Lorentz force and the left hand rule. Movement of charged particles in a magnetic field

Placed in a magnetic field conductorthrough which passed electricity, is affected by the force of Ampere $F_A$ , and its value can be calculated using the following formula:

$F_A=B\cdot I\cdot l\cdot sin\alpha$ (1)

where $I$ and $l$ - current strength and conductor length, $B$ – magnetic field induction, $\alpha$ - the angle between the directions of the current strength and magnetic induction. Why is this happening?

Lorentz force. Movement of a charged particle in a magnetic field.

Content

1 What is the Lorentz force - determining when it occurs, obtaining the formula
2 Determining the direction of the Lorentz force using the left hand rule
3 Movement of a charged particle in a magnetic field
4 Application of the Lorentz force in engineering

What is the Lorentz force - determining when it occurs, obtaining the formula

It is known that electric current is an ordered movement of charged particles. It has also been established that during movement in a magnetic field, each of these particles is subjected to the action of a force. For a force to occur, the particle must be in motion.

The Lorentz force is the force that acts on an electrically charged particle as it moves in a magnetic field.Its direction is orthogonal to the plane in which the vectors of particle velocity and magnetic field strength lie. The resultant of the Lorentz forces is the Ampère force. Knowing it, we can derive a formula for the Lorentz force.

The time required for the particle to pass through the segment of the conductor, $t = \frac {l}{v}$ , where $l$ - the length of the segment, $v$ is the speed of the particle. The total charge transferred during this time through the cross section of the conductor, $Q = I\cdot t$ . Substituting here the time value from the previous equation, we have

$Q = \frac {I\cdot l}{v}$ (2)

In the same time $F_A=F_L\cdot N$ , where $N$ is the number of particles in the considered conductor. Wherein $N = \frac {Q}{q}$ , where $q$ is the charge of one particle. Substituting the value into the formula $Q$ from (2), one can get:

$N = \frac {I\cdot l}{v\cdot q}$

In this way,

$F_A=F_L\cdot \frac {I\cdot l}{v\cdot q}$

Using (1), the previous expression can be written as

$B\cdot I\cdot l\cdot sin\alpha = F_L\cdot \frac {I\cdot l}{v\cdot q}$

After contractions and transfers, a formula appears for calculating the Lorentz force

$F_L = q\cdot v\cdot B\cdot sin\alpha$

Given that the formula is written for the force modulus, it must be written as follows:

$F_L = |q|\cdot v\cdot B\cdot sin\alpha$ (3)

Because the $sin\alpha = sin(180^{\circ} - \alpha)$ , then to calculate the Lorentz force modulus, it does not matter where the velocity is directed, - in the direction of the current strength or against, - and we can say that $\alpha$ is the angle formed by the particle velocity and magnetic induction vectors.

Writing a formula in vector form will look like this:

$\vec{F_L} = q\cdot [\vec{v}\times \vec{B}]$

$[\vec{v}\times \vec{B}]$ is a cross product, the result of which is a vector with modulus equal to $v\cdot B\cdot sin\alpha$ .

Based on formula (3), we can conclude that the Lorentz force is maximum in the case of perpendicular directions of the electric current and magnetic field, that is, when $\alpha = 90^{\circ}$ , and disappear when they are parallel ( $\alpha = 0^{\circ}$ ).

It must be remembered that in order to obtain the correct quantitative answer - for example, when solving problems - one should use the units of the SI system, in which magnetic induction is measured in teslas (1 T = 1 kg s⁻²·BUT⁻¹), force - in Newtons (1 N = 1 kg m/s²), current strength - in amperes, charge in coulombs (1 C = 1 A s), length - in meters, speed - in m / s.

Determining the direction of the Lorentz force using the left hand rule

Since the Lorentz force manifests itself as the Ampère force in the world of macroobjects, the left hand rule can be used to determine its direction.

Determination of the direction of action of the Lorentz force according to the rule of the left hand.

