with a positively charged electrode, an anode, the electrons will be attracted to it and move radially outward as indicated in Figure 1. On their way to the anode they will acquire a kinetic energy , where is the potential difference, or voltage, between the heated V filament, called the and the anode. cathode, Figure 1 Most of the electrons will strike the anode. However, if one cuts a narrow slit into the anode, those electrons that started out toward the slit will exit through it as a narrow beam with a kinetic energy Ek. The Magnetic Field According to a wire Ampere’s law Figure 2 Magnetic field of carrying a current is surrounded by I a straight wire. a magnetic field B, as shown in If the wire is bent into a circle the field lines from all Figure 2. sides reinforce each other at the center, creating an axial field (see Usually one will not use a single loop of wire to create a Figure 3). field but a coil with many turns. Figure 3 Magnetic field of a wire loop. If one uses two coaxial coils of radius that are a distance apart, R d as shown in Figure 4, the field at the center point between the coils will be nearly homogeneous. H. von Helmholtz (18211894) realized that there remains a free parameter, namely the coil separation that can still be adjusted. He showed that when d, = the result is a particularly homogeneous field in the central d R, region between the coils. Since that time Helmholtz coils have been used when there is a need for homogeneous magnetic fields. Figure 4 Magnetic field B of a pair of Helmholtz coils. One can show that the field in the center of a Helmholtz coil is given by where is the current flowing through both coils, is their mean I R radius, is the number of turns of wire in each coil, and N = 4 × 10-7 is the permeability constant. T∙m/A 0
with a positively charged electrode, an anode, the electrons will be attracted to it and move radially outward as indicated in Figure 1. On their way to the anode they will acquire a kinetic energy , where V is the potential difference, or voltage, between the heated filament, called the cathode, and the anode. Figure 1 Most of the electrons will strike the anode. However, if one cuts a narrow slit into the anode, those electrons that started out toward the slit will exit through it as a narrow beam with a kinetic energy Ek. Figure 2 Magnetic field of a straight wire. The Magnetic Field According to Ampere’s law a wire carrying a current I is surrounded by a magnetic field B, as shown in Figure 2. If the wire is bent into a circle the field lines from all sides reinforce each other at the center, creating an axial field (see Figure 3). Usually one will not use a single loop of wire to create a field but a coil with many turns. Figure 3 Magnetic field of a wire loop. If one uses two coaxial coils of radius R that are a distance d apart, as shown in Figure 4, the field at the center point between the coils will be nearly homogeneous. H. von Helmholtz (18211894) realized that there remains a free parameter, namely the coil separation d, that can still be adjusted. He showed that when d = R, the result is a particularly homogeneous field in the central region between the coils. Since that time Helmholtz coils have been used when there is a need for homogeneous magnetic fields. Figure 4 Magnetic field B of a pair of Helmholtz coils. One can show that the field in the center of a Helmholtz coil is given by where I is the current flowing through both coils, R is their mean radius, N is the number of turns of wire in each coil, and 0 = 4 × 10-7 T∙m/A is the permeability constant.