Edits to “Problem”, “Solution”, “Answer”
en/8.2.1.md
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| @@ -1,11 +1,47 @@ | |||
| − | ### | ||
| + | ### Problem | ||
| − | $8.2.1.$ [Insert the problem statement] | ||
| + | $8.2.1^*.$ $\textbf{a.}$ Determine the specific conductivity of a metal if the number of conduction electrons per unit volume of the metal is $n_e$, and the time between successive collisions of an electron with the crystal lattice ions is $\tau$. Immediately after a collision, any direction of the electron's velocity is equally probable.\ | ||
| + | $\textbf{b.}$ Estimate the average time between successive collisions of a conduction electron with the crystal lattice ions of copper. | ||
| ### Solution | |||
| − | Studio Cyborg Squad presents | ||
| + | $\textbf{a.}$ In the classical electron theory of conductivity, the motion of an electron between collisions with lattice ions is considered uniformly accelerated under the action of an external electric field $\vec{E}$. The acceleration of the electron is: | ||
| + | $$ \vec{a} = -\frac{e\vec{E}}{m_e} $$ | ||
| + | where $e$ is the elementary charge, and $m_e$ is the mass of the electron. | ||
| + | Since immediately after a collision any direction of velocity is equally probable, the average velocity of the random motion of electrons immediately after a collision is zero: $\langle \vec{v}_0 \rangle = 0$. | ||
| + | |||
| + | At any given time $t$ after the last collision, the average velocity of directed motion (drift velocity) $\upsilon_d$ is determined by averaging over all free flight times. With strict consideration of the flight time distribution (exponential law), the average value of the acquired velocity is: | ||
| + | $$ \upsilon_d = a\tau = \frac{eE\tau}{m_e} $$ | ||
| + | |||
| + | The electric current density $j$ is related to the drift velocity and the electron concentration $n_e$ by the relation: | ||
| + | $$ j = en_e\upsilon_d $$ | ||
| + | |||
| + | Substituting the expression for the drift velocity, we obtain: | ||
| + | $$ j = \frac{e^2 n_e \tau}{m_e} E $$ | ||
| + | |||
| + | According to Ohm's law in differential form, the current density is proportional to the electric field strength: | ||
| + | $$ j = \lambda E $$ | ||
| + | |||
| + | Comparing the two expressions, we find the required specific conductivity $\lambda$: | ||
| + | $$ \lambda = \frac{e^2 n_e \tau}{m_e} $$ | ||
| + | |||
| + | $\textbf{b.}$ To estimate the average time between collisions $\tau$ for copper, we express it from the obtained formula: | ||
| + | $$ \tau = \frac{\lambda m_e}{e^2 n_e} $$ | ||
| + | |||
| + | From reference data, the specific conductivity of copper is known: $\lambda \approx 5.9 \cdot 10^7 \, \text{S/m}$. | ||
| + | Let us estimate the concentration of conduction electrons $n_e$, assuming that there is one free electron per copper atom in the crystal lattice: | ||
| + | $$ n_e = \frac{\rho_{Cu} N_A}{M} $$ | ||
| + | where $\rho_{Cu} \approx 8900 \, \text{kg/m}^3$ is the density of copper, $M \approx 0.0635 \, \text{kg/mol}$ is the molar mass, and $N_A \approx 6.02 \cdot 10^{23} \, \text{mol}^{-1}$ is Avogadro's number. | ||
| + | $$ n_e = \frac{8900 \cdot 6.02 \cdot 10^{23}}{0.0635} \approx 8.4 \cdot 10^{28} \, \text{m}^{-3} $$ | ||
| + | |||
| + | Substituting the known constants ($e \approx 1.6 \cdot 10^{-19} \, \text{C}$, $m_e \approx 9.1 \cdot 10^{-31} \, \text{kg}$) into the formula for $\tau$: | ||
| + | $$ \tau = \frac{5.8 \cdot 10^7 \cdot 9.1 \cdot 10^{-31}}{(1.6 \cdot 10^{-19})^2 \cdot 8.4 \cdot 10^{28}} = \frac{52.78 \cdot 10^{-24}}{2.56 \cdot 10^{-38} \cdot 8.4 \cdot 10^{28}} \approx 2.45 \cdot 10^{-14} \, \text{s} $$ | ||
| + | |||
| + | $\textit{Note: A mathematical calculation based on modern reference data gives a value of $2.4 \cdot 10^{-14} \, \text{s}$. The answer provided in the textbook ($2.4 \cdot 10^{-15} \, \text{s}$) contains a typo by one order of magnitude.}$ | ||
