What Is a Magnetic Field Force?

Magnetic field force is a kind of acting force, including Lorentz force of magnetic field on moving charge and Ampere force of magnetic field on electric current. Magnetic field forces include two types, one is the force of a magnetic field on a conducting wire, and the other is the force of a magnetic field on a moving charge.

Magnetic force

Two types of extreme value problems related to magnetic field forces

Extreme value problems in magnetic fields are often related to magnetic field forces. Extreme value problems in magnetic fields can be divided into two types according to magnetic field forces. One is the extreme value problem related to ampere force, and the other is related to Lorentz. Force-related extremum issues. But no matter which type of extreme value problem is solved, the research object must be determined first, and the force analysis must be done well. Then, according to the force situation and the initial state, the movement process of the research object must be clarified, and then the corresponding physical laws must be used according to the movement process. Finally, Find the required physical quantity.

Extreme problems with Ampere participation

Whether the conducting wire is stationary or moving in the magnetic field, it may be affected by amperage. However, when the energized wire is in a stationary state, it will not generate electromotive force itself, and when in a moving state, the energized wire itself will also generate induced electromotive force due to electromagnetic induction. Therefore, when solving the extreme value problem of the energized wire in a moving state, you cannot forget This induced emf .

1.1 Extreme value problems when the energized wire is stationary

Example 1 As shown in FIG. 1, the copper rod has a mass of 0.1 kg and is stationary on a horizontal orbit with a distance of 8 cm. The dynamic friction factor between the two is 0.5. Now carry a 5A current from one track to another track. What is the minimum value of the magnetic induction intensity of the uniform magnetic field applied between the two guide rails to slide the copper rod?

Picture 1 The angle between the ampere force F and the horizontal direction is set as , and the force analysis diagram shown in FIG. 2 is drawn. then:

F = BIL. (1)

And the copper rod under the combined action of gravity, support, ampere and friction should meet:

Fcos- (mg-Fsin) = 0. (2)

Simultaneous (1), (2) solution:

Picture 2 1.2 Extremum Problems When Moving Conductive Wires

Example 2 As shown in Figure 3, the distance between two parallel metal rails placed horizontally is L = 0.25cm, the battery's electromotive force E = 6V, the internal resistance r = 0, the resistance R = 5, and the uniform magnetic field magnetic induction strength B vertical Straight down, after K is closed, the metal bar ab placed on the guide rail moves from the static position to the right under the action of the magnetic field force. The sliding friction between the metal rod and the guide rail f = 0.15N, in order to maximize the speed of the rod. , How big should B be? At this time, what is Vmax?

When analyzing the movement of the metal rod, an induced electromotive force is generated. If the direction of this electromotive force is opposite to that of the battery electromotive force, the current in the circuit is: I = (E-BLv) / R. (1) When the speed of the metal rod is maximum, the acceleration should be zero, then: BIL-f = 0. (2) Substituting (1) into (2) gives:

Picture 4

Picture 3

Extreme problem with Lorentz force

2.1 Extreme value problems when only Lorentz force is applied. When there is only one charged particle in the problem, it is often only one motion orbit to determine the extreme value problem of particles. When it is related to the motion of multiple charged particles, Multiple orbits are needed to determine the extremum of charged particles.

Example 3 As shown in FIG. 4, a sufficiently long rectangular area abcd has a uniform magnetic field with a magnetic induction intensity of B perpendicular to the paper surface. At the midpoint O of ab, a vertical magnetic field enters a charged particle with a velocity direction at an angle of 30 ° to the ab side and a velocity of v. It is known that the mass of the particle is m, the charge is e, and the length of the ab side is ab The edges are long enough that the particle gravity is negligible. Question: When all the electrons are emitted from the bc side, what is the value range of the electron incident velocity v?

