# THE FORCE AND THE MAGNETIC FIELD (2023)

• THE STRENGTH AND THE MAGNETIC FIELD.
• 30.1. the magnetic force
• Example: Issue 30.10
• 30.2. Lei de Biot-Savart
• Example: Problem 30.33
• Example: Problem 30.22
• 30.3. magnetic dipole
• Example: Problem 30.44

## 30.1. the magnetic force

So far, we have only considered the electrostatic forces acting on charges at rest. When charges are in motion, an additional force acts on them. This additional force is calledmagnetic force🇧🇷 The magnetic force between two charges q1yq2, moving with speed v1is V2, It's the same as (30.1)

where you0it's calledpermeability constantwhich is equal to 4[pi] x 10-7ns2/C2, and r is the distance between the two charges (see Figure 30.1). The ratio R between the magnetic force and the electric force is equal to (30.2) Figure 30.1. Vectors relevant to the definition of magnetic force.
Replacing numeric values ​​of [epsilon]0is that you0in equation (30.2), the relation R can be rewritten as (30.3)

where c is the speed of light in vacuum (c = 3 x 108IN). Clearly, the magnetic force is small compared to the electrical force unless the particle velocity is high (a significant fraction of the speed of light).

A magnetic field B can be associated with the magnetic force. The magnetic field at some point near a moving charge can be determined by placing a test charge at that point and moving it with a certain speed v. The test charge will experience, in addition to the electrical force, a magnetic force Fmagazine🇧🇷 By definition, the magnetic field B is related to the magnetic force Fmagazineacross (30.4)

A measure of the magnetic force acting on the test charge in various directions of v can be used to determine the magnetic field B. The magnetic force is always perpendicular to the velocity vector and the direction of the magnetic field. The unit of magnetic field strength is theTesla(T)🇧🇷 Comparing equation (30.1) and equation (30.4) we can determine the magnetic field generated by a point charge q2moving with a speed2: (30.5)

Similar to electric field lines, we can graph the magnetic field by field lines. The density of the field lines indicates the strength of the magnetic field. The tangent of the field lines indicates the direction of the magnetic field. Magnetic field lines form closed loops, that is, they don't start or end anywhere in the same way that electric field lines start and end at positive and negative charges. This immediately implies that the magnetic flux through an arbitrary closed surface equals zero: (30.6)

The principle of superposition is also valid for the magnetic field.

### Example: Issue 30.10

On the surface of a pulsar or neutron star, the magnetic field can reach 108T. Consider the electron in a hydrogen atom on the surface of the neutron star. The electron is at a distance of 0.53 x 10-10m of the proton and has a speed of 2.2 x 106IN. Compare the electrical force that the proton exerts on the electron to the magnetic force that the neutron star's magnetic field exerts on the electron. Is it reasonable to expect the magnetic field to strongly deform the hydrogen atom?

The electron in a hydrogen atom is at a distance r equal to 0.53 x 10-10m of the proton. The electric force acting on the electron is equal to (30.7)

The maximum magnetic force acting on the electron occurs when the direction of the electron is perpendicular to the direction of the magnetic field. The maximum magnetic force is equal to (30.8)

Comparing to Eq. (30.7) and Eq. (30.8), we conclude that the magnetic field is significantly stronger than the electric field, and we expect that the electron orbits are strongly affected by the strong magnetic field.

(Video) Magnetic Force

## 30.2. Lei de Biot-Savart

The definition of magnetic force showed that two moving charges experience a magnetic force. In other words, a moving charge produces a magnetic field which results in a magnetic force acting on all charges moving in this field.

A current flowing through a wire is equivalent to a set of electrons moving with a certain velocity along the direction of the wire. Each of the moving electrons produces a magnetic field which is given by equation (30.5). Consider a short segment of wire of length dL (see Figure 30.2). At any time, a dq charge will be located in this segment. The magnetic field, dB, generated by this charge at point P is equal to (30.9)

where v is the speed of the charge carriers. The time dt it takes for all the original charge carriers to leave segment dL is given by (30.10)

The current I in the wire can now be easily obtained (30.11) Figure 30.2. Calculation of the magnetic field produced by an electric current.
This equation can be rewritten as (30.12)

and replaced in equation (30.9): (30.13)

Equation (30.13) is calledLei de Biot-Savart.

