Physics
Moving Charges in a Magnetic Field
When a charged particle moves through a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field direction. This force causes the particle to move in a circular or helical path, depending on its initial velocity. The magnitude and direction of the force can be determined using the right-hand rule and the Lorentz force equation.
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10 Key excerpts on "Moving Charges in a Magnetic Field"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
21.3 | The Motion of a Charged Particle in a Magnetic Field 585 21.3 | The Motion of a Charged Particle in a Magnetic Field Comparing Particle Motion in Electric and Magnetic Fields The motion of a charged particle in an electric field is noticeably different from the motion in a magnetic field. For example, Figure 21.9a shows a positive charge moving between the plates of a parallel plate capacitor. Initially, the charge is moving perpendicular to the direction of the electric field. Since the direction of the electric force on a positive charge is in the same direction as the electric field, the particle is deflected sideways. Part b of the drawing shows the same particle traveling initially at right angles to a magnetic field. An application of RHR-1 shows that when the charge enters the field, the charge is deflected upward (not sideways) by the magnetic force. As the charge moves upward, the direction of the magnetic force changes, always remaining perpendicular to both the magnetic field and the velocity. Conceptual Example 2 focuses on the difference in how electric and magnetic fields apply forces to a moving charge. CONCEPTUAL EXAMPLE 2 | The Physics of a Velocity Selector A velocity selector is a device for measuring the velocity of a charged particle. The device operates by applying electric and magnetic forces to the particle in such a way that these forces balance. Figure 21.10a shows a particle with a positive charge 1q and a velocity v B that is per- pendicular to a constant magnetic field* B B . Figure 21.10b illustrates a velocity selector, which is a cylindrical tube that is located within the magnetic field. Inside the tube there is a parallel plate capacitor that produces an electric field E B (not shown) perpendicular to the magnetic field. The charged particle enters the left end of the tube, moving perpendicular to the magnetic field.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
21.3 | The Motion of a Charged Particle in a Magnetic Field 523 21.3 | The Motion of a Charged Particle in a Magnetic Field Comparing Particle Motion in Electric and Magnetic Fields The motion of a charged particle in an electric field is noticeably different from the motion in a magnetic field. For example, Figure 21.9a shows a positive charge moving between the plates of a parallel plate capacitor. Initially, the charge is moving perpendicular to the direction of the electric field. Since the direction of the electric force on a positive charge is in the same direction as the electric field, the particle is deflected sideways. Part b of the drawing shows the same particle traveling initially at right angles to a magnetic field. An application of RHR-1 shows that when the charge enters the field, the charge is deflected upward (not sideways) by the magnetic force. As the charge moves upward, the direction of the magnetic force changes, always remaining perpendicular to both the magnetic field and the velocity. Conceptual Example 2 focuses on the difference in how electric and magnetic fields apply forces to a moving charge. CONCEPTUAL EXAMPLE 2 | The Physics of a Velocity Selector A velocity selector is a device for measuring the velocity of a charged particle. The device operates by applying electric and magnetic forces to the particle in such a way that these forces balance. Figure 21.10a shows a particle with a positive charge 1q and a velocity v B that is per- pendicular to a constant magnetic field* B B . Figure 21.10b illustrates a velocity selector, which is a cylindrical tube that is located within the magnetic field. Inside the tube there is a parallel plate capacitor that produces an electric field E B (not shown) perpendicular to the magnetic field. The charged particle enters the left end of the tube, moving perpendicular to the magnetic field.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