You need to put your left hand so that the open palm is perpendicular to and towards the lines of the magnetic field, four fingers should be extended in the direction of the current strength, then the Lorentz force will be directed where the thumb points, which should be bent.

Movement of a charged particle in a magnetic field

In the simplest case, that is, when the vectors of magnetic induction and particle velocity are orthogonal, the Lorentz force, being perpendicular to the velocity vector, can only change its direction. The magnitude of the speed, therefore, and the energy will remain unchanged. This means that the Lorentz force acts by analogy with the centripetal force in mechanics, and the particle moves in a circle.

In accordance with Newton's II law ( $F = m\cdot a$ ) we can determine the radius of rotation of the particle:

$N = \frac {m\cdot v}{q\cdot B}$ .

It should be noted that with a change in the specific charge of the particle ( $\frac {q}{m}$ ) the radius also changes.

In this case, the rotation period T = $\frac {2\cdot \pi\cdot r}{v}$ = $\frac {2\cdot \pi\cdot m}{q\cdot B}$ . It does not depend on speed, which means that the mutual position of particles with different speeds will be unchanged.

Movement of a charged particle in a uniform magnetic field.

In a more complicated case, when the angle between the particle velocity and the magnetic field strength is arbitrary, it will move along a helical trajectory - translationally due to the velocity component directed parallel to the field, and along the circle under the influence of its perpendicular component.

Application of the Lorentz force in engineering

Kinescope

The kinescope, which stood until recently, when it was replaced by an LCD (flat) screen, in every TV set, could not work without the Lorentz force. To form a television raster on the screen from a narrow stream of electrons, deflecting coils are used, in which a linearly changing magnetic field is created. The horizontal coils move the electron beam from left to right and return it back, the personnel coils are responsible for the vertical movement, moving the beam running horizontally from top to bottom. The same principle is used in oscilloscopes - devices used to study alternating electrical voltage.

mass spectrograph

A mass spectrograph is a device that uses the dependence of the radius of rotation of a charged particle on its specific charge. The principle of its operation is as follows:

The source of charged particles, which pick up speed with the help of an artificially created electric field, is placed in a vacuum chamber in order to exclude the influence of air molecules. Particles fly out of the source and, having passed along the arc of a circle, hit the photographic plate, leaving traces on it. Depending on the specific charge, the radius of the trajectory changes and, therefore, the point of impact. This radius is easy to measure, and knowing it, you can calculate the mass of the particle. With the help of a mass spectrograph, for example, the composition of the lunar soil was studied.

Cyclotron

The independence of the period, and hence the frequency of rotation of a charged particle from its speed in the presence of a magnetic field, is used in a device called a cyclotron and designed to accelerate particles to high speeds. A cyclotron is two hollow metal half-cylinders - a dee (in shape, each of them resembles the Latin letter D) placed with straight sides towards each other at a short distance.

Cyclotron - application of the Lorentz force.

The dees are placed in a constant uniform magnetic field, and an alternating electric field is created between them, the frequency of which is equal to the frequency of rotation of the particle, determined by the magnetic field strength and specific charge. Getting twice during the period of rotation (during the transition from one dee to another) under the influence of an electric field, each time the particle accelerates, while increasing the radius of the trajectory, and at a certain moment, having gained the desired speed, flies out of the device through the hole. In this way, a proton can be accelerated to an energy of 20 MeV (megaelectronvolt).

Magnetron

A device called a magnetron, which is installed in each microwave oven, is another representative of devices using the Lorentz force. The magnetron is used to create a powerful microwave field, which heats the internal volume of the oven, where the food is placed. The magnets included in its composition correct the trajectory of the movement of electrons inside the device.

Earth's magnetic field

And in nature, the Lorentz force plays an extremely important role for humanity. Its presence allows the Earth's magnetic field to protect people from the deadly ionizing radiation of space. The field prevents charged particles from bombarding the surface of the planet, forcing them to change direction.