# Why -e(v/c)H is the magnetic force?

• Haorong Wu
In summary: This equation can be written in a more intuitive way as follows:#\vec{B}=\mu_0 \vec{H}+\vec{M}\left (\vec{E}+\frac{\vec{v}}{c}\right).
Haorong Wu
In a paper I am reading, it reads, "For these orbits the electric force ##-e E_r## almost balances the magnetic force ##-e \left ( v/c \right ) H_0##." where ##-e## is the charge of the electrons, ##v## is the speed of the electrons, ##H_0## is a homogeneous magnetic field, and ##c## is not clearly indicated, but I guess it is the speed of light.

However, according to Lorentz force, ##F=-e v B= -e v \mu _0 H##. I do not see any ##c## or ##\epsilon_0##, so why the paper says ##-e \left ( v/c \right ) H_0## is the magnetic force?

Thanks.

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From the following content, it seems that ##B=\frac {H_0} c##, which I am not familiar with, so what is ##H_0##?

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The presence of c in the definition of magnetic force is probably due to the use of cgs instead of the the more familiar MKS system. That is the charge is measured in statcoulombs instead of coulombs

Haorong Wu
It's ##\vec{B}##, not ##\vec{H}## which has to be used in the Lorentz force,
$$\vec{F}=q \left (\vec{E} + \frac{\vec{v}}{c} \times \vec{B} \right).$$
This is valid in Gaussian as well as Heaviside-Lorentz units.

In the SI the eq. reads
$$\vec{F}=Q (\vec{E} + \vec{v} \times \vec{B}).$$
There it's the more important to write correctly ##\vec{B}=\mu_0 \vec{H}## (in vacuo, with ##\mu_0## the permeability of the vacuum, which occurs in the SI as a conversion factor of units).

Haorong Wu
vanhees71 said:
It's B not H which has to be used in the Lorentz force

Are you sure? Does not a charged particle in a region with B = 0 but M <> 0 deflect?

gleem said:
The presence of c in the definition of magnetic force is probably due to the use of cgs instead of the the more familiar MKS system. That is the charge is measured in statcoulombs instead of coulombs

Thanks, gleem. You are right. The author used cgs system which I am not familiar with and it is a paper in 1960s so H may mean B in that time.

And thanks to other friends.

Are you sure? Does not a charged particle in a region with B = 0 but M <> 0 deflect?
How can ##\vec{B}=\vec{H}+\vec{M}=0## (Heaviside-Lorentz units)?

Good point.

## 1. What is the meaning of -e(v/c)H in the magnetic force equation?

-e(v/c)H represents the charge of a particle (e) multiplied by its velocity (v) divided by the speed of light (c) and the strength of the magnetic field (H). It is a mathematical representation of the force a charged particle experiences when moving through a magnetic field.

## 2. Why is the magnetic force dependent on the velocity of the particle?

The magnetic force is dependent on the velocity of the particle because it is a result of the interaction between the particle's charge and the magnetic field. When the particle moves, its charge creates a magnetic field around it, which then interacts with the external magnetic field, resulting in a force.

## 3. How does the direction of the magnetic force change with the direction of the magnetic field?

The direction of the magnetic force is always perpendicular to both the velocity of the particle and the direction of the magnetic field. If the particle is moving parallel to the magnetic field, there will be no magnetic force acting on it. If the particle is moving perpendicular to the magnetic field, the force will be at a maximum.

## 4. Why is the magnetic force considered a centripetal force?

The magnetic force can be considered a centripetal force because it acts towards the center of the circular motion of a charged particle in a magnetic field. This force is necessary to keep the particle moving in a circular path, preventing it from flying off in a straight line.

## 5. How does the strength of the magnetic field affect the magnitude of the magnetic force?

The strength of the magnetic field directly affects the magnitude of the magnetic force. The stronger the magnetic field, the greater the force will be on a charged particle moving through it. This can be seen in the magnetic force equation, where the strength of the magnetic field (H) is a factor in determining the overall force on the particle.

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