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