Picture 5 Analyze from the formula of the radius of the uniform motion of the charged particles in the magnetic field. The smaller the radius, the smaller the moving speed of the charged particles; the larger the radius, the greater the moving speed of the charged particles. Because the orbit radius across a point on the circumference is always perpendicular to the velocity direction at that point, it can be seen that the centers of the circles with the maximum orbit radius and the minimum orbit radius are on the perpendicular to the initial velocity direction. This can be drawn as

A schematic diagram shown in FIG. 5. When the orbit radius is the smallest, the electrons are emitted from M on the bc side, and when the orbit radius is the largest, the electrons are not emitted from the ad side. Assuming the maximum and minimum rates are v1 and v2, and the maximum radii are R1 and R2, respectively, the geometric relationship in the figure can be obtained:

R1 (1-sin) = 1/2. (1)

R2 (1 + sin) = 1/2. (2)

Since = 30 °, substituting into the above two formulas can be obtained:

R1 = 1, R2 = 1/3. (3)

From the Lorentz force as a centripetal force:

Bev = mv2 / R. (4)

From the above four formulas, the number of particles ejecting v from the side of bc can be obtained.

Value range is:

Picture 6 Example 4 As shown in Figure 6, in the first quadrant of the plane rectangular coordinate system, there is a beam of electrons with a width of d, where the velocity of each electron is v, all of which move parallel to the right axis of the Ox axis at a constant speed The mass of the electron is m and the amount of electricity is e. The coordinates of point A in the figure are (1,0) and 1> d. It is now required that this beam of electrons can pass point A after passing through a uniform magnetic field perpendicular to the xOy plane, and An electron with a distance of d from the x-axis passes through the point A perpendicular to the x-axis after being deflected by a magnetic field. Find: (1) the magnitude and direction of the magnetic induction intensity B of the magnetic field in this area. (2) Design a uniform magnetic field area with the smallest area that meets the above conditions, find its area, and draw the magnetic field boundary on the xOy plane.

Picture 7 Parsing

(1) As shown in Figure 7, first consider an electron with a distance d from the x-axis, which passes through the point A on the x-axis perpendicularly after being deflected by a magnetic field. The left-hand rule shows that the direction of the magnetic field is perpendicular to the paper surface and the orbit of the electron Radius R = d. (1) The Lorentz force of the electron is obtained as the centripetal force: Bev = mv ^ 2 / R. (2) Simultaneous solution of (1) and (2): B = mv / ed.

(2) It can be seen from Figure 7 that the electrons with a distance of d from the x-axis pass through the magnetic field and pass through A perpendicularly. The arc QA should be the upper boundary of the magnetic field region sought. To determine the lower boundary, we examine the motion of an electron with a distance of y from the x-axis. Let it enter a magnetic field from point P (x, y), and then pass through point A after deflection by the magnetic field. A circle with center and d as radius

Picture 8

Picture 9 2.2 Extreme problems under the combined action of Lorentz force and other external forces

The movement of the charge in this case is not as simple as the former. In some problems it is moving in a straight line, while in other problems it is moving in a curve. The motion of the charge depends on the specific situation.

Example 5 As shown in Figure 8, a small ball with a mass of 0.1g is charged with a power of 4 × 10-4C, and it is placed in the

On a long insulating straight rod, put the rod vertically into a uniform and electric field that is horizontal and parallel to each other. The starting field strength is E = 10N / C, and the magnetic induction strength is B = 0.5T. The dynamic friction coefficient between the ball and the stick is 0.2. Find the maximum speed at which the small ball slides along the stick.

Analyze the force analysis diagram shown in Figure 9. The ball is affected by gravity, electric field force, Lorentz force, elastic force and friction force. When the total force is zero, the speed of the ball falling down reaches the maximum. (F in Fig. 9 represents the combined force of the electric field force and the Lorentz force)

Picture 11

Picture 10

Picture 12 Example 6 As shown in Figure 10, a smooth cone with a vertex angle of 2 is fixed in a uniform magnetic field. A small ball edge with mass m and electric quantity q

What is the minimum radius of the cone surface for uniform circular motion?

The analysis is shown in Figure 11. There are three forces in the ball movement: gravity mg, support force FN, Lorentz force F = qvB.

By the following two formulas:

FNsin-mg = 0,

And qvB-FNcos = mv / r.

Obtained: mv-qBrv + rmgctg = 0.

Because the value range of v is a real number, therefore:

(qBr) -4mgrctg0,

Picture 13