### Example: Problem 30.33

Helmholtz coils are often used to generate reasonably uniform magnetic fields in laboratories. These coils consist of two thin circular rings of wire parallel to each other and a common axis, the z-axis. The rings have a radius R and are separated by a distance that is also R. These rings carry equal currents in the same direction. Find the magnetic field at any point on the z axis. Figure 30.3. Calculation of the magnetic field produced by a ring.
The first step in calculating the field of a pair of Helmholtz coils is to calculate the magnetic field produced by each ring. Assume that the ring is located in the x-y plane and we are interested in the field at point P, a distance z above the x-y plane (see Figure 30.3). The ring's net magnetic field at point P will be directed along the z axis. The magnitude dB of the magnetic field produced by a small segment of the ring with length dL is equal to (30.14)

To obtain equation (30.14), we use the fact that, for any point on the ring, the position vector r is perpendicular to the dL direction. The z component of the magnetic field dB is equal to (30.15)

The magnitude of the position vector r is related to R and z: (30.16)

The angle a is also related to R and z:

(Video) Magnetic Force and Magnetic Field | Don't Memorise (30.17)

Combining equations (30.15), (30.16) and (30.17) we get (30.18)

Integrating equation (30.18) over the entire ring we obtain for the total field generated by the ring (30.19)

Figure 30.4 shows the magnetic field generated by a coil of radius 1 m located at z = 0 m.

To find the field generated by a pair of Helmholtz coils, we assume that the coils are centered at z = 0 and at z = R. The magnetic field generated by the coil located at z = 0 is given by Eq. (30.19). The magnetic field generated by the coil located at z = R is given by (30.20)

The total field on the axis of a pair of Helmholtz coils is equal to the sum of the field generated by coil 1 and the field generated by coil 2: (30.21)

The total magnetic field generated by a pair of Helmholtz coils is shown in Fig. 30.5, where the contributions of the two coils are also shown individually. We observe that the field is quite homogeneous between the coils (0 < z < R). Figure 30.4. Magnetic field generated by a coil with R = 1 m. Figure 30.5. Magnetic field generated by a pair of Helmholtz coils.

### Example: Problem 30.22

A very long wire is bent at a right angle near its midpoint. One of its branches is along the positive x axis and the other along the positive y axis (see Figure 30.6). The wire carries a current I. What is the magnetic field at a point in the first quadrant of the x-y plane? Figure 30.6. Problem 30.22 Figure 30.7. Field generated by wire.
The first step in solving this problem is to observe the magnetic field produced by this single wire (see Figure 30.7). The direction of the magnetic field generated by a short segment of the wire points away from the paper. The magnitude of the field, dB, is equal to (30.22)

The x position of the considered segment is determined by the angle a: (30.23)

o (30.24)

From equation (30.24) we can obtain a relationship between dx and da:

(Video) Magnetism, Magnetic Field Force, Right Hand Rule, Ampere's Law, Torque, Solenoid, Physics Problems (30.25)

And more, (30.26)

Substituting Eq. (30.25) and Eq. (30.26) in Eq. (30.22), we get (30.27)

The total field can be obtained by integrating Eq. (30.27) on the wire. The limits of integration are (30.28)

y (30.29)

The result of the integration is (30h30)

The vertical wire field can be obtained in a similar way: (30.31)

The magnitude of the total field is therefore equal to (30.32)

## 30.3. magnetic dipole

The magnetic field at the axis of a current loop was discussed in Problem 30.33. At large distances from the current loop (z >> R), the field is approximately equal to (30.33)

which shows that the strength of the magnetic field decreases as 1/z3This dependence of magnetic field strength on distance is similar to the observed dependence for the electric field strength of an electric dipole:

(Video) Magnetic Force on a Moving Charge In a Magnetic Field (30.34)

Equation (30.33) is often rewritten as (30.35)

Where (30.36)

is called the magnetic dipole moment of the loop. In general, the dipole moment of a current loop is equal to (30.37)

There are magnetic dipole moments for objects as small as electrons and as large as Earth.

### Example: Problem 30.44

A quantity of charge Q is uniformly distributed on a paper disk of radius R. The disk rotates about its axis with angular velocity [omega]. Find the magnetic dipole moment of the disk.

The first step in solving this problem is to determine the dipole moment of the edge of the disk, of radius r and width dr. The amount of charge in this ring is equal to (30.38)

The angular speed of the disk is [omega] and its period T is equal to (30.39)

During a period, charge dq will pass through any point on the ring. The current dI is therefore equal to (30.40)

The magnetic dipole moment du of the ring is equal to (30.41)

The total dipole moment of the disk can be found by integrating Eq. (30.41) between r = 0 and r = R:

(Video) The Electromagnetic field, how Electric and Magnetic forces arise (30.42)

Send your comments, questions and/or suggestions via email towolfs@nsrl.rochester.eduand/or visit theMain pagethe Frank Wolfs.

## FAQs

### How is force related to magnetic field? ›

What Is the Force Due to a Magnetic Field? Magnetic fields can exert a force on an electric charge only if it moves, just as a moving charge produces a magnetic field. This force increases with both an increase in charge and magnetic field strength. Moreover, the force is greater when charges have higher velocities.

