Help deriving expression of current density

In summary, the expression for current density (j) is derived from quantum mechanics and takes into account the charge and velocity of electrons as well as the fermi-dirac distribution function. The condition k_{x} > 0 means we are only considering electrons with positive momentum in the x-direction, and the integration is limited to the region x_{0}<x<x_{1}, which represents an energy barrier. The height of this barrier can affect the value of the current density.
  • #1
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Hey I got an assignment where I have to get an expression of the current density to be:

j = -e[itex]\int[/itex] [itex]\frac{dk}{4\pi ^{3}}[/itex] v[itex]_{x}[/itex]f(k)

It says k[itex]_{x}[/itex] >0 underneath the intergrale. The current is flowing to the right (velocity of v[itex]_{x}[/itex] > 0)

Where f(k) is the fermi-dirac distribution function and the interval where I have to find the Current density is in the region x[itex]_{0}[/itex]<x<x[itex]_{1}[/itex] (energy barrier). The height of the barrier is W.

Plz help me?!
 
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  • #2


Hello, it looks like you are working on a physics or engineering assignment. The expression for current density that you have been given is correct and is derived from the basic principles of quantum mechanics. Let me break it down for you.

First, the expression for current density (j) is given by:

j = -e\int \frac{dk}{4\pi ^{3}} v_{x}f(k)

Where e is the charge of an electron, v_{x} is the velocity of the electron in the x-direction, and f(k) is the fermi-dirac distribution function. This function takes into account the occupation of energy levels by electrons in a system, and is dependent on the temperature and energy of the system.

The integral sign indicates that we are summing over all possible values of k, which is the wave vector of the electron. This is because the current density is a measure of the flow of electrons, which is dependent on their momentum.

Now, the condition k_{x} > 0 means that we are only considering electrons with positive momentum in the x-direction. This makes sense because you mentioned that the current is flowing to the right, so the electrons must have a positive velocity in the x-direction.

The integration is also limited to the region x_{0}<x<x_{1}, which is the energy barrier you mentioned. This means that we are only considering electrons within this region, which is important in determining the current density.

Finally, the height of the energy barrier (W) will affect the value of the current density. This is because the barrier will act as a potential barrier for the electrons, and they will need to have enough energy to overcome it and continue flowing.

I hope this helps to clarify the expression for current density and how it is related to the given parameters. If you have any further questions, please let me know. Keep up the good work!
 

1. What is current density?

Current density is a measure of the flow of electric charge per unit area. It is typically represented by the symbol J and is given in units of amperes per square meter (A/m²).

2. Why is it important to derive the expression for current density?

Deriving the expression for current density helps us understand the behavior and properties of electric currents in different materials. It also allows us to calculate and predict the flow of electric charge in a given system, which is crucial in many applications such as electrical engineering and physics research.

3. How is the expression for current density derived?

The expression for current density is derived using Ohm's law, which states that the current flowing through a material is directly proportional to the electric field and inversely proportional to the material's resistance. By rearranging this equation, we can obtain the expression for current density, which is J = σE, where σ is the material's conductivity and E is the electric field strength.

4. What factors affect the expression for current density?

The expression for current density is affected by the material's conductivity, the magnitude and direction of the electric field, and the material's dimensions. Temperature can also play a role in altering the conductivity of a material, thus affecting the current density.

5. Can the expression for current density be applied to all materials?

No, the expression for current density is only applicable to materials that obey Ohm's law, also known as Ohmic materials. These materials have a linear relationship between the electric field and the resulting current, and their conductivity remains constant regardless of the applied electric field.

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