kV/m and B = 0.034 T. If the electric field is turned off, the particle moves in a circular path of radius r = 2.7 cm. What might the particle be?

Protons move in a circle of radius 5.10 cm in a 0.625-T magnetic field. What value of electric field could make their paths straight? In what direction must the electric field point?

Protons with momentum 3.8 X 10_16k g -m /s are magnetically steered clockwise in a circular path 2.0 km in diameter at Fermi National Accelerator Laboratory in Illinois. Determine the magnitude and direction of the field in the magnets surrounding the beam pipe.

A proton and an electron have the same kinetic energy upon entering a region of constant magnetic field. What is the ratio of the radii of their circular paths?

Two stiff parallel wires a distance d apart in a horizontal plane act as rails to support a light metal rod of mass m (perpendicular to each rail), Fig. 27-49. A magnetic field B, directed vertically upward (outward in diagram), acts throughout. At t = 0, a constant current I begins to flow through the system. Determine the speed of the rod, which starts from rest at t = 0, as a function of time (a) assuming no friction between the rod and the rails, and (b) if the coefficient of friction is . (c) In which direction does the rod move, east or west, if the current through it heads north?FIGURE 27-49 Looking down on a rod sliding on rails. Problems 59 and 60.

Suppose the rod in Fig. 2 7-49 (Problem 59) has mass m = 0.40 kg and length 22 cm and the current through it is I = 36 A . If the coefficient of static friction is fxs = 0.50, determine the minimum magnetic field B (not necessarily vertical) that will just cause the rod to slide. Give the magnitude of B and its direction relative to the vertical (outwards towards us).

Near the equator, the Earths magnetic field points almost horizontally to the north and has magnitude B = 0.50 X 10_4T. What should be the magnitude and direction for the velocity of an electron if its weight is to be exactly balanced by the magnetic force?

Calculate the magnetic force on an airplane which has acquired a net charge of 1850 jjlC and moves with a speed of 120 m/s perpendicular to the Earths magnetic field of 5.0 X 10-5 T.

A motor run by a 9.0-V battery has a 20 turn square coil with sides of length 5.0 cm and total resistance 24 fl. When spinning, the magnetic field felt by the wire in the coil is 0.020 T. What is the maximum torque on the motor?

Estimate the approximate maximum deflection of the electron beam near the center of a CRT television screen due to the Earths 5.0 X 10-5 T field. Assume the screen is 18 cm from the electron gun, where the electrons are accelerated (a) by 2.0 kV, or (b) by 28 kV. Note that in color TV sets, the beam must be directed accurately to within less than 1 mm in order to strike the correct phosphor. Because the Earths field is significant here, mu-metal shields are used to reduce the Earths field in the CRT. (See Section 23-9.)

The rectangular loop of wire shown in Fig. 27-22 has mass m and carries current I. Show that if the loop is oriented at an angle 0 1 (in radians), then when it is released it will e x e c u te s irn n le h a rm o n ic m o tio n ab o u t 0 = 0. C a lc u la te

The cyclotron (Fig. 27-50) is a device used to accelerate elementary particles such as protons to high speeds. Particles starting at point A with some initial velocity travel in circular orbits in the magnetic field B. The particles are accelerated to higher speeds each time they pass in the gap between the metal dees, where there is an electric field E. (There is no electric field within the hollow metal dees.) The electric field changes direction each half-cycle, due to an ac voltage V = V0 sin 2irft, so that the particles are increased in speed at each passage through the gap. (a) Show that the frequency / of the voltage must be / = Bq/2'irm, where q is the charge on the particles and m their mass, (b) Show that the kinetic energy of the particles increases by 2qVQ each revolution, assuming that the gap is small, (c) If the radius of the cyclotron is 0.50 m and the magnetic field strength is 0.60 T, what will be the maximum kinetic energy of accelerated protons in MeV? FIGURE 27-50 A cyclotron. Problem 66.

Magnetic fields are very useful in particle accelerators for beam steering; that is, magnetic fields can be used to change the beams direction without altering its speed (Fig. 27-51). Show how this could work with a beam of protons. What happens to protons that are not moving with the speed that the magnetic field is designed for? If the field extends over a region 5.0 cm wide and has a magnitude of 0.38 T, by approximately what angle will a beam of protons traveling at 0.85 X 107m/s be bent? FIGURE 27-51 Problem 67.

