Current in a Wire: Density of Free Electrons

In summary, the homework problem involves finding the density of free electrons in a metal wire given its diameter, current, and drift velocity. The equation I = nev_{d}A is used, where A is calculated using the wire's diameter. The final answer, expressed in m^-3 to two significant figures, is 6.94 x 10^28.
  • #1
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[SOLVED] Current in a Wire

Homework Statement


A metallic wire has a diameter of 4.12 mm. When the current in the wire is 8 A, the drift velocity is 5.4*10^-5 m/s.

What is the density of free electrons in the metal?

I = 8
n = density??
A = (2.06)^2 * 3.14
v = 5.4 x 10^-5
q = 1.6x10^-19

Express your answer numerically in m^-3 to two significant figures.

Homework Equations



I = ne[tex]v_{d}[/tex]A

A = [tex]\pi r^2[/tex]

The Attempt at a Solution



A = [tex]\pi r^2[/tex] = [tex]\pi (2.06x10^-3)^2[/tex] = 1.3 x 10^-5

So manipulating the equation, I get n = I / evA

n = [tex]\frac{8}{(1.6x10^-19)(5.4x10^5)(1.3 x 10^-5)}[/tex] = 6.94 x 10^28

and since I need it in millimeters, I multiply it by 1000 (since I did the calculations in SI) and get 6.94 x 10 ^31, but it says it's wrong. Any ideas where I went wrong?
 
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  • #2
The question you've posted says express the answer in m-3. Does the answer need to be in meters cubed or millimeters cubed?
 
  • #3
it's to the power of -3 so m^-3 is millimetres
 
  • #4
it is not 10^-3 m it is 1/(m^3) where m^3 is the unit of volume (cubic meter)
 
  • #5
so the answer would be 1 / 6.94 x 10^28 ?
 
  • #6
The variable n is a number density or the number of electrons per unit volume so it is already expressed in terms of m-3
 
  • #7
Thank You.
 

Related to Current in a Wire: Density of Free Electrons

What is current and how is it related to the density of free electrons in a wire?

Current is the flow of electric charge through a material. In a wire, this flow is made possible by the movement of free electrons. The density of free electrons in a wire determines how much current can flow through it.

How does the density of free electrons vary in different types of wires?

The density of free electrons can vary depending on the material of the wire. Metals, such as copper, have a high density of free electrons, making them good conductors of electricity. Insulators, such as rubber, have a low density of free electrons, making it difficult for current to flow through them.

Can the density of free electrons be changed in a wire?

The density of free electrons in a wire is intrinsic to the material and cannot be changed. However, the number of free electrons that are able to move and contribute to current flow can be altered by applying an external electric field or changing the temperature of the wire.

How does the density of free electrons affect the resistance of a wire?

The density of free electrons is directly related to the resistance of a wire. A higher density of free electrons means that there are more pathways for current to flow, resulting in lower resistance. A lower density of free electrons means that there are fewer pathways for current to flow, resulting in higher resistance.

Is the density of free electrons the only factor that affects current flow in a wire?

No, the density of free electrons is just one of many factors that can affect current flow in a wire. Other factors include the material of the wire, its length and cross-sectional area, and the presence of any impurities or defects. Additionally, external factors such as temperature and applied voltage can also impact current flow in a wire.

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