Drift Velocity, TCR, and Atomic/ Weight Percentages

[EugP]

Homework Statement

I'm taking a metrials class, and there are 2 questions I'm not sure how to do. I would see the professor first, but he refuses helping students until the assignments is submitted.

1) A circular tungsten (W) wire that is 0.1m long and 1mm in diameter has 9V dropped across it. Using a spreadsheet program such as Excel, generate a plot of the drift velocity as a function of temperature from 23C - 300C. Assume that density is temperature independent. Assume that W has 4 valence electrons per atom. Include in your plot, a write-up of the equations used.

2) What are the maximum atomic and weight percentages of Cu that can be added to Au without exceeding a resistivity that is twice that of pure gold? What are the maximum atomic and weight percentages of Au that can be added to pure Cu without exceeding twice the resistivity of pure copper?

Homework Equations

He never actually taught us any equations, just explained the theory, but this is what I found in the book:
1)
Drift Velocity
$$v _{dx} = \frac{e\tau}{m_e} E_x$$

Drift Mobility
$$\mu _d = \frac{\sigma}{en}$$

Conductivity
$$\sigma = en\mu _d$$

Number of atoms per unit volume
$$n = \frac{dN_A}{M_{at}}$$

The Attempt at a Solution

1) I don't know where to use the 9V in any of the equations. But just by using what I have, and some other equations I found online, I got:

$$n = \frac{dN_A}{M_{at}} = \frac{(19.25)(6.02 \times 10^{23})}{183.84} = 63.04 \times 10^{21}$$

$$\sigma = \frac{1}{\rho} = \frac{1}{AT} = \frac{1}{\pi (0.1cm)^2 T} = \frac{4}{\pi T}$$

$$\mu _d = \frac{\sigma}{en} = \frac{4}{\pi T (1.6\times 10^{-19})(63.04\times 10^{21})} = \frac{126.23\times 10^{-6}}{T}$$

At this point, I have no idea what to do, or whether what I've done is even correct so far. Please help.

2) For this problem, the only information I have is that the percentages must add up to 100, which is obvious. Any help here would be great.

 

[nate808]

Dear student,

Thank you for reaching out for help with your materials class assignment. I understand that you are having difficulty with two questions and are unable to get assistance from your professor until the assignment is submitted. I will do my best to provide you with guidance and equations that may be helpful in solving these problems.

1) To start, we can use the equation for drift velocity (v_{dx} = \frac{e\tau}{m_e} E_x) to calculate the drift velocity as a function of temperature. In this case, the electric field (E_x) is equal to the voltage (9V) divided by the length of the wire (0.1m). This gives us an electric field of 90V/m. Now, we can use the equation for drift mobility (\mu _d = \frac{\sigma}{en}) to calculate the conductivity (\sigma) at each temperature. Since we are assuming that density is temperature independent, we can use the equation for number of atoms per unit volume (n = \frac{dN_A}{M_{at}}) to calculate the number of atoms per unit volume at each temperature. Finally, we can use the equation for conductivity (\sigma = en\mu _d) to calculate the conductivity at each temperature.

Once we have the conductivity at each temperature, we can use the equation for resistivity (\rho = \frac{1}{\sigma}) to calculate the resistivity at each temperature. This will give us a plot of resistivity as a function of temperature.

2) For this problem, we can use the equation for resistivity (\rho = \frac{1}{\sigma}) to calculate the resistivity of pure gold and pure copper. Then, we can use the equation for resistivity (\rho = \frac{1}{\sigma}) again to calculate the resistivity of a mixture of gold and copper. We can then set this equal to twice the resistivity of pure gold and solve for the maximum atomic and weight percentages of Cu that can be added without exceeding this resistivity. We can repeat this process for the maximum atomic and weight percentages of Au that can be added to pure Cu.

I hope this helps guide you in solving these problems. If you need further assistance, please do not hesitate to reach out for help from your professor or a tutor. Good luck with your assignment!

 

[Blueshift5]

I can understand your frustration with your professor's refusal to help until the assignment is submitted. However, I will do my best to assist you with your questions.

For the first question, you are on the right track with using the equations for drift velocity, drift mobility, conductivity, and number of atoms per unit volume. To incorporate the 9V, you can use Ohm's Law, which states that the current (I) flowing through a material is equal to the voltage (V) divided by the resistance (R). In this case, the resistance can be calculated using the resistivity (ρ) and the length and cross-sectional area of the wire. So, the equation would be I = V/R = V(πr^2)/ρl, where r is the radius of the wire and l is the length. From there, you can use the equation for drift velocity to calculate the velocity as a function of temperature. Your plot should show an increase in drift velocity with increasing temperature, as the increased thermal energy allows for faster movement of the electrons.

For the second question, you will need to use the equation for resistivity, which is ρ = (1/σ) = (1/neμd), where n is the number of charge carriers per unit volume, e is the charge of an electron, and μd is the drift mobility. You can set this equation equal to twice the resistivity of pure gold or copper and solve for the maximum atomic and weight percentages of Cu or Au that can be added without exceeding this resistivity. This will require some algebraic manipulation and substitution of values for n and μd, which can be found in the given information.

I hope this helps guide you in the right direction for solving these problems. Remember, as a scientist, it is important to understand the theory and equations behind a problem, but also to think critically and creatively in order to solve it. Good luck with your assignments!