Question

Review. Consider the Bohr model of the hydrogen atom, with the electron in the ground state. The magnetic field at. the nucleus produced by the orbiting electron has a value of 12.5 T . (See Problem 4 in Chapter 30 . ) The proton can have its magnetic moment aligned in either of two directions perpendicular to the plane of the electron's orbit. The interaction of the proton's magnetic moment with the electron's magnetic field causes a difference in energy between the states with the two different orientations of the proton's magnetic moment. Find that energy difference in electron volts.

Answer 1

Step-by-step explanation:

To find the energy difference between the two different orientations of the proton's magnetic moment in electron volts, we need to use the equation:

ΔE = μB

Where:

ΔE is the energy difference

μ is the magnetic moment of the proton

B is the magnetic field at the nucleus

Given that the magnetic field at the nucleus is 12.5 T, we need to determine the value of the proton's magnetic moment (μ). The magnetic moment of a proton is given by the formula:

μ = γ * S

Where:

γ is the gyromagnetic ratio, which is a constant for protons and is equal to 2.675 × 10^8 T^-1 s^-1

S is the spin of the proton, which is equal to 1/2 for protons

Therefore, the proton's magnetic moment (μ) is:

μ = (2.675 × 10^8 T^-1 s^-1) * (1/2) = 1.3375 × 10^8 T^-1 s^-1

Now we can calculate the energy difference (ΔE):

ΔE = μB = (1.3375 × 10^8 T^-1 s^-1) * (12.5 T) = 1.671875 × 10^9 eV

Rounding to an appropriate number of significant figures, the energy difference is approximately 1.67 × 10^9 eV.

Therefore, the energy difference between the two different orientations of the proton's magnetic moment is approximately 1.67 × 10^9 electron volts (eV).