Motion of a charged particle in a magnetic field: formulas. Motion of charged particles in a uniform magnetic field
As is known, it is common practice to characterize an electric field by the magnitude of the force with which it acts on a test unit. The magnetic field is traditionally characterized by the force with which it acts on a conductor with a “unit” current. However, when it flows, an ordered movement of charged particles in a magnetic field occurs. Therefore, we can determine the magnetic field B at some point in space from the point of view of the magnetic force FBwhich the field exerts on a particle as it moves in it at a speed v.
General properties of magnetic force
Experiments in which the movement of charged particles in a magnetic field was observed give the following results:
- F valueBThe magnetic force acting on the particle is proportional to the charge q and the velocity v of the particle.
- If the motion of a charged particle in a magnetic field occurs parallel to the vector of this field, then the force acting on it is zero.
- When the particle velocity vector is any Angle θ 0 with a magnetic field, then the force acts in the direction perpendicular to v and B; that is, FBperpendicular to the plane formed by v and B (see figure below).
- F value and directionBdepends on the velocity of the particle and the magnitude and direction of the magnetic field B.
- The direction of the force acting on the positive charge is opposite to the direction of the same force acting on the negative charge moving in the same direction.
- The magnitude of the magnetic force acting on a moving particle is proportional to the sin θ of the angle θ between the vectors v and B.
We can summarize the above observations by recording the magnetic force as FB= qv x B.
When a charged particle moves in a magnetic field, the Lorentz force FBfor positive q, it is directed along the vector product v x B. It is, by definition, perpendicular to both v and B. We consider this equation to be the working definition of a magnetic field at some point in space.That is, it is defined in terms of the force acting on the particle as it moves. Thus, the motion of a charged particle in a magnetic field can be briefly defined as movement under the action of this force.
A charge moving at a speed v in the presence of both the electric field E and magnetic B is affected by both the electric force qE and magnetic qv x B. The total effect applied to it is FL= qE + qv x B. It is customarily called it: the full power of Lorentz.
Motion of charged particles in a uniform magnetic field
We now consider a particular case of a positively charged particle moving in a uniform field, with an initial velocity vector perpendicular to it. Suppose that the B field vector is directed per page. The figure below shows that the particle moves in a circle in a plane perpendicular to B.
The motion of a charged particle in a magnetic field around the circumference occurs because the magnetic force FBdirected at right angles to v and B and has a constant qvB. As the force deflects the particles, the directions v and FBchange continuously as shown in the figure. Since fBalways directed towards the center of the circle, it changes only the direction of v, and not its value.As shown in the figure, the movement of a positively charged particle in a magnetic field occurs counterclockwise. If q is negative, the rotation will occur clockwise.
Dynamics of the circular motion of a particle
What parameters characterize the above described motion of a charged particle in a magnetic field? Formulas for their definition we can get if we take the previous equation and equate FBthe centrifugal force required to maintain a circular path of motion:
That is, the radius of the circle is proportional to the momentum mv of the particle and inversely proportional to the magnitude of its charge and the magnetic field. Particle angular velocity
The period with which a charged particle moves in a magnetic field in a circle is equal to the length of a circle divided by its linear velocity:
These results show that the angular velocity of a particle and the period of circular motion do not depend on the linear velocity or on the radius of the orbit. The angular velocity ω is often called cyclotron.frequency (circular), because charged particles circulate with it in an accelerator type called a cyclotron.
Motion of a particle at an angle to the magnetic field vector
If the particle velocity vector v forms some arbitrary angle with respect to the vector B, then its trajectory is a helix. For example, if a uniform field is directed along the x axis, as shown in the figure below, then there is no component of the magnetic force FBin this direction. As a result, the acceleration component ax= 0, and the x-component of the velocity of the particle is constant. However, the magnetic force FB= qv x B causes a change in the time components of the velocity vyand vz. As a result, a charged particle moves in a magnetic field along a helix, the axis of which is parallel to the magnetic field. The projection of the trajectory on the yz plane (when viewed along the x axis) is a circle. Its projections on the xy and xz planes are sinusoids! The equations of motion remain the same as in a circular path, provided that v is replaced by ν⊥=√(νat2+ νz2).
Inhomogeneous magnetic field: how particles move in it
The motion of a charged particle in a magnetic field, which is inhomogeneous, occurs along complex trajectories. Thus, in a field whose magnitude is amplified at the edges of its region of existence and weakened in its middle, as, for example, shown in the figure below, a particle can oscillate back and forth between the end points.The charged particle starts at one end of the helix, wound along the lines of force, and moves along it until it reaches the other end, where it turns its way back. This configuration is known as a “magnetic bottle” because charged particles can be trapped in it. It was used to confine the plasma, a gas consisting of ions and electrons. Such a plasma confinement scheme can play a key role in controlling nuclear fusion, a process that will provide us with an almost endless source of energy. Unfortunately, the "magnetic bottle" has its own problems. If a large number of particles are trapped, collisions between them cause them to leak out of the system.
How the Earth affects the movement of cosmic particles
Van Allen near-Earth belts consist of charged particles (mainly electrons and protons) surrounding the Earth in the form of toroidal regions (see the figure below). The motion of a charged particle in the magnetic field of the Earth occurs in a spiral around the lines of force from pole to pole, covering this distance in a few seconds. These particles come mostly from the Sun, but some come from stars and other celestial objects.For this reason, they are called cosmicrays. Most of them are deflected by the Earth’s magnetic field and never reach the atmosphere. However, some of the particles fall into the trap, it is they who make up the Van Allen belt. When they are above the poles, they sometimes collide with atoms in the atmosphere, as a result of which they emit visible light. So there are beautiful auroras in the northern and southern hemispheres. They tend to occur in the polar regions, because this is where the Van Allen belts are located closest to the surface of the Earth.
