@@ -87,12 +87,19 @@ While most materials have $R_{\rm H}>0$, interestingly some materials are found
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@@ -87,12 +87,19 @@ While most materials have $R_{\rm H}>0$, interestingly some materials are found
## Exercises
## Exercises
### Exercise 1: Motion of free electron due to electric field
### Exercise 1: Motion of an electron in a magnetic and an electric field.
We first consider an electron in free space, moving in a plane perpendicular to a magnetic field ${\bf B}$ with velocity ${\bf v}$.
1. What is the shape of the motion of the electron? Calculate the characteristic frequency and time-period $T_c$ of this motion for $B=1$ Tesla.
2. Formulate a differential equation for ${\bf v}$.
3. Now we add an electric $E$ that is perpendicular to ${\bf B}$. Sketch the motion of the electron.
4. Adjust the differential equation of found in (2) to include ${\bf E}$.
We now consider an electron in a metal.
5. Drude theory assumes a scattering time $\tau$. Include this time into the differential equation for the velocity you formulated in 4.
6. Let's assume $\tau\ggT_c$. What would the shape of the motion of the electron be in this limit? What would the shape be when $\tau\ll T_c$? Which of these two limits holds in a typical metal?
1. Consider N free electrons in copper. Explain their motion under no electric field and while applying electric field in the direction from left to right side of the conductor.
2. In the case of externally applied electric field, let the time taken between two scattering events (electron colliding with themselves and with positive ions) be $t-t_i$. What is the average velocity of electrons.
3. From the average velocity derived in (b), can you formulate a factor that explains how fast the electrons drift in a certain direction due to external electric field??
4. Does high carrier mobility alone signify that the metal has high electrical conductivity??
### Exercise 2: Temperature dependence of resistance
### Exercise 2: Temperature dependence of resistance
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@@ -102,19 +109,13 @@ While most materials have $R_{\rm H}>0$, interestingly some materials are found
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@@ -102,19 +109,13 @@ While most materials have $R_{\rm H}>0$, interestingly some materials are found
4. In the Drude model, how does the electrical resistivity change as a function of temperature. Sketch ρ(T).
4. In the Drude model, how does the electrical resistivity change as a function of temperature. Sketch ρ(T).
5. Compare your plot of ρ(T) against the plot sketched in the lecture notes and please provide your judgement on the temperature dependence of resistivity.
5. Compare your plot of ρ(T) against the plot sketched in the lecture notes and please provide your judgement on the temperature dependence of resistivity.
### Exercise 3: Motion of a free electron in a magnetic and an electric field
### Exercise 3: Classical Hall effect
1. Consider a free electron moving in a magnetic field $B$ with velocity $v$. What is the shape of the motion the electron performs?
2. What is the characteristic frequency of this motion?
3. How does the motion change while also adding an electric field? Sketch the trajectory.
### Exercise 4: Classical Hall effect
1. In the hall [conductor](https://en.wikipedia.org/wiki/Hall_effect#/media/File:Hall_Effect_Measurement_Setup_for_Electrons.png), consider a magnetic field $B_z$ applied along z and current I flows along x-direction. Having in mind the electron motion under electric and magentic fields, can you formulate a steady state condition at which the net force on the charges due to these two fields is zero along the y-direction.
1. In the hall [conductor](https://en.wikipedia.org/wiki/Hall_effect#/media/File:Hall_Effect_Measurement_Setup_for_Electrons.png), consider a magnetic field $B_z$ applied along z and current I flows along x-direction. Having in mind the electron motion under electric and magentic fields, can you formulate a steady state condition at which the net force on the charges due to these two fields is zero along the y-direction.
2. If the Hall voltage $V_H$ and current I are measured in an experiment, calculate the magnetic field. Provide a potential commercial application for the hall effect.
2. If the Hall voltage $V_H$ and current I are measured in an experiment, calculate the magnetic field. Provide a potential commercial application for the hall effect.
3. Given the hall coefficient and electrical conductivity of copper, calculate the carrier mobility.
3. Given the hall coefficient and electrical conductivity of copper, calculate the carrier mobility.
4. From the hall voltage (or hall coefficient) can you qualitatively determine the type of charge carriers. Is there a case at which the hall effect based on Drude model doesn’t make sense?
4. From the hall voltage (or hall coefficient) can you qualitatively determine the type of charge carriers. Is there a case at which the hall effect based on Drude model doesn’t make sense?
## Exercise 5: Shortcomings of Drude model
## Exercise 4: Shortcomings of Drude model
1. Calculate the Seebeck coefficient of Sodium and Copper using Drude model at room temperature.
1. Calculate the Seebeck coefficient of Sodium and Copper using Drude model at room temperature.
2. Compare it against the experimental values (Sodium = $-5*10^-6 V/K$, Copper = $-1.6*10^-6 V/K$) and discuss the vulnerabilities of Drude model. Please justify your opinions.
2. Compare it against the experimental values (Sodium = $-5*10^-6 V/K$, Copper = $-1.6*10^-6 V/K$) and discuss the vulnerabilities of Drude model. Please justify your opinions.
