New solution
en/11.2.19.md
+41 −0
| @@ -0,0 +1,41 @@ | |||
| + | ### Statement | ||
| + | |||
| + | $11.2.19.$ [Insert the problem statement] | ||
| + | |||
| + | ### Solution | ||
| + | |||
| + | ### Condition | ||
| + | $11.2.19.$ | ||
| + | When charges are accelerated, eddy electric fields are generated, the intensity of which, if radiation is neglected, is proportional to the acceleration. | ||
| + | Therefore, a charge moving with acceleration $a$ is acted upon by these electric fields by a force $F = m_{e}a$. The proportionality constant $m_{e}$ can be called the electromagnetic mass of the charge. | ||
| + | a. How many times is the electromagnetic mass of a conduction electron in a long solenoid with a radius of $0.1 m$ and a number of turns per unit length of $10^3 m^{-1}$ greater than the mass of a free electron? The cross-section of the solenoid wire is $1 mm^2$, and the number of conduction electrons per unit volume of the solenoid material is $10^{23} cm^{−3}$. | ||
| + | b. What parameters should the solenoid have to ensure that the electromagnetic mass of an electron in the solenoid is equal to the mass of a free electron? The number of conduction electrons per unit volume of the solenoid material is $10^{23} cm^{−3}$. | ||
| + | ### Solution | ||
| + | a) | ||
| + | First, let's define the quantities from the problem statement: $n$ is the electron concentration, $S$ is the cross-sectional area, $r$ is the radius, and $N$ is the number of turns per unit length. | ||
| + | Let's assume that an alternating current $I$ flows through the solenoid, and at time $t$, its value is: | ||
| + | $$I = nS\upsilon e$$ | ||
| + | Let's differentiate the expression with respect to time: | ||
| + | $$\frac{d I}{dt} = nSae$$ | ||
| + | Let's assume that the field inside the solenoid is uniform and equal to: | ||
| + | $$B = \mu_0 I N$$ | ||
| + | Differentiate with respect to time: | ||
| + | $$\frac{dB}{dt} = \mu_0 nSaeN$$ | ||
| + | Write Faraday's law of electromagnetic induction for a current loop: | ||
| + | $$E 2 \pi r = \mu_0 nSaeN \pi r^2$$ | ||
| + | An electric field is a force acting on a charge, or, based on the condition: | ||
| + | $$E = \frac{m_e a}{e}$$ | ||
| + | Substitute, convert, and obtain the desired ratio: | ||
| + | $$\frac{m_e}{m_{free}}=\frac{\mu_0 nS e^2Nr}{2} \approx 1.8 \cdot 10^{11}$$ | ||
| + | b) All that remains is to express the product of the solenoid parameters and calculate: | ||
| + | $$NSr \approx 5.7 \cdot 10^{-16} m^2$$ | ||
| + | Apparently, Savchenko has made a mistake | ||
| + | #### Answer | ||
| + | a) | ||
| + | $$\frac{m_e}{m_{free}}=\frac{\mu_0 nS e^2Nr}{2} \approx 1.8 \cdot 10^{11}$$ | ||
| + | |||
| + | б)$$NSr \approx 5.7 \cdot 10^{-16} м^2$$ | ||
| + | |||
| + | #### Answer | ||
| + | |||
| + | [Insert a concise answer or boxed result] | ||
| @@ -0,0 +1,41 @@ | |||
| ### Statement | |||
| $11.2.19.$ [Insert the problem statement] | |||
| ### Solution | |||
| ### Condition | |||
| $11.2.19.$ | |||
| When charges are accelerated, eddy electric fields are generated, the intensity of which, if radiation is neglected, is proportional to the acceleration. | |||
| Therefore, a charge moving with acceleration $a$ is acted upon by these electric fields by a force $F = m_{e}a$. The proportionality constant $m_{e}$ can be called the electromagnetic mass of the charge. | |||
| a. How many times is the electromagnetic mass of a conduction electron in a long solenoid with a radius of $0.1 m$ and a number of turns per unit length of $10^3 m^{-1}$ greater than the mass of a free electron? The cross-section of the solenoid wire is $1 mm^2$, and the number of conduction electrons per unit volume of the solenoid material is $10^{23} cm^{−3}$. | |||
| b. What parameters should the solenoid have to ensure that the electromagnetic mass of an electron in the solenoid is equal to the mass of a free electron? The number of conduction electrons per unit volume of the solenoid material is $10^{23} cm^{−3}$. | |||
| ### Solution | |||
| a) | |||
| First, let's define the quantities from the problem statement: $n$ is the electron concentration, $S$ is the cross-sectional area, $r$ is the radius, and $N$ is the number of turns per unit length. | |||
| Let's assume that an alternating current $I$ flows through the solenoid, and at time $t$, its value is: | |||
| $$I = nS\upsilon e$$ | |||
| Let's differentiate the expression with respect to time: | |||
| $$\frac{d I}{dt} = nSae$$ | |||
| Let's assume that the field inside the solenoid is uniform and equal to: | |||
| $$B = \mu_0 I N$$ | |||
| Differentiate with respect to time: | |||
| $$\frac{dB}{dt} = \mu_0 nSaeN$$ | |||
| Write Faraday's law of electromagnetic induction for a current loop: | |||
| $$E 2 \pi r = \mu_0 nSaeN \pi r^2$$ | |||
| An electric field is a force acting on a charge, or, based on the condition: | |||
| $$E = \frac{m_e a}{e}$$ | |||
| Substitute, convert, and obtain the desired ratio: | |||
| $$\frac{m_e}{m_{free}}=\frac{\mu_0 nS e^2Nr}{2} \approx 1.8 \cdot 10^{11}$$ | |||
| b) All that remains is to express the product of the solenoid parameters and calculate: | |||
| $$NSr \approx 5.7 \cdot 10^{-16} m^2$$ | |||
| Apparently, Savchenko has made a mistake | |||
| #### Answer | |||
| a) | |||
| $$\frac{m_e}{m_{free}}=\frac{\mu_0 nS e^2Nr}{2} \approx 1.8 \cdot 10^{11}$$ | |||
| б)$$NSr \approx 5.7 \cdot 10^{-16} м^2$$ | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||