21.3.1 Solve problems dealing with the circular motion of charged particles in a magnetic field. 21.3.2 Solve problems related to the concept of the velocity selector. Electric fields and magnetic fields can exert forces on electric charges, but their effects are different. Here in Section 21.3, we describe the motion of charges in electric and magnetic fields and we ignore the effects of gravity unless otherwise specified. Animated Figure 21.3.1(a) illustrates a positive charge that is moving in the positive y direction. It enters a region in which there is a uniform electric field that is in the positive x direction. The force on the charge q F E = is in the positive x direction. According to Newton’s second law, the charge experiences a constant acceleration in the positive x direction. The y component of the velocity is constant. Run the animation to see the motion of the charge. 21.3 THE MOTION OF CHARGED PARTICLES IN A MAGNETIC FIELD Learning Objectives Animated Figure 21.3.1 (a) A charge in a constant electric field experiences a constant acceleration. (b) A charge in a magnetic field experiences an acceleration that is perpendicular to its velocity—a centripetal acceleration. I N T E R A C T I V E F E A T U R E The motion of a charge in a magnetic field is quite different. In Animated Figure 21.3.1(b), the magnetic field is uniform and directed out of the screen. (Directions “into the screen” and “out of the screen” are denoted by the symbols × and i , respectively.) According to the magnetic force right-hand rule, a positive charge moving in the posi- tive y direction will experience a magnetic force in the positive x direction as it enters the magnetic field region. The magnetic force on a charge, however, is perpendicular to the velocity. Thus, as the direction of the velocity of the charge changes, so does the direction of the force.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
As the charge moves upward, the direction of the magnetic force changes, always remaining perpendicular to both the magnetic field and the velocity. Conceptual Example 2 focuses on the difference in how electric and magnetic fields apply forces to a moving charge. – – – – E B v v F +q +q (a) (b) – + + + + + + + + + + + + F FIGURE 21.9 (a) The electric force F → that acts on a positive charge is parallel to the electric field E → . (b) The magnetic force F → is perpendicular to both the magnetic field B → and the velocity v → . *In many instances it is convenient to orient the magnetic field B → so its direction is perpendicular to the page. In these cases it is customary to use a dot to symbolize the magnetic field pointing out of the page (toward the reader); this dot symbolizes the tip of the arrow representing the B → vector. A region in which a magnetic field is directed into the page is drawn as a series of crosses that indicate the tail feathers of the arrows representing the B → vectors. Therefore, regions in which a magnetic field is directed out of the page or into the page are drawn as shown below: · · · · × × × × · · · · × × × × · · · · × × × × Out of page Into page CONCEPTUAL EXAMPLE 2 The Physics of a Velocity Selector A velocity selector is a device for measuring the velocity of a charged particle. The device operates by applying electric and magnetic forces to the particle in such a way that these forces balance. Figure 21.10a shows a particle with a positive charge +q and a velocity v → that is perpendicular to a constant magnetic field* B → . Figure 21.10b illustrates a velocity se- lector, which is a cylindrical tube that is located within the magnetic field. Inside the tube there is a parallel plate capacitor that produces an elec- tric field E → (not shown) perpendicular to the magnetic field. The charged particle enters the left end of the tube, moving perpendicular to the mag- netic field.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
The following two conditions must be met for a charge to experience a magnetic force when placed in a magnetic field. 1. The charge must be moving, because no magnetic force acts on a stationary charge. 2. The velocity of the moving charge must have a component that is perpendicular to the direction of the magnetic field. To examine the second condition, consider figure 21.7, which shows a positive test charge +q 0 moving with a velocity v through a magnetic field B. The field is produced by magnets not shown in the drawing and is assumed to be constant in both magnitude and direction. If the charge moves parallel or antiparallel to the field, as in figure 21.7a, the charge experiences no magnetic force. If, however, the charge moves perpendicular to the field, as in figure 21.7b, the charge experiences the maximum possible force F max . In general, if a charge moves at an angle * with respect to the field (see figure 21.7c), only the velocity component v sin , which is perpendicular to the field, gives rise to a magnetic force. This force F is smaller than the maximum possible force. The component of the velocity that is parallel to the magnetic field yields no force. *The angle between the velocity of the charge and the magnetic field is chosen so that it lies in the range 0 ≤ ≤ 180°. 