A

Step 1: Given the potential energy function:
U = K log(r)
where K is a constant and r is the separation between the electron and proton.

Step 2: According to the Bohr model, the total energy E of an electron in a hydrogen atom is the sum of its kinetic energy (T) and potential energy (U):
E = T + U.
Step 3: The kinetic energy of an electron in Bohr's orbit can be expressed as:
T = \frac{1}{2} mv^2 = \frac{K_e e^2}{2r} for a traditional model, where K_e is the Coulomb's constant, and e is the charge of the electron.
Step 4: For the modified potential energy, we can set up the effective potential energy for the electron as:
U = K log(r).
Step 5: We need to also account for the centrifugal potential due to angular momentum (L):
U_{eff} = K log(r) + \frac{L^2}{2mr^2}.
Using the quantization of angular momentum, we know that L = n \hbar, where n is the principal quantum number.
Step 6: Therefore, substituting L:
U_{eff} = K log(r) + \frac{(n\hbar)^2}{2mr^2}.
Step 7: To find the radius of the n-th Bohr's orbit, we can minimize the effective potential:
\frac{dU_{eff}}{dr} = 0.
Taking the derivative and setting it to zero gives us the equilibrium condition.
Step 8: Solving we get:
\frac{K}{r} - \frac{(n \hbar)^2}{mr^3} = 0
\Rightarrow K r^2 = (n\hbar)^2 / m
\Rightarrow r_n = \sqrt{\frac{(n\hbar)^2}{K m}}.
Step 9: To find energy levels, we plug radius r_n back into our expression for energy:
E_n = T + U = \frac{K}{2 r_n} + K log(r_n).
Step 10: Substituting the expression for r_n we find energy levels as a function of n.
Therefore, the radius of the n-th Bohr's orbit is: r_n = \sqrt{\frac{(n\hbar)^2}{K m}} and energies can be derived from substituting back.
Thus, the respective radius is: \sqrt{\frac{(n\hbar)^2}{K m}} and energies follow accordingly based on further simplifications.