Chapter 20

\(\bullet\) 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 ion having charge \(+6 e\) is traveling horizontally to the left at 8.50 \(\mathrm{km} / \mathrm{s}\) when it enters a magnetic field that is perpendicular to its velocity and deflects it downward with an initial magnetic force of \(6.94 \times 10^{-15} \mathrm{N} .\) What are the direction and magnitude of this field? Illustrate your method of solving this problem with a diagram.

\(\bullet\) A proton traveling at 3.60 \(\mathrm{km} / \mathrm{s}\) suddenly enters a uniform magnetic field of 0.750 \(\mathrm{T}\) , traveling at an angle of \(55.0^{\circ}\) with the field lines (Figure 20.57\()\) . (a) Find the magnitude and direction of the force this magnetic field exerts on the proton. (b) If you can vary the direction of the proton's velocity, find the magnitude of the maximum and minimum forces you could achieve, and show how the velocity should be oriented to achieve these forces. (c) What would the answers to part (a) be if the proton were replaced by an electron traveling in the same way as the proton?

\(\cdot\) A particle having a mass of 0.195 g carries a charge of \(-2.50 \times 10^{-8} \mathrm{C} .\) The particle is given an initial horizontal northward velocity of \(4.00 \times 10^{4} \mathrm{m} / \mathrm{s} .\) What are the magnitude and direction of the minimum magnetic field that will balance the earth's gravitational pull on the particle?

At a given instant, a particle with a mass of \(5.00 \times 10^{-3} \mathrm{kg}\) and a charge of \(3.50 \times 10^{-8} \mathrm{C}\) has a velocity with a magnitude of \(2.00 \times 10^{5} \mathrm{m} / \mathrm{s}\) in the \(+y\) direction. It is moving in a uniform magnetic field that has magnitude 0.8 \(\mathrm{T}\) and is in the \(-x\) direction. What are (a) the magnitude and direction of the magnetic force on the particle and (b) its resulting acceleration?

If the magnitude of the magnetic force on a proton is \(F\) when it is moving at \(15.0^{\circ}\) with respect to the field, what is the magnitude of the force (in terms of \(F\) ) when this charge is moving at \(30.0^{\circ}\) with respect to the field?

\(\bullet \mathrm{A}^{9}\) Be nucleus containing four protons and five neutrons has a mass of \(1.50 \times 10^{-26} \mathrm{kg}\) and is traveling vertically upward at 1.35 \(\mathrm{km} / \mathrm{s} .\) If this particle suddenly enters a horizontal magnetic field of 1.12 T pointing from west to east, find the magnitude and direction of its acceleration vector the instant after it enters the field.

A particle with a charge of \(-2.50 \times 10^{-8} \mathrm{C}\) is moving with an instantaneous velocity of magnitude 40.0 \(\mathrm{km} / \mathrm{s}\) in the \(x y-\) plane at an angle of \(50^{\circ}\) counterclockwise from the \(+x\) axis. What are the magnitude and direction of the force exerted on this particle by a magnetic field with magnitude 2.00 T in the (a) \(-x\) direction, and (b) \(+z\) direction?

\(\bullet\) 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?

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?

\(\bullet\) In a cloud chamber experiment, a proton enters a uniform 0.250 T magnetic field directed perpendicular to its motion. You measure the proton's path on a photograph and find that it follows a circular arc of radius 6.13 \(\mathrm{cm} .\) How fast was the proton moving?

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 \(\mathrm{T}\) 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 deuteron particle (the nucleus of an isotope of hydrogen consisting of one proton and one neutron and having a mass of \(3.34 \times 10^{-27} \mathrm{kg}\) ) moving horizontally enters a uniform, vertical, 0.500 T magnetic field and follows a circular arc of radius 55.6 \(\mathrm{cm} .\) (a) How fast was this deuteron moving just before it entered the magnetic field and just after it came out of the field? (b) What would be the radius of the arc followed by a proton that entered the field with the same velocity as the deuteron?