578 Physics FIGURE 21.7 (a) No magnetic force acts on a charge moving with a velocity v that is parallel or antiparallel to a magnetic field B. (b) The charge experiences a maximum force F max when the charge moves perpendicular to the field. (c) If the charge travels at an angle with respect to B, only the velocity component perpendicular to B gives rise to a magnetic force F, which is smaller than F max . This component is v sin . B –v v +q 0 +q 0 (a) B B v +q 0 (b) F max 90° B B B +q 0 (c) F θ θ v sin B B B v FIGURE 21.8 Right‐hand rule no.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
This ability may be related to the presence in the lobsters of the mineral magnetite, a magnetic material used for compass needles. 21.2 The Force That a Magnetic Field Exerts on a Moving Charge When a charge is placed in an electric field, it experiences an electric force, as Section 18.6 discusses. When a charge is placed in a magnetic field, it also experiences a force, provided that certain conditions are met, as we will see. The magnetic force, like all the forces we have studied (e.g., the gravitational, elastic, and electric forces), may contribute to the net force that causes an object to accelerate. Thus, when present, the magnetic force must be included in Newton’s second law. The following two conditions must be met for a charge to experience a magnetic force when placed in a magnetic field: 1. The charge must be moving, because no magnetic force acts on a stationary charge. 2. The velocity of the moving charge must have a component that is perpendicular to the direction of the magnetic field. To examine the second condition, consider Figure 21.7 , which shows a positive test charge +q 0 moving with a velocity → v through a magnetic field → B. The field is produced North magnetic pole Magnetic axis Rotational axis North geographic pole S N FIGURE 21.5 The earth behaves magnetically almost as if a bar magnet were located near its center. The axis of this fictitious bar magnet does not coincide with the earth’s rotational axis; the two axes are currently about 9.4° apart. FIGURE 21.6 Spiny lobsters use the earth’s magnetic field to navigate and determine their geographic position. D.P. Wilson/FLPA/Science Source *At present it is not known with certainty what causes the earth’s magnetic field. The magnetic field seems to originate from electric currents that in turn arise from electric charges circulating within the liquid outer region of the earth’s core. Section 21.7 discusses how a current produces a magnetic field.
eBook - PDF- Juda Shohet(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
3 The Motion of Isolated Charged Particles A. THE MOTION OF A CHARGED PARTICLE INA MAGNETIC FIELD We shall now begin the study of the motion of isolated charged particles in electric, magnetic, and gravitational fields. In many situations (low plasma density and low background gas pressure) the collision frequency is low enough so that over the time of interest for a particular problem, the single-particle approach has significant application. The fundamental equation of motion of a single charged particle in an electric, magnetic, and gravitational field is d F = m — = ψ{Ε + v x B) + mg (3.1) In Eq. (3.1), E is the electric field seen at the particle, B is the magnetic field present, and g is the gravitational acceleration vector. Since B is always perpendicular to v, the magnetic field can do no work on the charge. Hence, the energy of a charged particle is unaffected by the magnetic field alone. However, if the magnetic field should change rapidly in time, an induced electric field may produce a change in energy of the particle. The conditions under which any or all of the three terms in Eq. (3.1) must be considered depend upon the magnitudes of the fields and the time of interaction. We shall be interested in steady state solutions to Eq. (3.1), and we will include the effects of time-varying fields of constant amplitude. The solution to Eq. (3.1) will yield much information. We shall be able to obtain the velocity as a function of an applied electric field (the mobility). This result is especially interesting when a dc magnetic field is included. For 39 40 3 The Motion of Isolated Charged Particles the first time, we will observe analytically that the mobility is a function of direction with respect to the magnetic field, showing that a plasma becomes anisotropic in a magnetic field. Subsequently, we will generalize the single-particle results to a collection of particles, each of which may also be considered isolated.