Sometimes, however, solar activity causes a greater number of charged particles entering these belts and significantly distorts the normal magnetic field lines associated with the Earth. In these situations, the aurora can sometimes be seen at lower latitudes.
In many experiments in which the movement of charged particles occurs in a uniform magnetic field, it is important that all particles move at almost the same speed. This can be achieved by applying a combination of electric field and magnetic field, oriented as shown in the figure below.The homogeneous electric field is directed vertically downwards (in the plane of the page), and the same magnetic field is applied in a direction perpendicular to the electric (per page).For positive q magnetic force FB= qv x B is directed up, and the electric force qE is down. When the values of two fields are chosen so that qE = qvB, then the particle moves in a straight horizontal line through the field region. From the expression qE = qvB, we find that only particles with a velocity v = E / B pass without deviation through mutually perpendicular electric and magnetic fields. Force fBacting on particles moving at a speed greater than v = E / B turns out to be more electric, and they are deflected upwards. Those of them that move at a lower speed deviate downward.
This deviceseparates ions in accordance with the ratio of their mass to charge. According to one version of this device, known as the Bainbridge mass spectrometer, the ion beam first passes through the velocity selector and then enters the second field B0, also homogeneous and having the same direction as the field in the selector (see figure below). After entering it, the motion of a charged particle in a magnetic field occurs in a semicircle of radius r before impact into the photographic plate R.If the ions are positively charged, the beam is deflected upwards, as shown in the figure. If the ions are negatively charged, the beam will deflect downwards. From the expression for the radius of the circular trajectory of the particle, we can find the ratio m / q
and then, using the equation v = E / B, we find that
Thus, we can determine m / q by measuring the radius of curvature, knowing the fields of values B, B0, and E. In practice, it usually measures the masses of various isotopes of a given ion, since they all carry one charge q. Thus, the mass ratio can be determined even if q is unknown. A variation of this method was used by J.J. Thomson (1856-1940) in 1897 to measure the ratio e / mefor electrons.
It can accelerate charged particles to very high speeds. Both electric and magnetic forces play a key role here. The resulting high-energy particles are used to bombard atomic nuclei, and thereby produce nuclear reactions of interest to researchers. A number of hospitals use cyclotron equipment for the production of radioactive substances for diagnosis and treatment.
A schematic representation of the cyclotron is shown in Fig. below. Particles move inside two semi-cylindrical containers D 1 and D 2, called deants.The high-frequency variable potential difference is applied to the dees separated by a gap, and the uniform magnetic field is directed along the axis of the cyclotron (the south pole of its source is not shown in the figure).
A positive ion released from a source at point P near the center of the device in the first dual moves along a semicircular trajectory (shown by a dotted red line in the figure) and arrives back into the slot at time T / 2, where T is the time of one complete revolution inside two deants .
The frequency of the applied potential difference is regulated in such a way that the polarities of the diantes are reversed at the moment of time when the ion exits from one dianth. If the applied potential difference is adjusted so that at this moment D2gets lower electrical potential than D1by qΔV, then the ion is accelerated in the gap before entering D2, and its kinetic energy is increased by the value of qΔV. Then he moves around D2along a semicircular trajectory of a larger radius (because its speed has increased).
After some time T / 2, he again enters the gap between the deants. At this point, the polarities of the duents change again, and another “blow” is given to the ion through the gap.The movement of a charged particle in a magnetic field in a spiral continues, so that with each pass of one duant, the ion gets additional kinetic energy equal to qΔV. When the radius of its trajectory becomes close to the radius of the dees, the ion leaves the system through the exit slit. It is important to note that the work of the cyclotron is based on the fact that T does not depend on the velocity of the ion and the radius of the circular trajectory. We can get an expression for the kinetic energy of an ion when it leaves the cyclotron, depending on the radius R of the dees. We know that the speed of a circular motion of a particle is ν = qBR / m. Therefore, its kinetic energy
When the ion energy in the cyclotron exceeds about 20 MeV, relativistic effects come into play. We note that T increases, and that moving ions do not remain in phase with an applied potential difference. Some accelerators solve this problem by changing the period of the applied potential difference, so that it remains in phase with moving ions.
When a conductor with a current is placed in a magnetic field, an additional potential difference is created in a direction perpendicular to the direction of the current and the magnetic field. This phenomenon, first observed by Edwin Hall (1855-1938) in 1879, is known asHall.It is always observed when a charged particle moves in a magnetic field. This leads to the deflection of charge carriers on one side of the conductor as a result of the magnetic force they experience. The Hall effect gives information about the sign of charge carriers and their density, it can also be used to measure the magnitude of magnetic fields.
The device for observing the Hall effect consists of a flat conductor with a current I in the x direction, as shown in the figure below.A uniform field B is applied in the y direction. If charge carriers are electrons moving along the x axis with a drift velocity vdthen they experience an upward (considering negative q) magnetic force FB= qvdx B, are deflected upwards and accumulate at the top edge of the flat conductor, resulting in an excess of positive charge at the bottom edge.This charge accumulation at the edges increases until the electrical force resulting from charge separation balances the magnetic force on the carriers. When this equilibrium is reached, the electrons no longer deflect upward.A sensitive voltmeter or potentiometer connected to the upper and lower edges of the conductor can measure the potential difference known as the Hall voltage.