Chapter 28

A long, straight wire carries a current of 2.5 A. a) What is the strength of the magnetic field at a distance of \(3.9 \mathrm{~cm}\) from the wire? b) If the wire still carries \(2.5 \mathrm{~A}\), but is used to form a long solenoid with 32 turns per centimeter and a radius of \(3.9 \mathrm{~cm}\) what is the strength of the magnetic field at the center of the solenoid?

A particle detector utilizes a solenoid that has 550 turns of wire per centimeter. The wire carries a current of 22 A. A cylindrical detector that lies within the solenoid has an inner radius of \(0.80 \mathrm{~m} .\) Electron and positron beams are directed into the solenoid parallel to its axis. What is the minimum momentum perpendicular to the solenoid axis that a particle can have if it is to be able to enter the detector?

An electron has a spin magnetic moment of magnitude \(\mu=9.285 \cdot 10^{-24} \mathrm{~A} \mathrm{~m}^{2}\). Consequently, it has energy associated with its orientation in a magnetic field. If the difference between the energy of an electron that is "spin up" in a magnetic field of magnitude \(B\) and the energy of one that is "spin down" in the same magnetic field (where "up" and "down" refer to the direction of the magnetic field) is \(9.460 \cdot 10^{-25} \mathrm{~J}\), what is the field magnitude, \(B\) ?

When a magnetic dipole is placed in a magnetic field, it has a natural tendency to minimize its potential energy by aligning itself with the field. If there is sufficient thermal energy present, however, the dipole may rotate so that it is no longer aligned with the field. Using \(k_{\mathrm{B}} T\) as a measure of the thermal energy, where \(k_{\mathrm{B}}\) is Boltzmann's constant and \(T\) is the temperature in kelvins, determine the temperature at which there is sufficient thermal energy to rotate the magnetic dipole associated with a hydrogen atom from an orientation parallel to an applied magnetic field to one that is antiparallel to the applied field. Assume that the strength of the field is \(0.15 \mathrm{~T}\)

If you want to construct an electromagnet by running a current of 3.0 A through a solenoid with 500 windings and length \(3.5 \mathrm{~cm}\) and you want the magnetic field inside the solenoid to have magnitude \(B=2.96 \mathrm{~T}\), you can insert a ferrite core into the solenoid. What value of the relative magnetic permeability should this ferrite core have in order to make this work?

What is the magnitude of the magnetic field inside a long, straight tungsten wire of circular cross section with diameter \(2.4 \mathrm{~mm}\) and carrying a current of \(3.5 \mathrm{~A}\), at a distance of \(0.60 \mathrm{~mm}\) from its central axis?

Consider a model of the hydrogen atom in which an electron orbits a proton in the plane perpendicular to the proton's spin angular momentum (and magnetic dipole moment) at a distance equal to the Bohr radius, \(a_{0}=5.292 \cdot 10^{-11} \mathrm{~m} .\) (This is an oversimplified classical model.) The spin of the electron is allowed to be either parallel to the proton's spin or antiparallel to it; the orbit is the same in either case. But since the proton produces a magnetic field at the electron's location, and the electron has its own intrinsic magnetic dipole moment, the energy of the electron differs depending on its spin. The magnetic field produced by the proton's spin may be modeled as a dipole field, like the electric field due to an electric dipole discussed in Chapter 22 Calculate the energy difference between the two electronspin configurations. Consider only the interaction between the magnetic dipole moment associated with the electron's spin and the field produced by the proton's spin.

Consider an electron to be a uniformly dense sphere of charge, with a total charge of \(-e=-1.60 \cdot 10^{-19} \mathrm{C}\) spinning at an angular frequency, \(\omega\). a) Write an expression for its classical angular momentum of rotation, \(L\) b) Write an expression for its magnetic dipole moment, \(\mu\). c) Find the ratio, \(\gamma_{e}=\mu / L\), known as the gyromagnetic ratio.

You are standing at a spot where the magnetic field of the Earth is horizontal, points due northward, and has magnitude \(40.0 \mu \mathrm{T}\). Directly above your head, at a height of \(12.0 \mathrm{~m},\) a long, horizontal cable carries a steady \(\mathrm{DC}\) current of 500.0 A due northward. Calculate the angle \(\theta\) by which your magnetic compass needle is deflected from true magnetic north by the effect of the cable. Don't forget the sign of \(\theta-\) is the deflection eastward or westward?

The magnetic dipole moment of the Earth is approximately \(8.0 \cdot 10^{22} \mathrm{~A} \mathrm{~m}^{2}\). The source of the Earth's magnetic field is not known; one possibility might be the circulation culating ions move a circular loop of radius \(2500 \mathrm{~km}\). What “current" must they produce to yield the observed field?