| + | |||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | $\textbf{a.}$ $\lambda = \frac{e^2 n_e \tau}{m_e}$ | ||
| + | |||
| + | $\textbf{b.}$ $\tau \approx 2.4 \cdot 10^{-14} \, \text{s}$ | ||
| @@ -1,11 +1,47 @@ | |||
| ### |
### Problem | ||
| $8.2.1.$ [Insert the problem statement] | $8.2.1^*.$ $\textbf{a.}$ Determine the specific conductivity of a metal if the number of conduction electrons per unit volume of the metal is $n_e$, and the time between successive collisions of an electron with the crystal lattice ions is $\tau$. Immediately after a collision, any direction of the electron's velocity is equally probable.\ | ||
| $\textbf{b.}$ Estimate the average time between successive collisions of a conduction electron with the crystal lattice ions of copper. | |||
| ### Solution | ### Solution | ||
| Studio Cyborg Squad presents | $\textbf{a.}$ In the classical electron theory of conductivity, the motion of an electron between collisions with lattice ions is considered uniformly accelerated under the action of an external electric field $\vec{E}$. The acceleration of the electron is: | ||
| $$ \vec{a} = -\frac{e\vec{E}}{m_e} $$ | |||
| where $e$ is the elementary charge, and $m_e$ is the mass of the electron. | |||
| Since immediately after a collision any direction of velocity is equally probable, the average velocity of the random motion of electrons immediately after a collision is zero: $\langle \vec{v}_0 \rangle = 0$. | |||
| At any given time $t$ after the last collision, the average velocity of directed motion (drift velocity) $\upsilon_d$ is determined by averaging over all free flight times. With strict consideration of the flight time distribution (exponential law), the average value of the acquired velocity is: | |||
| $$ \upsilon_d = a\tau = \frac{eE\tau}{m_e} $$ | |||
| The electric current density $j$ is related to the drift velocity and the electron concentration $n_e$ by the relation: | |||
| $$ j = en_e\upsilon_d $$ | |||
| Substituting the expression for the drift velocity, we obtain: | |||
| $$ j = \frac{e^2 n_e \tau}{m_e} E $$ | |||
| According to Ohm's law in differential form, the current density is proportional to the electric field strength: | |||
| $$ j = \lambda E $$ | |||
| Comparing the two expressions, we find the required specific conductivity $\lambda$: | |||
| $$ \lambda = \frac{e^2 n_e \tau}{m_e} $$ | |||
| $\textbf{b.}$ To estimate the average time between collisions $\tau$ for copper, we express it from the obtained formula: | |||
| $$ \tau = \frac{\lambda m_e}{e^2 n_e} $$ | |||
| From reference data, the specific conductivity of copper is known: $\lambda \approx 5.9 \cdot 10^7 \, \text{S/m}$. | |||
| Let us estimate the concentration of conduction electrons $n_e$, assuming that there is one free electron per copper atom in the crystal lattice: | |||
| $$ n_e = \frac{\rho_{Cu} N_A}{M} $$ | |||
| where $\rho_{Cu} \approx 8900 \, \text{kg/m}^3$ is the density of copper, $M \approx 0.0635 \, \text{kg/mol}$ is the molar mass, and $N_A \approx 6.02 \cdot 10^{23} \, \text{mol}^{-1}$ is Avogadro's number. | |||
| $$ n_e = \frac{8900 \cdot 6.02 \cdot 10^{23}}{0.0635} \approx 8.4 \cdot 10^{28} \, \text{m}^{-3} $$ | |||
| Substituting the known constants ($e \approx 1.6 \cdot 10^{-19} \, \text{C}$, $m_e \approx 9.1 \cdot 10^{-31} \, \text{kg}$) into the formula for $\tau$: | |||
| $$ \tau = \frac{5.8 \cdot 10^7 \cdot 9.1 \cdot 10^{-31}}{(1.6 \cdot 10^{-19})^2 \cdot 8.4 \cdot 10^{28}} = \frac{52.78 \cdot 10^{-24}}{2.56 \cdot 10^{-38} \cdot 8.4 \cdot 10^{28}} \approx 2.45 \cdot 10^{-14} \, \text{s} $$ | |||
| $\textit{Note: A mathematical calculation based on modern reference data gives a value of $2.4 \cdot 10^{-14} \, \text{s}$. The answer provided in the textbook ($2.4 \cdot 10^{-15} \, \text{s}$) contains a typo by one order of magnitude.}$ | |||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | $\textbf{a.}$ $\lambda = \frac{e^2 n_e \tau}{m_e}$ | ||
| $\textbf{b.}$ $\tau \approx 2.4 \cdot 10^{-14} \, \text{s}$ | |||