# Motion of charged particle in magnetic field

The ions and electrons of a plasma interacting with the Earth's magnetic field generally follow its magnetic field lines. These represent the force that a north magnetic pole would experience at any given point.

### 21.4: Motion of a Charged Particle in a Magnetic Field

Denser lines indicate a stronger force. Plasmas exhibit more complex second-order behaviors, studied as part of magnetohydrodynamics. Thus in the "closed" model of the magnetosphere, the magnetopause boundary between the magnetosphere and the solar wind is outlined by field lines. Not much plasma can cross such a stiff boundary. This simple definition assumes a noon-midnight plane of symmetry, but closed fields lacking such symmetry also must have cusps, by the fixed point theorem.

The amount of solar wind energy and plasma entering the actual magnetosphere depends on how far it departs from such a "closed" configuration, i. As discussed further below, that extent depends very much on the direction of the Interplanetary Magnetic Field, in particular on its southward or northward slant.

Trapping of plasmae. A particle interacting with this B field experiences a Lorentz Force which is responsible for many of the particle motion in the magnetosphere. Furthermore, Birkeland currents and heat flow are also channeled by such lines — easy along them, blocked in perpendicular directions. Indeed, field lines in the magnetosphere have been likened to the grain in a log of wood [ citation needed ]which defines an "easy" direction along which it easily gives way.

The simplest magnetic field B is a constant one— straight parallel field lines and constant field intensity. In such a field, if an ion or electron enters perpendicular to the field lines, it can be shown to move in a circle the field only needs to be constant in the region covering the circle.

One gets. That velocity just stays constant as long as the field doesand adding the two motions together gives a spiral around a central guiding field line. If the field curves or changes, the motion is modified, but the general character of spiraling around a central field line persists: hence the name " guiding center motion. Because the magnetic force is perpendicular to the velocity, it performs no work and requires no energy—nor does it provide any.

Thus magnetic fields like the Earth's can profoundly affect particle motion in them, but need no energy input to maintain their effect. Particles may also get steered around, but their total energy remains the same. The spacing between field lines is an indicator of the relative strength of the magnetic field. Where magnetic field lines converge the field grows stronger, and where they diverge, weaker. The "very nearly" qualifier sets it apart from true constants of motion, such as energy, reducing it to merely an "adiabatic invariant.

Suppose the field line guiding a particle, the axis of its spiral path, belongs to a converging bundle of lines, so that the particle is led into an increasingly larger B.

However, as noted before, the total energy of a particle in a "purely magnetic" field remains constant.A particle with a positive charge Q begins at rest. Describe the motion of the particle after switching on both a homogeneous electric field with direction corresponding to the z axis and a homogeneous magnetic field with direction corresponding to the x axis. There is a Lorentz force acting on a charged particle in an electromagnetic field.

Both the electric and magnetic fields act on the particle with forces. The force of the electrical field is parallel to the electric field vector and also to the z axis.

The magnetic force is perpendicular to the magnetic field vector which is parallel to the x axis. The net force of both the electric and magnetic forces acts in the yz plane. There is no force acting on the particle in the direction of the x axis. The electrical force acts in the direction of the electric field, in our case it is the direction of the z axis.

### Magnetosphere particle motion

The magnetic force is perpendicular to the magnetic field which has the same direction as the x axis, so the magnetic force acts in the yz plane. The expression for the Lorentz force acting on a charged particle in an electromagnetic field is:. But we know that the constant must be zero, since the particle begins at rest. So as we assumed the motion will be only in the yz plane.

We have now a non-homogeneous linear differential equation with constant coefficients. Its further solution can be found in the following section. Its solution has several steps. First we have to solve the corresponding homogeneous equation, i. Then we should find a particular solution, which is a random solution of the non-homogeneous equation. The general solution of a non-homogeneous equation is the sum of its particular solution and the general solution of the corresponding homogeneous equation.

Constants in the general solution will be found from the initial conditions. We have now got a general solution of the homogeneous equation, where C and D are real constants because we are looking only for real solutions of the given differential equation. The particular solution is one random solution of a non-homogenous differential equation, i. Actually, this often works very well. In our case equation 1 has on its right side a constant. We will get a general solution of a non-homogenous equation from the sum of its particular solution and the general solution of the relevant homogeneous equation, so.

