Physics 2420 In-Class Problems: Magnetic Force

1. The earth's magnetic field is approximately 5 ´ 10^–5 T. If an electron is traveling perpendicular to the field at 1000 m/s, determine its cyclotron radius and frequency. Ignore the effects of gravity and the electric field of the atmosphere. The mass of an electron is 9.11 ´ 10^–31 kg.
2. A beam of protons moves in a circle of radius 0.25 m. The beam moves perpendicular to a 0.30 T magnetic field. (a) What is the speed of each proton? (b) Determine the cyclotron frequency of the protons. (c) Determine the magnitude of the centripetal force on each proton. Protons have mass 1.673 ´ 10^–27 kg.
3. A velocity selector with an electric field of 4.50 ´ 10³ V/m and a magnetic field of 0.100 T is used to select the speed of an ion of charge +e before it enters a mass spectrometer. A 0.400–T magnetic field then bends the ion into a circular path of radius 0.230 m. What is the mass of the ion? What is the atomic mass of the ion? What element is the ion? (Atomic mass is the mass in grams of one mole of the substance. N_A = 6.022 ´ 10²³)
4. Suppose that an ion source produces doubly ionized gold ions (Au⁺⁺), each with a mass of 3.27 ´ 10^–25 kg. The ions are passed through a velocity selector with E = 5.00 ´ 10³ V/m and B = 0.150 T. Then, a 0.500–T magnetic field causes the ions to follow a circular path. Determine the radius of the path.
5. Consider a thin wire carrying a constant current I. The wire has length L and lies in the xy plane at an angle of 45° to the x axis. The magnetic field in the area is B = (2xy, 2z+3, 5–yz). Determine the total magnetic force acting on the thin wire.
6. An arc of thin wire of radius R lies in the positive quadrant of the xy plane. It carries a current I clockwise. The magnetic field in the area is B = (2xy, 2z+3, 5–yz). Determine the total magnetic force acting on the thin wire.



7. A flat ribbon of length L and width W lies in the yz plane as shown below. It carries surface current K = kCz. The magnetic field in the area is B = (2xy, 2z+3, 5–yz). Determine the total magnetic force acting on the ribbon.



8. A semi–circular ribbon has inner radius R and outer radius R + a. It lies in the xy plane centered on the origin. The ribbon carries a surface current K = Crθ , where θ is the polar coordinate unit vector. The magnetic field in the area is B = (2xy, 2z+3, 5–yz). Determine the total magnetic force acting on the ribbon.



9. A flat disk of radius R lies in the xy plane as shown below. It carries surface charge density Σ . The disk is given angular velocity Ω φ where φ is the cylindrical rotational unit vector. Since the charge is moving, we now have a surface current density K = Σ sΩ φ where s is the distance from the centre of the disk. The magnetic field in the area is B = (2xy, 2z+3, 5–yz). Determine the total magnetic force acting on the disk.



10. A sphere of radius R is shown below. It carries volume charge density ρ . The sphere is given angular velocity Ω φ about the z axis where φ is the spherical rotational unit vector. Since the charge is moving, we now have a volume current density J = ρ rsin(θ )Ω φ where r is the radial distance from the centre of the sphere and angle θ is measured from the z-axis. The magnetic field in the area is B = (5, –y, z). Determine the total magnetic force acting on the sphere.



11. A tapered cylinder has its central axis oriented in the z direction. Its base is in the xy plane. The base has radius R, the top has radius a, and the cylinder has height h. A constant current I enters the base. The magnetic field in the area is B = (2xy, 2z+3, 5–yz). Determine the total magnetic force acting on the cylinder.



12. The force on a straight cylindrical wire of length L carrying a uniform current I in a magnetic field B₀ perpendicular to the wire is F = ILB₀. Show that you obtain exactly the same result even if the current density is not uniform but has the form J = z J(s,φ ) where z is the unit vector along the cylindrical axis, s is the cylindrical radial distance from the centre of the wire, and φ is the cylindrical angle. Hint: consider what I is in this case.

Questions? mike.coombes@kwantlen.ca