A square loop of aluminum wire is 20.0 cm on a side. It is to carry 15.0 A and rotate in a uniform 1.35-T magnetic field as shown in Fig. 27-52. (a) Determine the minimum diameter of the wire so that it will not fracture from tension or shear. Assume a safety factor of 10. * (See Table 12-2.) (b) What is the resistance of a single loop of this wire? FIGURE 2 7 -5 2

A sort of projectile launcher is shown in Fig. 27-53. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar touching the rails. A 1.8 T magnetic field is perpendicular to the plane of the circuit. If the rails are a distance d = 24 cm apart, and the bar has a mass of 1.5 g, what constant current flow is needed to accelerate the bar from rest to 25 m/s in a distance of 1.0 m? In what direction must the field point?

(a) What value of magnetic field would make a beam of electrons, traveling to the right at a speed of 4.8 X 106m/s, go undeflected through a region where there is a uniform electric field of 8400 V/m pointing vertically up? (b) What is the direction of the magnetic field if it is known to be perpendicular to the electric field? (c) What is the frequency of the circular orbit of the electrons if the electric field is turned off?

In a certain cathode ray tube, electrons are accelerated horizontally by 25 kV. They then pass through a uniform magnetic field B for a distance of 3.5 cm, which deflects them upward so they reach the top of the screen 22 cm away, 11 cm above the center. Estimate the value of B.

Zeeman effect. In the Bohr model of the hydrogen atom, the electron is held in its circular orbit of radius r about its proton nucleus by electrostatic attraction. If the atoms are placed in a weak magnetic field B, the rotation frequency of electrons rotating in a plane perpendicular to B is changed by an amount A f = eB Airm where e and m are the charge and mass of an electron. (a) Derive this result, assuming the force due to B is much less than that due to electrostatic attraction of the nucleus. (b) What does the sign indicate?

A proton follows a spiral path through a gas in a magnetic field of 0.018 T, perpendicular to the plane of the spiral, as shown in Fig. 27-54. In two successive loops, at points P and Q, the radii are 10.0 mm and 8.5 mm, respectively. Calculate the change in the kinetic energy of the proton as it travels from P to Q.

The net force on a current loop whose face is perpendicular to a uniform magnetic field is zero, since contributions to the net force from opposite sides of the loop cancel. However, if the field varies in magnitude from one side of the loop to the other, then there can be a net force on the loop. Consider a square loop with sides whose length is a, located with one side at x = b in the xy plane (Fig. 27-55). A magnetic field is directed along z, with a magnitude that varies with x according to B = B0 [ l If the current in the loop circulates counterclockwise (that is, the magnetic dipole moment of the loop is v along the z axis), find an expression for the net force on the loop. FIGURE 27-55 Problem 74.

The power cable for an electric trolley (Fig. 27-56) carries a horizontal current of 330 A toward the east. The Earths magnetic field has a strength 5.0 X 10-5 T and makes an angle of dip of 22 at this location. Calculate the magnitude and direction of the magnetic force on a 5.0-m length of this cable. /= ^ 0 A FIGURE 27-56 Problem 75.

A uniform conducting rod of length d and mass m sits atop a fulcrum, which is placed a distance d/4 from the rods left-hand end and is immersed in a uniform magnetic field of magnitude B directed into the page (Fig. 27-57). An object whose mass M is 8.0 times greater than the rods mass is hung from the rods left-hand end. What current (direction and magnitude) should flow through the rod in order for it to be balanced (i.e., be at rest horizontally) on the fulcrum? (Flexible connecting wires which exert negligible force on the rod are not shown.) FIGURE 27-57 Problem 76

In a simple device for measuring the magnitude B of a magnetic field, a conducting rod (length d = 1.0 m, mass m = 150 g) hangs from a friction-free pivot and is oriented so that its axis of rotation is aligned with the direction of the magnetic field to be measured. Thin flexible wires (which exert negligible force on the rod) carry a current I = 12 A, which causes the rod to deflect an angle 0 with respect to the vertical, where it remains at rest (Fig. 27-58). (a) Is the current flowing upward (toward the pivot) or downward in Fig. 27-58? (b) If 6 = 13, determine B. (c) What is the largest magnetic field magnitude that can be measured using this device? FIGURE 27-58 Problem 77.