Final answer:
To calculate the radius of the n-th orbit and the minimum possible speed of an electron moving perpendicular to a magnetic field, we can use Bohr's quantization rule adapted for an electron in a magnetic field and solve using the known values and fundamental constants.
Step-by-step explanation:
The question involves the motion of an electron in a magnetic field and invokes Bohr's quantization rule to calculate the radius of the nth orbit and the minimum possible speed of the electron. Bohr's quantization rule, applied to electron orbits around an atomic nucleus, states that the angular momentum of an electron is quantized and is given by ℓ = nh/2π, where ℓ is the angular momentum, h is Planck's constant, n is the principal quantum number, and π is pi. However, we can adapt this principle by considering the angular momentum of an electron moving in a circular path due to the magnetic force acting as the centripetal force.
The magnetic force on a moving charge is F = qvB sin(θ), where q is the charge of the electron, v is its velocity, and B is the magnetic field strength. When the electron moves perpendicular to the magnetic field, θ is 90 degrees, and sin(θ) is 1, leading to the force equation simplifying to F = qvB.
As the electron moves in a circle, the magnetic force provides the necessary centripetal force Fₒ = mv²/r, with ℓ = μvr and μ = q/2πμe, where m is the mass of the electron, μ is the permeability of free space, and μe is the electron magnetic moment. Equating the two expressions for the force (qvB = mv²/r) and solving for r gives us the radius of the circular path. To get the radius of the nth orbit and the minimum possible speed, we apply the quantization rule ℓ = nh/2π and solve for r and v. The resulting expressions for the radius r and speed v are dependent on the values of n, B, h, and fundamental constants like the charge and mass of an electron.