eBook - PDFMaxwell's Equations and Their Consequences
Elementary Electromagnetic Theory
- B. H. Chirgwin, C. Plumpton, C. W. Kilmister(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
CHAPTER 14 THE MOTION OF CHARGED PARTICLES 14.1. Introduction The motion of electrically charged particles in electromagnetic and gravitational fields is of interest in various studies of astronomical and engineering problems. During the past five decades research into the motion of charged particles in the terrestrial magnetic and gravitational fields has greatly improved our understanding of the structure and extent of the terrestrial magnetic field. In electronics, devices such as the magnetron were designed after theoretical studies similar to those illustrated in the examples of this chapter; in thermonuclear research, studies of magnetic bottles, wells and traps are of prime importance. However, much care must be taken when attempts are made to generalize the results for a single particle to the case of an ionized gas, which consists of a multitude of charged particles, in a magnetic field, since the interactions between charged particles can modify or completely change the calculated (and observed) results concerning a single particle. In fact the essential feature of the motion of a charged particle in a magnetic field is the tendency of the particle to spiral around the magnetic field lines. This is regarded as the counterpart of the magnetohydrodynamic (or ionized-gas) phenomenon in which a highly (electrically) conducting material can flow freely along the lines of magnetic force but motion of the material perpendicular to the magnetic field carries the magnetic field lines with the material. [This result is similar to the classical hydrodynamic result that vorticity lines are frozen into a perfect fluid.] In this chapter, we first give an account of the non-relativistic motion of a charged particle in uniform crossed electric and magnetic fields. This is followed by a brief account of the flow of charged particles when space charge density is taken into account. Finally relativistic corrections are 569
eBook - PDF- George Schmidt(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Motion of Charged Particles in Electromagnetic Fields 2-1. The Static Magnetic Field For one particle moving in a magnetic field with velocity v, (1-7) reduces to As the force is perpendicular to the velocity, no work is done by the magnetic field. Indeed, the scalar multiplication of (2-1) by v yields showing that the kinetic energy of the particle, in an arbitrary magnetic field, is a constant of motion. Let us restrict ourselves for the moment to the special case where the magnetic field lines are straight and parallel (but the field is not necessarily uniform). Denoting vector components parallel to the field with the sub-script || and those perpendicular to it with _L, we obtain m — = q (vxB ) (2-1) ( 2 -2 ) V=V||+V1 (2-3) and (2-1) becomes (2-4) since v(|x Bvanishes. Equation(2-4) splits into a ||-component equation and a _L-component equation dy and dv II dy, q . -J + — ^= -(vj. x B) dt dt m = 0 (v || = const) £ -2 · < ν , χ Β ) at m 5 (2-5) (2-6) 6 II. Motion of Charged Particles Since the right side of (2-6) is perpendicular to v± , the left side is a centri-petal acceleration. It can be written ^ ( -r ) = ^ ( v ± xB) (2-7) where r is the local radius of curvature of the particle path (Fig. 2-1). Its F ig . 2-1. Particle moving in a magnetic field of straight and parallel field lines. The field intensity varies in the plane perpendicular to B. value is, from (2-7), r = mvL qB ( 2 -8 ) In the special case of a uniform magnetic field, B = const, and considering the constancy of v± from (2-2) and (2-5), the radius of curvature R = mv I qB (2-9) is also a constant. In a uniform magnetic field, therefore, the particle moves in a circle with the so-called cyclotron or gyroradius R in the perpendicular plane, while it moves with a constant velocity along the field lines.
eBook - PDFQuantum Mechanics
A Modern Development
- Leslie E Ballentine(Author)
- 1998(Publication Date)
- WSPC(Publisher)
Chapter 11 Charged Particle in a Magnetic Field The theory of the motion of a charged particle in a magnetic field presents several difficult and unintuitive features. The derivation of the quantum theory does not require the classical theory; nevertheless it is useful to first review the classical theory in order to show that some of these unintuitive features are not peculiar to the quantum theory, but rather that they are characteristic of motion in a magnetic field. 11.1 Classical Theory The electric and magnetic fields, E and B, enter the Lagrangian and Hamil-tonian forms of mechanics through the vector and scalar potentials, A and : E= ~ V0 ~^fr (llla) B = V x A . (11.1b) (The speed of light c appears only because of a conventional choice of units.) The potentials are not unique. The fields E arid B are unaffected by the replacement 1 8Y A ^ A ' = A + V X , ^ 4! = **--£> ( 1L2 ) where — x( x > 0 i s a n arbitrary scalar function. This change of the potentials, called a gauge transformation, has no effect upon any physical result. It thus appears, in classical mechanics, that the potentials are only a mathematical construct having no direct physical significance. The Lagrangian for a particle of mass M and charge q in an arbitrary electromagnetic fields is £(x, v, t) = ^ -q (x, t) + ± v A (x, t), (11.3) tL C 307 308 Ch. 11: Charged Particle in a Magnetic Field where x and v = dx./dt are the position and velocity of the particle.
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