\(\bullet\) 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 in a direction perpendicular to its original direction (Fig. \(20.60 ) .\) The beam travels a distance of 1.18 cm while in the field. What is the magnitude of the magnetic field?

A uniform magnetic field bends an electron in a circular arc of radius \(R .\) What will be the radius of the arc (in terms of \(R )\) if the field is tripled?

\(\bullet\) A beam of protons is accelerated through a potential dif- ference of 0.745 \(\mathrm{kV}\) and then enters a uniform magnetic field traveling perpendicular to the field. (a) What magnitude of field is needed to bend these protons in a circular arc of diameter 1.75 \(\mathrm{m} ?\) (b) What magnetic field would be needed to produce a path with the same diameter if the particles were electrons having the same speed as the protons?

\(\bullet\) A 3.25 g bullet picks up an electric charge of 1.65\(\mu C\) as it travels down the barrel of a rifle. It leaves the barrel at a speed of 425 \(\mathrm{m} / \mathrm{s}\) , traveling perpendicular to the earth's magnetic field, which has a magnitude of \(5.50 \times 10^{-4} \mathrm{T} .\) Calculate (a) the magnitude of the magnetic force on the bullet and (b) the magnitude of the bullet's acceleration due to the magnetic force at the instant it leaves the rifle barrel.

An electron in the beam of a TV picture tube is accelerated through a potential difference of 2.00 \(\mathrm{kV}\) . It then passes into a magnetic field perpendicular to its path, where it moves in a circular arc of diameter 0.360 \(\mathrm{m} .\) What is the magnitude of this field?

(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?

\(\bullet\) Singly ionized (onne 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 \(\mathrm{T}\) . 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?

\(\bullet\) 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 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 \(1.99 \times 10^{-26} \mathrm{kg}(12 \mathrm{C})\) 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 \(^{13} \mathrm{Cions}\) at the detec- tor at the end of the semicircle? Is this distance large enough to be easily observed?

A straight vertical wire carries a current of 1.20 \(\mathrm{A}\) down- ward in a region between the poles of a large electromagnet where the field strength is 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 this wire if the magnetic-field direction is (a) toward the east, (b) toward the south, (c) \(30.0^{\circ}\) south of west?

Magnetic force on a lightning bolt. Currents during lightning strikes can be up to \(50,000\) A (or more!). We can model such a strike as a \(50,000\) A vertical current perpendicular to the earth's magnetic field, which is about \(\frac{1}{2}\) gauss. What is the force on each meter of this current due to the earth's magnetic field?

A horizontal rod 0.200 \(\mathrm{m}\) long carries a current through a uniform horizontal magnetic field of magnitude 0.067 T that points perpendicular to the rod. If the magnetic force on this rod is measured to be \(0.13 \mathrm{N},\) what is the current flowing through the rod?

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 under normal household conditions?

\(\bullet\) Between the poles of a powerful magnet is a cylindrical uniform magnetic field with a diameter of 3.50 \(\mathrm{cm}\) and a strength of 1.40 \(\mathrm{T}\) . A wire carries a current through the center of the field at an angle of \(65.0^{\circ}\) to the magnetic field lines. If the wire experiences a magnetic force of \(0.0514 \mathrm{N},\) what is the current flowing in it?

A rectangular 10.0 \(\mathrm{cm}\) by 20.0 \(\mathrm{cm}\) circuit carrying an 8.00 \(\mathrm{A}\) current is oriented with its plane parallel to a uniform 0.750 T magnetic field (Figure 20.62\()\) . (a) Find the magnitude and direction of the magnetic force on each segment \((a b, b c,\) etc. \()\) of this circuit. Illustrate your answers with clear diagrams. (b) Find the magnitude of the net force on the entire circuit.

\(\cdot\) The plane of a 5.0 \(\mathrm{cm}\) by 8.0 \(\mathrm{cm}\) rectangular loop of wire is parallel to a 0.19 T magnetic field, and the loop carries a cur- rent of 6.2 \(\mathrm{A}\) . (a) What torque acts on the loop? (b) What is the magnetic moment of the loop?

