Question regarding the flux of a field.

In summary, the flux of a field through a surface is defined as the normal component of the field times the surface area. It is interpreted in a physical way as the net charge flowing through the surface in the unit of time.
  • #1
Qubix
82
1
I'm currently studying Classical Electrodynamics, and I see everywhere that the flux of a field through a surface is defined as the normal component of the field times the surface area. They interpret this in a physical way as the net charge flowing through the surface in the unit of time. My question is what is the net charge? Is it charge going out - charge coming in?
 
Physics news on Phys.org
  • #2
Qubix said:
They interpret this in a physical way as the net charge flowing through the surface in the unit of time.

Assuming you're reading a real textbook, you're probably either misreading or misinterpreting it. The electric field (for example) and its flux do not represent a flow of electric charge.
 
  • #3
That's true, but the physical sense of the surface integral over or the divergence of a vector field is indeed the flux through a surface.

Let's look at a fluid with density [itex]\rho(t,\vec{x})[/itex], which gives you at an instant of time, [itex]t,[/itex], the number of particles per unit volume at position [itex]\vec{x}[/itex]. Further, the fluid is characterized by a velocity field, [itex]\vec{v}(t,\vec{x})[/itex]. Now take a fixed volume, [itex]V[/itex], in space with the surface, [itex]\partial V[/itex]. Along the surface we define the normal vectors [itex]\mathrm{d} \vec{F}(\vec{x})[/itex] such that this normal is always pointing out of the volume. The length of this vector is that of the infinitesimal surface element around [itex]\vec{x}[/itex].

Now, the question is, how many particles of the fluid per unit time flow through the surface of the volume. To answer that question we think about an infinitesimal layer of fluid inside the volume along the boundary which flowing out within the infinitesimal time, [itex]\mathrm{d} t[/itex]. The volume element of this fluid around [itex]\vec{x}[/itex] on the surface obviously is given by [itex]\mathrm{d} V= \mathrm{d} t \vec{v}(t,\vec{x}) \cdot \mathrm{d} \vec{F}(\vec{x}).[/itex]. Since the density (particles per volume) of the fluid at this place is [itex]\rho(t,\vec{x})[/itex], the total number of particles flowing out of the surface is given by

[tex]N_{\text{flux}}=\int_{\partial V} \mathrm{d} \vec{F}(\vec{x}) \cdot \vec{v}(t,\vec{x}) \rho(t,\vec{x}).[/tex]

Locally thus the flow through the surface is characterized by the current density,

[tex]\vec{j}(t,\vec{x})=\rho(t,\vec{x}) \vec{v}(t,\vec{x}).[/tex]

Of course, the infinitesimal scalar product [itex]\mathrm{d} \vec{F} \cdot \vec{j}[/itex] can be as well negative as positive, which means not more than that the velocity of the volume element under consideration is directed inwards. Thus, inward flow is counted negative, and outward flow positive.

If we make the volume element very small and locate around a positive [itex]\vec{x}[/itex], we can get the local form of the total flux with help of Gauss's theorem,

[tex]N_{\text{flux}}=\int_{\partial V} \mathrm{d} \vec{F} \cdot \vec{j} = \int_{V} \mathrm{d} ^3 \vec{x} \vec{\nabla} \cdot \vec{j}.[/tex]

To get the local form, just make the volume smaller and smaller and divide by this volume. If the volume becomes smaller than the typical length scale at which [itex]\vec{\nabla} \cdot \vec{j}[/itex] changes considerably, you can take this quantity out of the integral, and dividing out the volume gives

[tex]\frac{\mathrm{d} N_{\text{flux}}}{\mathrm{d} V}=\vec{\nabla} \cdot \vec{j}.[/tex]

Also very often one needs to formulate a conservation law in local form. For our fluid example one very often has the situation that the total particle number is conserved (as long as there's no chemical reaction of any kind). Then the change of the total number of particles inside the volume per unit time, i.e.,

[tex]\frac{\mathrm{d} N_V}{\mathrm{d} t}=\int_{V} \mathrm{d}^2 \vec{x} \partial_t \rho(t,\vec{x})[/tex]

must be given by the number of particles flowing through the surface. Since we count particles moving inward negative, we have to set this equal to the total negative flux:

[tex]\frac{\mathrm{d} N_V}{\mathrm{d} t}=-N_{\text{flux}}=-\int_{\partial V} \mathrm{d} \vec{F} \cdot \vec{j}=-\int_V \mathrm{d}^3 \vec{x} \vec{\nabla} \cdot \vec{j}.[/tex]

Using an infinitesimal volume, we get in the same way as above the local conservation law in form of the continuity equation,

[tex]\partial_t \rho(t,\vec{x})=-\vec{\nabla} \cdot \vec{j}(t,\vec{x}).[/tex]

Of course, this is only the most intuitive example for the use of vector calculus to fluxes. In electrodynamics this physical situation is applied not to the number of particles but to the electric-charge density and electric current density, which fulfill the continuity equation, because electric charge is a conserved quantity.

