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| ### Statement |
| ### Statement |
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| $11.2.19.$ [Insert the problem statement] |
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| ### Solution |
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| ### Condition |
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| $11.2.19.$ |
| $11.2.19.$ |
| When charges are accelerated, eddy electric fields are generated, the intensity of which, if radiation is neglected, is proportional to the acceleration. |
| 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. |
| 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 $10^{23} cm^{−3}$. | | 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 $10^{23} cm^{−3}$. | | 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 | | ### Solution |
| a) | | 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. | | 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: | | Let's assume that an alternating current $I$ flows through the solenoid, and at time $t$, its value is: |
| $$I = nS\upsilon e$$ | | $$I = nS\upsilon e$$ |
| Let's differentiate the expression with respect to time: | | Let's differentiate the expression with respect to time: |
| $$\frac{d I}{dt} = nSae$$ | | $$\frac{d I}{dt} = nSae$$ |
| Let's assume that the field inside the solenoid is uniform and equal to: | | Let's assume that the field inside the solenoid is uniform and equal to: |
| $$B = \mu_0 I N$$ | | $$B = \mu_0 I N$$ |
| Differentiate with respect to time: | | Differentiate with respect to time: |
| $$\frac{dB}{dt} = \mu_0 nSaeN$$ | | $$\frac{dB}{dt} = \mu_0 nSaeN$$ |
| Write Faraday's law of electromagnetic induction for a current loop: | | Write Faraday's law of electromagnetic induction for a current loop: |
| $$E 2 \pi r = \mu_0 nSaeN \pi r^2$$ | | $$E 2 \pi r = \mu_0 nSaeN \pi r^2$$ |
| An electric field is a force acting on a charge, or, based on the condition: | | An electric field is a force acting on a charge, or, based on the condition: |
| $$E = \frac{m_e a}{e}$$ | | $$E = \frac{m_e a}{e}$$ |
| Substitute, convert, and obtain the desired ratio: | | Substitute, convert, and obtain the desired ratio: |
| $$\frac{m_e}{m_{free}}=\frac{\mu_0 nS e^2Nr}{2} \approx 1.8 \cdot 10^{11}$$ | | $$\frac{m_e}{m_{free}}=\frac{\mu_0 nS e^2Nr}{2} \approx 1.8 \cdot 10^{11}$$ |
| b) All that remains is to express the product of the solenoid parameters and calculate: | | b) All that remains is to express the product of the solenoid parameters and calculate: |
| $$NSr \approx 5.7 \cdot 10^{-16} m^2$$ | | $$NSr \approx 5.7 \cdot 10^{-16} m^2$$ |
| Apparently, Savchenko has made a mistake | | Apparently, Savchenko has made a mistake |
| #### Answer | | #### Answer |