The constants C and D can be determined from initial conditions. This component can be expressed as. The component of velocity v z we may determine from equation 4 by inserting constants C and D. By solution of the differential equation we have got the component of velocity v y. The components of position y and z we obtain by integration over time of the respective components of velocity.

The trajectory of a charged particle in a homogeneous electromagnetic field is thus described by the following equations.

Combining those motions leads to motion on a cycloid a cycloid curve. Remark: The motion of a particle in a uniform electromagnetic field could be compared to the motion of the tire valve of a moving wheel. Both of these motions have a cycloid trajectory. After switching on both an electric field in the z direction and a homogeneous magnetic field in the x direction, the charged particle will move on a cycloid given by the equations.

Task list filter? Choose required ranks and required tasks.A charged particle experiences a force when moving through a magnetic field. What happens if this field is uniform over the motion of the charged particle? What path does the particle follow? In this section, we discuss the circular motion of the charged particle as well as other motion that results from a charged particle entering a magnetic field. If the field is in a vacuum, the magnetic field is the dominant factor determining the motion.

## Motion of a Charged Particle in a Uniform Magnetic Field

Since the magnetic force is perpendicular to the direction of travel, a charged particle follows a curved path in a magnetic field. The particle continues to follow this curved path until it forms a complete circle.

Another way to look at this is that the magnetic force is always perpendicular to velocity, so that it does no work on the charged particle. The direction of motion is affected but not the speed. Here, r is the radius of curvature of the path of a charged particle with mass m and charge qmoving at a speed v that is perpendicular to a magnetic field of strength B.

The time for the charged particle to go around the circular path is defined as the period, which is the same as the distance traveled the circumference divided by the speed. Based on this and Equation, we can derive the period of motion as.

If the velocity is not perpendicular to the magnetic field, then we can compare each component of the velocity separately with the magnetic field.

The component of the velocity perpendicular to the magnetic field produces a magnetic force perpendicular to both this velocity and the field:.

The component parallel to the magnetic field creates constant motion along the same direction as the magnetic field, also shown in Equation. The parallel motion determines the pitch p of the helix, which is the distance between adjacent turns.

## 11.4: Motion of a Charged Particle in a Magnetic Field

This distance equals the parallel component of the velocity times the period:. While the charged particle travels in a helical path, it may enter a region where the magnetic field is not uniform. In particular, suppose a particle travels from a region of strong magnetic field to a region of weaker field, then back to a region of stronger field. The particle may reflect back before entering the stronger magnetic field region.

This is similar to a wave on a string traveling from a very light, thin string to a hard wall and reflecting backward.A charged particle experiences a force when moving through a magnetic field.

Motion of a Charged Particle in a Uniform Magnetic Field - Physics4students

What happens if this field is uniform over the motion of the charged particle? What path does the particle follow? In this section, we discuss the circular motion of the charged particle as well as other motion that results from a charged particle entering a magnetic field. The simplest case occurs when a charged particle moves perpendicular to a uniform -field Figure 8. If the field is in a vacuum, the magnetic field is the dominant factor determining the motion.

Since the magnetic force is perpendicular to the direction of travel, a charged particle follows a curved path in a magnetic field. The particle continues to follow this curved path until it forms a complete circle. Another way to look at this is that the magnetic force is always perpendicular to velocity, so that it does no work on the charged particle. The direction of motion is affected but not the speed. Figure 8. In this situation, the magnetic force supplies the centripetal force.

Noting that the velocity is perpendicular to the magnetic field, the magnitude of the magnetic force is reduced to. The time for the charged particle to go around the circular path is defined as the period, which is the same as the distance traveled the circumference divided by the speed. If the velocity is not perpendicular to the magnetic field, then we can compare each component of the velocity separately with the magnetic field.

The component of the velocity perpendicular to the magnetic field produces a magnetic force perpendicular to both this velocity and the field:. This distance equals the parallel component of the velocity times the period:. While the charged particle travels in a helical path, it may enter a region where the magnetic field is not uniform. In particular, suppose a particle travels from a region of strong magnetic field to a region of weaker field, then back to a region of stronger field.