Is magnetic field and force the same? ›

Magnetic field is the strength of magnetism created by a magnet, whereas the magnetic force is the force due to two magnetic objects. The concepts of magnetic field and magnetic force are widely used in fields such as classical mechanics, electromagnetic theory, field theory and various other applications.

What is the force law of a magnetic field? ›

Lorentz force, the force exerted on a charged particle q moving with velocity v through an electric field E and magnetic field B. The entire electromagnetic force F on the charged particle is called the Lorentz force (after the Dutch physicist Hendrik A. Lorentz) and is given by F = qE + qv × B.

Is force proportional to magnetic field? ›

The strength of the magnetic force on a charge is proportional to the magnetic field through which the charge is moving – This is not surprising, as it was also true for the electric force and field, but more to the point, it really is a result of our definition of magnetic field.

What is the relation between force and field? ›

Explanation: The electric field strength is calculated by the formula E= F/q, where 'F' is the force and 'q' is the test charge. And according to Coulomb's law force due to point charges is given by the formula.

How are force and field related? ›

Field is force per unit charge. The field tells you what the force would be on a particle with a charge of 1, if such a particle was there. Force is what the actual force is on whatever particle is there.

What is magnetic force also called? ›

This force is termed as the Lorentz Force. It is the combination of the electric and magnetic force on a point charge due to electromagnetic fields.

Is a magnetic field a force or energy? ›

Magnetism is a force, but it has no energy of its own,” says David Cohen-Tanugi, vice president of the MIT Energy Club and a John S. Hennessy Fellow in MIT's Materials Science and Engineering department. Still, he adds, “magnetism is extremely useful for converting energy from one form to another.

Is force opposite to magnetic field? ›

Right Hand Rule

The force is in the opposite direction for a negative charge moving in the direction shown. One fact to keep in mind is that the magnetic force is perpendicular to both the magnetic field and the charge velocity, but that leaves two possibilities.

What is magnetic force in simple words? ›

magnetic force, attraction or repulsion that arises between electrically charged particles because of their motion. It is the basic force responsible for such effects as the action of electric motors and the attraction of magnets for iron.

### Where does the force that creates a magnetic field come from? ›

Although it may seem like magic, that force comes from tiny particles called electrons inside an atom. In certain types of metals, electrons spin around and pair off in different ways than they do for other types of materials. That activity is what creates the magnetic field.

What are the 4 characteristics of magnetic force? ›

General Properties of Magnetic Lines of Force

They never cross one another. They all have the same strength. Their density decreases (they spread out) when they move from an area of higher permeability to an area of lower permeability. Their density decreases with increasing distance from the poles.

What do magnetic forces depend on? ›

The force acting on an electrically charged particle in a magnetic field depends on the magnitude of the charge, the velocity of the particle, and the strength of the magnetic field.

Why force is perpendicular to magnetic field? ›

The superficial answer is simply that the Lorentz (magnetic) force is proportional to v×B, where v is the particle velocity and B is the magnetic field. Since the vector cross product is always at right angles to each of the vector factors, the force is perpendicular to v.

Is a magnetic field a force field? ›

Magnetism and magnetic fields are one aspect of the electromagnetic force, one of the four fundamental forces of nature.

How do you increase the force of a magnetic field? ›

The force on a current-carrying conductor in a magnetic field can be increased by increasing the flow of current in the conductor and also by increasing externally applied magnetic field.

How are a field and a force different? ›

In other words force describes the actual effect on a charge; field describes the potential effect. Field is measured in N/C (newtons per coulomb), or alternatively V/m (volts per meter).

Is a field a force? ›

In physics, a field is a region where every point in spacetime is well defined, and the concept of fields is used to describe the action of forces, called a force field.

What creates a magnetic field? ›

Magnetic fields are produced by moving electric charges. Everything is made up of atoms, and each atom has a nucleus made of neutrons and protons with electrons that orbit around the nucleus. Since the orbiting electrons ≠are tiny moving charges, a small magnetic field is created around each atom.

What are the 4 force fields? ›

The four basic forces are the gravitational force, the electromagnetic force, the weak nuclear force, and the strong nuclear force.

### What are the 3 force fields? ›

Currently, polarizable functional forms used in polarizable force fields can be classified into three categories: the fluctuating charge model, the induced dipole model, and the classical Drude oscillator model. These models are briefly introduced below.

What are 5 examples of magnetic force? ›

Examples of magnetic force is a compass, a motor, the magnets that hold stuff on the refrigerator, train tracks, and new roller coasters. All moving charges give rise to a magnetic field and the charges that move through its regions, experience a force.

What are the 2 types of magnetic forces? ›

The two types of magnetic forces are:
• Attractive force: Unlike poles (north-south) of a magnet attract each other.
• Repulsive force: Like poles (north-north; south-south) of a magnet repel each other.

What are the two magnetic forces called? ›

Magnetic dipole–dipole interaction.

