The figure gives the potential energy U of a magnetic dipole in an external magnetic field B, as a function of angle between the directions of B and the dipole moment. The vertical axis scale is set by Us = 2.2 × 10-4 J. The dipole can be rotated about an axle with negligible friction so as to change . Counterclockwise rotation from @= 0 yields positive values of p, and clockwise rotations yield negative values. The dipole is to be released at angle =0 with a rotational kinetic energy of 6.3 x 10-4 J, so that it rotates counterclockwise. To what maximum value of will it rotate? Number i Units -180° U (10-4J) U₁ -U₂ ° (degrees) 0 180°

Doubly-ionized particles with an atomic mass of 59.4 u are moving due N at a speed of 7.78 x 105 m/s when they enter a uniform magnetic field of strength, B = 35.5 mT. The ions, which are moving perpendicular to the field, follow a circular path of radius, r, and travel north a distance, h = 2.02 m, before leaving the field at an angle, 0, as shown in the figure. (a) What is the radius of the ions' circular path? m (b) At what angle do the ions leave the field? 0 = h 2

Problem 10: A magnetic needle with magnetic moment μ-0.065 Am2 is placed in a uniform magnetic field B0.65 T as shown in the figure. The angle between the direction of the magnetic moment and the direction of the magnetic field is θ 250 Randomized Variables μ= 0.065 Arn B- 0.65T θ 250 Δ Part (a) Express the potential energy U in terms oft, B, θ. | cos(p) sin(p) | cos(α) sin(a) cos(0) sin(0) ( HOMI 4 5 6 BACKSPACE CLEAR Submit Hint I give up! Hints: 1% deduction per hint. Hints remaining: 1 Feedback: 1% deduction per feedback. Part (b) Calculate the numerical value of U in J. Part (c) If θ can be changed, which value has the minimum potential energy? Part (d) If θ can be changed, which value has the maximum potential energy in degrees? Δ Part (e) Ignore the mass of the needle, express the work needed to change the angle from θ to α in terms oftı, B, θ, α Part (f) If α-1500, calculate the numerical value of the work in J.

A student makes a short electromagnet by winding 390 turns of wire around a wooden cylinder of diameter d = 4.1 cm. The coil is connected to a battery producing a current of 4.1A in the wire. (a) What is the magnitude of the magnetic dipole moment of this device? (b) At what axial distance z>> dwill the magnetic field have the magnitude 4.8 µT (approximately one-tenth that of Earth's magnetic field)? (a) Number i Units (b) Number i Units

The image below shows a mass spectrometer, an analytical instrument used to identify the various molecules in a sample by measuring their charge-to-mass ratio e/m. The sample is ionized, the positive ions are accelerated (starting from rest) through a potential difference ΔV, and they then enter a region of uniform magnetic field. The field bends the ions into circular trajectories, but after just half a circle they either strike the wall or pass through a small opening to a detector. As the accelerating voltage is slowly increased, different ions reach the detector and are measured. Typical design values are a magnetic field strength B = 0.627 T and a spacing between the entrance and exit holes d = 5.83 cm. a)What accelerating potential difference ΔV is required to detect N2+? b)What accelerating potential difference ΔV is required to detect O2+? c)What accelerating potential difference ΔV is required to detect CO+?

The magnetic moment of the Earth is approximately 8.00 x 1022 A ⋅ m2. Imagine that the planetary magnetic field were caused by the complete magnetization of a huge iron deposit with density 7 900 kg/m3 and approximately 8.50 x 1028 iron atoms/m3. (a) How many unpaired electrons, each with a magnetic moment of 9.27 x 10-24 A ⋅ m2, would participate? (b) At two unpaired electrons per iron atom, how many kilograms of iron would be present in the deposit?

A circular coil of 205 turns has a radius of 1.84 cm. (a) Calculate the current that results in a magnetic dipole moment of magnitude 3.44 A-m2. (b) Find the maximum magnitude of the torque that the coil, carrying this current, can experience in a uniform 46.8 mT magnetic field. (a) Number Units (b) Number Units

Suppose that the dipole moment associated with an iron atom of an iron bar is 2.8 x 1023 J/T. The bar is 5.2 cm long and has a cross-sectional area of 1.5 cm2. Two-thirds of the atoms have their dipole moment in one directions and remaining one-third have their dipole moment in the opposite direction. What torque (in N*m) must be exerted to hold this magnet perpendicular to an external field of 1.2 T? (The density of iron is 7.9 g/cm3 and its molar mass is 55.9 g/mol. Avogadro's number is 6.022*1023 atoms/mol.)

A magnetic dipole with a dipole moment of magnitude 0.016 J/T is released from rest in a uniform magnetic field of magnitude 58 mT. The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is 0.60 mJ. (a) What is the initial angle between the dipole moment and the magnetic field? _____________°(b) What is the angle between the dipole moment and the magnetic field when the dipole is next (momentarily) at rest? ______________°

A permanent magnet has a magnetic dipole moment of 0.160 A · m2. The magnet is in the presence of an external uniform magnetic field (provided by current-carrying coils) with a magnitude of 0.0800 T, which makes an angle of 17.0° with the orientation of the permanent magnet.   (a)What is the magnitude of the torque (in N · m) on the permanent magnet?       (b)What is the potential energy (in J) of the system consisting of the permanent magnet and the magnetic field provided by the coils?