The particle may reflect back before entering the stronger magnetic field region. This is similar to a wave on a string traveling from a very light, thin string to a hard wall and reflecting backward. If the reflection happens at both ends, the particle is trapped in a so-called magnetic bottle. Auroraelike the famous aurora borealis northern lights in the Northern Hemisphere Figure 8.

Aurorae have also been observed on other planets, such as Jupiter and Saturn. A research group is investigating short-lived radioactive isotopes. They need to design a way to transport alpha-particles helium nuclei from where they are made to a place where they will collide with another material to form an isotope.

The direction of the magnetic field is shown by the RHR Your fingers point in the direction ofand your thumb needs to point in the direction of the force, to the left. Therefore, since the alpha-particles are positively charged, the magnetic field must point down.

First, point your thumb up the page. In order for your palm to open to the left where the centripetal force and hence the magnetic force points, your fingers need to change orientation until they point into the page.

This is the direction of the applied magnetic field.

The period of the charged particle going around a circle is calculated by using the given mass, charge, and magnetic field in the problem. This works out to be. However, for the given problem, the alpha-particle goes around a quarter of the circle, so the time it takes would be. This time may be quick enough to get to the material we would like to bombard, depending on how short-lived the radioactive isotope is and continues to emit alpha-particles.

If we could increase the magnetic field applied in the region, this would shorten the time even more. The path the particles need to take could be shortened, but this may not be economical given the experimental setup. A uniform magnetic field of magnitude is directed horizontally from west to east. At what angle must the magnetic field be from the velocity so that the pitch of the resulting helical motion is equal to the radius of the helix?In physics specifically in electromagnetism the Lorentz force or electromagnetic force is the combination of electric and magnetic force on a point charge due to electromagnetic fields.

A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of. Variations on this basic formula describe the magnetic force on a current-carrying wire sometimes called Laplace forcethe electromotive force in a wire loop moving through a magnetic field an aspect of Faraday's law of inductionand the force on a charged particle which might be traveling near the speed of light relativistic form of the Lorentz force.

Historians suggest that the law is implicit in a paper by James Clerk Maxwellpublished in The force F acting on a particle of electric charge q with instantaneous velocity vdue to an external electric field E and magnetic field Bis given by in SI units [1] : [6]. In terms of cartesian components, we have:. In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as:.

A positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to both the instantaneous velocity vector v and the B field according to the right-hand rule in detail, if the fingers of the right hand are extended to point in the direction of v and are then curled to point in the direction of Bthen the extended thumb will point in the direction of F.

This article will not follow this nomenclature: In what follows, the term "Lorentz force" will refer to the expression for the total force. The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the Laplace force. The Lorentz force is a force exerted by the electromagnetic field on the charged particle, that is, it is the rate at which linear momentum is transferred from the electromagnetic field to the particle.

Associated with it is the power which is the rate at which energy is transferred from the electromagnetic field to the particle. That power is. Notice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle.

For a continuous charge distribution in motion, the Lorentz force equation becomes:. Next, the current density corresponding to the motion of the charge continuum is.

The total force is the volume integral over the charge distribution:. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the energy flux flow of energy per unit time per unit distance in the fields to the force exerted on a charge distribution.

See Covariant formulation of classical electromagnetism for more details. If we separate the total charge and total current into their free and bound parts, we get that the density of the Lorentz force is.

In this way, the Lorentz force can explain the torque applied to a permanent magnet by the magnetic field. The density of the associated power is. The above-mentioned formulae use SI units which are the most common among experimentalists, technicians, and engineers.

In cgs-Gaussian unitswhich are somewhat more common among theoretical physicists as well as condensed matter experimentalists, one has instead. Although this equation looks slightly different, it is completely equivalent, since one has the following relations: [1].

In practice, the subscripts "cgs" and "SI" are always omitted, and the unit system has to be assessed from context. Early attempts to quantitatively describe the electromagnetic force were made in the midth century.

It was proposed that the force on magnetic poles, by Johann Tobias Mayer and others in[10] and electrically charged objects, by Henry Cavendish in[11] obeyed an inverse-square law. However, in both cases the experimental proof was neither complete nor conclusive. It was not until when Charles-Augustin de Coulombusing a torsion balancewas able to definitively show through experiment that this was true.

The modern concept of electric and magnetic fields first arose in the theories of Michael Faradayparticularly his idea of lines of forcelater to be given full mathematical description by Lord Kelvin and James Clerk Maxwell. Thomson was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields.