A circular coil of wire 8.6 \(\mathrm{cm}\) in diameter has 15 turns and carries a current of 2.7 \(\mathrm{A}\) . The coil is in a region where the magnetic field is 0.56 \(\mathrm{T}\) (a) What orientation of the coil gives the maximum torque on the coil, and what is this maximum torque? (b) For what orientation of the coil is the magnitude of the torque 71\(\%\) of the maximum found in part (a)?

A solenoid having 165 turns and a cross-sectional area of 6.75 \(\mathrm{cm}^{2}\) carries a current of 1.20 A. If it is placed in a uniform 1.12 T magnetic field, find the torque this field exerts on the solenoid if its axis is oriented (a) perpendicular to the field, (b) parallel to the field, (c) at 35. \(0^{\circ}\) with the field.

\bullet A circular coil of 50 loops and diameter 20.0 \(\mathrm{cm}\) is lying flat on a tabletop, and carries a clockwise current of 2.50 A. A magnetic field of 0.450 \(\mathrm{T}\) , directed to the north and at an angle of \(45.0^{\circ}\) from the vertical down through the coil and into the tabletop is turned on. (a) What is the torque on the coil, and (b) which side of the coil (north or south) will tend to rise from the tabletop?

You want to produce a magnetic field of magnitude \(5.50 \times 10^{-4} \mathrm{T}\) at a distance of 0.040 \(\mathrm{m}\) from a long, straight wire's center. (a) What current is required to produce this field? (b) With the current found in part (a), how strong is the magnetic field 8.00 \(\mathrm{cm}\) from the wire's center?

Household magnetic fields. Home circuit breakers typically have current capacities of around 10 A. How large a magnetic field would such a current produce 5.0 \(\mathrm{cm}\) from a long- wire's center? How does this field compare with the strength of the earth's magnetic field?

\(\bullet\) (a) How large a current would a very long, straight wire have to carry so that the magnetic field 2.00 \(\mathrm{cm}\) from the wire is equal to 1.00 \(\mathrm{G}\) (comparable to the earth's northward-pointing magnetic field)? (b) If the wire is horizontal with the current running from east to west, at what locations would the magnetic field of the wire point in the same direction as the horizontal component of the earth's magnetic field? (c) Repeat part (b) except with the wire vertical and the current going upward.

Magnetic sensitivity of electric fish. In a problem dealing with electric fish in Chapter 19, we saw that these fish navigate by responding to changes in the current in seawater. This current is due to a potential difference of around 3.0 \(\mathrm{V}\) generated by the fish and is about 12 \(\mathrm{mA}\) within a centimeter or so from the fish. Receptor cells in the fish are sensitive to the current. since the current is at some distance from the fish, the sensitivity of these cells suggests that they might be responding to the magnetic field created by the current. To get some estimate of how sensitive the cells are, we can model the current as that of a long, straight wire with the receptor cells 2.0 \(\mathrm{cm}\) away. What is the strength of the magnetic field at the receptor cells?

\(\cdot\) In a conventional cheap flashlight, a straight copper strip runs along the tube of the flashlight to connect the bulb to the negative terminal of the battery at the bottom of the tube. If this strip carries a current of 0.65 A while you're holding the flashlight, what is the magnitude of the magnetic field at the surface of your hand, 0.30 \(\mathrm{cm}\) from the strip? (You can treat the strip as a long, thin, straight wire.) How does your answer compare to the earth's magnetic field?

\(\bullet\) If the magnetic field due to a long, straight current-carrying wire has a magnitude \(B\) at a distance \(R\) from the wire's center, how far away must you be (in terms of \(R\) ) for the magnetic field to decrease to \(B / 3 ?\)

\(\bullet\) A current in a long, straight wire produces a magnetic field of 8.0\(\mu\) t at 2.0 \(\mathrm{cm}\) from the wire's center. Answer the following questions without finding the current: (a) What is the magnetic field strength 4.0 \(\mathrm{cm}\) from the wire's center? (b) How far from the wire's center will the field be 1.0\(\mu \mathrm{T} ?\) (c) If the current were doubled, what would the field be 2.0 \(\mathrm{cm}\) from the wire's center?

