Final answer:
To find the current i, we use the Lorentz force acting on the electron moving in a magnetic field B created by the solenoid, and relate it to the centripetal force of the electron's circular motion, allowing us to solve for i.
Step-by-step explanation:
The question asks us to calculate the current i in a long solenoid that carries a specific number of turns per centimeter and has an electron moving in a circular path within it. This type of problem involves knowledge of electromagnetism, specifically the magnetic field inside a solenoid and the force exerted on a moving charge in a magnetic field.
To find the current, we use the Lorentz force equation F = qvB for an electron moving perpendicular to the magnetic field lines of the solenoid, where q is the charge of the electron, v is its velocity, and B is the magnetic field. We can relate the magnetic field B to the current i using the formula for the magnetic field inside a long solenoid, B = μ0ni, where μ0 is the permeability of free space and n is the number of turns per unit length of the solenoid.
The centripetal force required for the electron's circular motion is provided solely by the magnetic Lorentz force, thus we set F = mv2/r, where m is the mass of the electron, v is its velocity, and r is the radius of the circle. By equating this to qvB, and substituting B = μ0ni, we can solve for the current i.