Interested in determining the electromagnetic behavior of the charged particles in cathode raysThomson published a paper in wherein he gave the force on the particles due to an external magnetic field as [5].

Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the displacement currentincluded an incorrect scale-factor of a half in front of the formula.Force due to both electric and magnetic forces will influence the motion of charged particles.

However, the resulting change to the trajectory of the particles will differ qualitatively between the two forces. Below we will quickly review the two types of force and compare and contrast their effects on a charged particle.

Recall that in a static, unchanging electric field E the force on a particle with charge q will be:. Where F is the force vector, q is the charge, and E is the electric field vector. Note that the direction of F is identical to E in the case of a positivist charge q, and in the opposite direction in the case of a negatively charged particle.

This electric field may be established by a larger charge, Q, acting on the smaller charge q over a distance r so that:. It should be emphasized that the electric force F acts parallel to the electric field E.

The curl of the electric force is zero, i. A consequence of this is that the electric field may do work and a charge in a pure electric field will follow the tangent of an electric field line. In contrast, recall that the magnetic force on a charged particle is orthogonal to the magnetic field such that:. The direction of F can be easily determined by the use of the right hand rule. Right Hand Rule : Magnetic fields exert forces on moving charges.

This force is one of the most basic known. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by v and B and follows right hand rule—1 RHR-1 as shown. The magnitude of the force is proportional to q, v, B, and the sine of the angle between v and B. If the particle velocity happens to be aligned parallel to the magnetic field, or is zero, the magnetic force will be zero. This differs from the case of an electric field, where the particle velocity has no bearing, on any given instant, on the magnitude or direction of the electric force.

The angle dependence of the magnetic field also causes charged particles to move perpendicular to the magnetic field lines in a circular or helical fashion, while a particle in an electric field will move in a straight line along an electric field line. A further difference between magnetic and electric forces is that magnetic fields do not net work, since the particle motion is circular and therefore ends up in the same place. We express this mathematically as:.

The Lorentz force is the combined force on a charged particle due both electric and magnetic fields, which are often considered together for practical applications. If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field Bthen it will experience a force:.Electric and magnetic forces both affect the trajectory of charged particles, but in qualitatively different ways. Force due to both electric and magnetic forces will influence the motion of charged particles.

However, the resulting change to the trajectory of the particles will differ qualitatively between the two forces. Below we will quickly review the two types of force and compare and contrast their effects on a charged particle. Recall that in a static, unchanging electric field E the force on a particle with charge q will be:. Where F is the force vector, q is the charge, and E is the electric field vector.

Note that the direction of F is identical to E in the case of a positivist charge q, and in the opposite direction in the case of a negatively charged particle.

This electric field may be established by a larger charge, Q, acting on the smaller charge q over a distance r so that:. It should be emphasized that the electric force F acts parallel to the electric field E. The curl of the electric force is zero, i. A consequence of this is that the electric field may do work and a charge in a pure electric field will follow the tangent of an electric field line. In contrast, recall that the magnetic force on a charged particle is orthogonal to the magnetic field such that:.

The direction of F can be easily determined by the use of the right hand rule. Right Hand Rule : Magnetic fields exert forces on moving charges. This force is one of the most basic known. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by v and B and follows right hand rule—1 RHR-1 as shown.

The magnitude of the force is proportional to q, v, B, and the sine of the angle between v and B. If the particle velocity happens to be aligned parallel to the magnetic field, or is zero, the magnetic force will be zero.

This differs from the case of an electric field, where the particle velocity has no bearing, on any given instant, on the magnitude or direction of the electric force.

The angle dependence of the magnetic field also causes charged particles to move perpendicular to the magnetic field lines in a circular or helical fashion, while a particle in an electric field will move in a straight line along an electric field line. A further difference between magnetic and electric forces is that magnetic fields do not net work, since the particle motion is circular and therefore ends up in the same place.

We express this mathematically as:. The Lorentz force is the combined force on a charged particle due both electric and magnetic fields, which are often considered together for practical applications. If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field Bthen it will experience a force:.

We mentioned briefly above that the motion of charged particles relative to the field lines differs depending on whether one is dealing with electric or magnetic fields.