But it also appears in more abstract form, where there is no direct interpretation in the sense of flux. Then such laws may describe the sources of fields. E.g., Gauss's Law states that the surface integral over the electric field gives the total charge contained inside the volume (in Heaviside-Lorentz units; if you use the SI there appears a factor [itex]1/\epsilon_0[/itex] or if you use Gauss units a factor [itex]4 \pi[/itex]):

[tex]Q_V=\int_{\partial V} \mathrm{d} \vec{F} \cdot \vec{E},[/tex]

which reads in local form

[tex]\rho=\vec{\nabla} \cdot \vec{E},[/tex]

which is one of the basic laws of electrodynamics (i.e., one of the Maxwell equations).

For a magnetic field, there's also such a law. Since up to now nobody has found evidence for magnetic charges, one assumes there is none in nature (also this would have the profound consequence of charge quantization in quantum electrodynamics, which cannot be derived from first principles but must be assumed as an independent principle), i.e., the analogue of Gauss's Law for the magnetic field reads

[tex]\int_V \mathrm{d}^3 \vec{x} \vec{\nabla} \cdot \vec{B}=0 \quad \text{or}\quad \vec{\nabla} \cdot \vec{B}=0.[/tex]
 
  • #4
vanhees, thank you for your detailed answer, it is really clear now :)
 
  • #5


The concept of flux in a field is a fundamental concept in classical electrodynamics and is essential in understanding the behavior of electromagnetic fields. The flux of a field through a surface is defined as the amount of field passing through that surface. In other words, it is the measure of the flow of the field through a given area.

In the case of electromagnetic fields, the flux is calculated by taking the normal component of the field (perpendicular to the surface) and multiplying it by the surface area. This interpretation can be thought of as the net charge flowing through the surface in a unit of time. However, it is important to note that this interpretation is not entirely accurate.

The concept of net charge is often used in the context of electric circuits, where there is a flow of charge from one point to another. In this case, the net charge is the difference between the amount of charge entering a point and the amount of charge leaving that point. However, in the context of flux, we are not considering a closed system where charges are flowing in and out. Instead, we are looking at the flow of the field itself.

Therefore, the net charge in the context of flux is not the difference between charge going in and charge coming out. It is simply a measure of the amount of field passing through a given surface. This concept is important to understand in order to properly interpret and apply the equations and principles of classical electrodynamics.
 

1. What is flux?

Flux is a measure of the flow of a physical quantity through a given surface. It is represented by the symbol Φ and is calculated by integrating the dot product of the field and the surface over the surface.

2. What is the difference between electric and magnetic flux?

Electric flux is the measure of the flow of electric field through a surface, while magnetic flux is the measure of the flow of magnetic field through a surface. They are both represented by the same symbol Φ, but have different units and physical interpretations.

3. How is flux related to Gauss's Law?

Gauss's Law states that the electric flux through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of free space. This relationship allows us to calculate the electric field of a charge distribution by knowing the electric flux through a surface enclosing the charge.

4. What factors affect the flux of a field?

The flux of a field is affected by the strength and direction of the field, the orientation and size of the surface, and the relative position of the surface with respect to the field. It also depends on the type of field, as electric and magnetic fields have different properties that affect their flux.

5. How is flux used in real-world applications?

Flux is used in various real-world applications, such as in the design of electrical circuits, motors, and generators. It is also used in industries like geology and meteorology to measure the flow of fluids, such as water and air. Flux is also utilized in medical imaging techniques like MRI to map out the flow of fluids in the body.

Similar threads

Replies
25
Views
1K
  • Electromagnetism
Replies
30
Views
2K
Replies
1
Views
1K
  • Electromagnetism
Replies
3
Views
1K
Replies
35
Views
2K
Replies
35
Views
1K
  • Electromagnetism
Replies
4
Views
851
Replies
2
Views
574
  • Electromagnetism
Replies
16
Views
988
Back
Top