A long, straight telephone cable contains six wires, each carrying a current of 0.300 A. The distances between wires can be neglected. (a) If the currents in all six wires are in the same direction, what is the magnitude of the magnetic field 2.50 \(\mathrm{m}\) from the cable? (b) If four wires carry currents in one direction and the other two carry currents in the opposite direction, what is the magnitude of the field 2.50 m from the cable?

Two long parallel transmission lines 40.0 \(\mathrm{cm}\) apart carry 25.0 \(\mathrm{A}\) and 75.0 A currents. Find all locations where the net magnetic field of the two wires is zero if these currents are in (a) the same direction, (b) opposite directions.

\(\cdot\) Two high-current transmission lines carry currents of 25 \(\mathrm{A}\) and 75 \(\mathrm{A}\) in the same direction and are suspended parallel to each other 35 \(\mathrm{cm}\) apart. If the vertical posts supporting these wires divide the lines into straight 15 \(\mathrm{m}\) segments, what magnetic force does each segment exert on the other? Is this force attractive or repulsive?

\(\bullet\) An electric bus operates by drawing current from two parallel overhead cables, at a potential difference of \(600 \mathrm{V},\) and spaced 55 \(\mathrm{cm}\) apart. When the power input to the bus's motor is at its maximum power of \(65 \mathrm{hp},(\) a) what current does it draw and (b) what is the attractive force per unit length between the cables?

A circular metal loop is 22 \(\mathrm{cm}\) in diameter. (a) How large a current must flow through this metal so that the magnetic field at its center is equal to the earth's magnetic field of \(0.50 \times\) \(10^{-4} \mathrm{T}\) (b) Show how the loop should be oriented so that it can cancel the earth's magnetic field at its center.

A closely wound circular coil with a diameter of 4.00 \(\mathrm{cm}\) has 600 turns and carries a current of 0.500 A. What is the magnetic field at the center of the coil?

\(\bullet\) A closely wound circular coil has a radius of 6.00 \(\mathrm{cm}\) and carries a current of 2.50 A. How many turns must it have if the magnetic field at its center is \(6.39 \times 10^{-4} \mathrm{T} ?\)

Currents in the brain. The magnetic field around the head has been measured to be approximately \(3.0 \times 10^{-8}\) gauss. Although the currents that cause this field are quite complicated, we can get a rough estimate of their size by modeling them as a single circular current loop 16 \(\mathrm{cm}\) (the width of a typical head) in diameter. What is the current needed to produce such a field at the center of the loop?

A closely wound, circular coil with radius 2.40 \(\mathrm{cm}\) has 800 turns. What must the current in the coil be if the magnetic field at the center of the coil is 0.0580 \(\mathrm{T}\) ?

\(\bullet\) Two circular concentric loops of wire lie on a tabletop, one inside the other. The inner loop has a diameter of 20.0 \(\mathrm{cm}\) and carries a clockwise current of 12.0 \(\mathrm{A}\) , as viewed from above, and the outer wire has a diameter of 30.0 \(\mathrm{cm} .\) What must be the magnitude and direction (as viewed from above) of the current in the outer loop so that the net magnetic field due to this combination of loops is zero at the common center of the loops?

\(\cdot\) A solenoid contains 750 coils of very thin wire evenly wrapped over a length of 15.0 \(\mathrm{cm} .\) Each coil is 0.800 \(\mathrm{cm}\) in diameter. If this solenoid carries a current of \(7.00 \mathrm{A},\) what is the magnetic field at its center?

\(\bullet\) As a new electrical technician, you are designing a large solenoid to produce a uniform 0.150 T magnetic field near its center. You have enough wire for 4000 circular turns, and the solenoid must be 1.40 m long and 2.00 \(\mathrm{cm}\) in diameter. What current will you need to produce the necessary field?