Do humans have a magnetic force? ›

Today, two hundred years later, we know that the human body is indeed magnetic in the sense that the body is a source of magnetic fields, but this body magnetism is very different from that imagined by Mesmer.

Is magnetism a real force? ›

Magnetism is the force exerted by magnets when they attract or repel each other. Magnetism is caused by the motion of electric charges. Every substance is made up of tiny units called atoms. Each atom has electrons, particles that carry electric charges.

Does everything have a magnetic force? ›

Magnetic fields are everywhere

Naturally occurring magnetic fields are seen everywhere in the universe. They were first observed on Earth thousands of years ago, through their interaction with magnetized minerals like lodestone, and used for navigation long before people had any understanding of their nature or origin.

Are force and magnetic field perpendicular? ›

The magnetic field does not point along the direction of the source of the field; instead, it points in a perpendicular direction. In addition, the magnetic force acts in a direction that is perpendicular to the direction of the field.

What are the effects of force? ›

Effects of force with examples :
• It can make a stationary object move. Example: pushing a box at rest on the table brings the box in motion.
• It can stop a moving object. ...
• It can change the speed of a moving object. ...
• It can change the direction of a moving object. ...
• It can change the shape or size of an object.

Where is the magnetic field the strongest? ›

The magnetic field is strongest near the poles of the magnet. It is equally strong at both the poles. The magnetic field lines are closer near the poles. The strength of the magnetic field depends on how close the magnetic field lines are.

### What are the 3 rules of magnetism? ›

These are for (1) long, straight wires, (2) free moving charges in magnetic fields, and (3) the solenoid rule – which are loops of current. Calling these "rules" is the right name. They are not laws of nature, but conventions of humankind.

What are the 3 main magnetic elements? ›

Since then only three elements on the periodic table have been found to be ferromagnetic at room temperature—iron (Fe), cobalt (Co), and nickel (Ni).

What two factors affect magnetic force? ›

The factors affecting the strength of a magnetic field at a point due to a straight current-carrying conductor are the magnitude of the electric current and the perpendicular distance between the point and the conductor.

What are the four main factors that will affect magnetic force? ›

Explanation: The answer is "all of these affect the strength of a magnet" The proximity to the object, the size of the object, the material of the object it is sticking to, and the temperature of the object all affect magnetic pull.

What makes a magnetic force stronger? ›

The force generated by the aligned atoms creates a magnetic field. A larger piece of iron would have more atoms to align, potentially resulting in a stronger magnetic field than a smaller piece of the same material.

How do you find the direction of force in a magnetic field? ›

The right hand rule states that: to determine the direction of the magnetic force on a positive moving charge, point your right thumb in the direction of the velocity (v), your index finger in the direction of the magnetic field (B), and your middle finger will point in the direction of the the resulting magnetic force ...

Does magnetic force depend on charge? ›

The magnitude of the magnetic force between them depends on how much charge is in how much motion in each of the two objects and how far apart they are. The direction of the force depends on the relative directions of motion of the charge in each case.

Why do magnetic fields have direction? ›

And the reason for that is because at the point of intersection. If we were to keep a magnetic compass a magnetic meter. Than this field line will make a point in this direction. But this field line will make a point in this direction.

Which field is force? ›

Hence, the two field forces in Physics are Gravitational force and electric force.

Is force inversely proportional to magnetic field? ›

CONCEPT: Coulomb's law of magnetism: It states that, If two magnetic poles of strengths m1 and m2 are held apart at a distance r then the force of attraction or repulsion between the two poles is directly proportional to the product of their pole strengths and inversely proportional to the distance square between them.

### What is magnetic field directly proportional to? ›

The magnetic field due to current flowing in a ling straight conductor is directly proportional to the current and inversely proportional to the distance of the point of observation from the conductor.

What causes magnetic force? ›

Magnetism is the force exerted by magnets when they attract or repel each other. Magnetism is caused by the motion of electric charges. Every substance is made up of tiny units called atoms. Each atom has electrons, particles that carry electric charges.

What is difference between field and force? ›

In other words force describes the actual effect on a charge; field describes the potential effect. Field is measured in N/C (newtons per coulomb), or alternatively V/m (volts per meter).

What type of force is magnetism? ›

Magnetic forces are non contact forces; they pull or push on objects without touching them. Magnets are only attracted to a few 'magnetic' metals and not all matter. Magnets are attracted to and repel other magnets.

## Videos

1. 21.5 The Force on a Current in a Magnetic Field
(Physics Demos)
2. Magnetic force on a charge | Physics | Khan Academy
3. Magnets and Magnetic Fields
(Professor Dave Explains)
4. Magnetic Forces and Magnetic Fields
(NPS Physics)
5. Magnetic Force on a Current Carrying Wire
(The Organic Chemistry Tutor)
6. Magnetism: Crash Course Physics #32
(CrashCourse)
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