# Ampère's circuital law and the free current density

• gop
In summary, the conversation discusses Ampere's law, current density, and the equation for free current density in the context of a point charge. While the equation J_f(x) = q*v'*delta(x-x') is a representation of the free current density at a specific point, the full set of Maxwell's equations must be used to accurately simulate the movement of a point charge in a system.
gop
Hi

In Ampere's circuit law we have the current density which is separated in bound and free current. Bound current is due to the atom's internal structure and free current is well that is the question actually.
Suppose I have a point charges governed by maxwell's equations and I try to solve/simulate the movement of this charge what is the free current density I have to put into the equation?

I think it is something like
$$J_f(x) = q*v'*delta(x-x')$$
(where x' and v' is the position and velocity of the particle)
But I can't find a explicit equation for J in any textbook. So I'm somewhat confused.

thx

Hello,

First of all, I would like to clarify that Ampere's circuit law is a simplified version of the more general Ampere's law, which states that the line integral of the magnetic field around a closed loop is equal to the total current passing through the loop. This law is a consequence of Maxwell's equations, which describe the fundamental relationship between electric and magnetic fields.

In terms of current density, the bound current is due to the displacement of charges within a material, while the free current is due to the movement of free charges (such as electrons) through a material. In your example of a point charge, the free current density would indeed be represented by the equation J_f(x) = q*v'*delta(x-x'), where q is the charge of the particle and v' is its velocity.

However, it is important to note that this equation represents the current density at a specific point in space, and not the entire current density in a given system. In order to fully simulate the movement of a point charge, you would need to consider the contributions of all the other charges in the system, both bound and free.

To do this, you would need to use the full set of Maxwell's equations, which includes the continuity equation that relates the current density to the charge density and the time derivative of the electric field. This equation takes into account the movement of all charges in the system and would allow you to accurately simulate the movement of your point charge.

I hope this helps clarify your confusion. It is always important to consider the full set of equations and the context of the system when trying to understand a specific aspect of physics. Good luck with your simulations!

Hello,

You are correct in your understanding that Ampère's circuital law deals with the current density, which is a measure of the flow of electric charge per unit area. This law states that the circulation of the magnetic field around a closed loop is equal to the current passing through the loop, taking into account the bound and free currents. Bound currents are due to the internal structure of atoms and are typically associated with materials that have a magnetic moment, such as ferromagnetic materials. Free currents, on the other hand, are due to the movement of electric charges and are present in conductors, for example.

In order to solve or simulate the movement of a point charge, you would need to take into account both the bound and free currents in the system. The equation you have written for the free current density appears to be correct, where q is the charge of the particle and v' is its velocity. However, the exact equation for J may vary depending on the specific situation and the type of material you are studying. It is important to note that in most cases, the bound current can be neglected when studying the movement of a single point charge, as its contribution to the overall current density is usually much smaller compared to the free current.

I recommend consulting a textbook on electromagnetism or seeking guidance from an expert in the field for a more specific equation for J in your particular case. I hope this helps clarify your confusion. Good luck with your research!

## 1. What is Ampère's circuital law?

Ampère's circuital law is a fundamental principle in electromagnetism that describes the relationship between the magnetic field and the electric current flowing through a closed loop. It states that the line integral of the magnetic field around a closed loop is equal to the total current passing through the loop.

## 2. How is Ampère's circuital law related to Gauss's law?

Ampère's circuital law is often compared to Gauss's law in electrostatics because they both involve the concept of a closed loop. However, while Gauss's law relates the electric field to the charge enclosed by a closed surface, Ampère's law relates the magnetic field to the electric current passing through a closed loop.

## 3. What is the free current density in Ampère's circuital law?

The free current density, also known as the conduction current density, refers to the flow of electric charges through a material. In Ampère's circuital law, the free current density is used to calculate the total current passing through the closed loop in order to determine the magnetic field.

## 4. What is the significance of Ampère's circuital law in practical applications?

Ampère's circuital law is a crucial tool in understanding and predicting the behavior of magnetic fields in various systems. It is used in the design and analysis of electrical circuits, motors, generators, and other devices. It also has applications in fields such as electronics, telecommunications, and power generation.

## 5. Are there any limitations to Ampère's circuital law?

While Ampère's circuital law is a powerful and widely applicable principle, it does have some limitations. It is only valid for steady-state situations, meaning that the electric currents and magnetic fields must be constant over time. It also assumes that the magnetic field is generated solely by the electric current and does not take into account other factors such as magnetic materials or changing electric fields.

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