**Difference in Energies of Protons in a Magnetic Field**

In order to calculate the difference in energies of protons oriented with and against a magnetic field, we need to understand the concept of magnetic field and proton behavior. Let's break down the calculation and explanation into the following sections:

**1. Magnetic Field**
A magnetic field is a region in space where a magnetic force can be observed. It is represented by magnetic field lines that indicate the direction and strength of the field. The strength of a magnetic field is measured in units of Tesla (T).

**2. Proton Behavior in a Magnetic Field**
When a proton is placed in a magnetic field, it experiences a force due to its positive charge and the magnetic field. This force causes the proton to move in a circular path perpendicular to the magnetic field lines. The radius of this circular path is determined by the speed of the proton and the strength of the magnetic field.

**3. Magnetic Moment of a Proton**
The magnetic moment of a proton (μ) is a property that describes its interaction with a magnetic field. The magnetic moment is given by the equation:

μ = γ * spin
where γ is the gyromagnetic ratio (a constant) and spin is the spin angular momentum of the proton.

**4. Energy Difference Calculation**
The energy difference between a proton oriented with and against a magnetic field can be calculated using the equation:

ΔE = -μ * B
where ΔE is the energy difference, μ is the magnetic moment of the proton, and B is the magnetic field strength.

**5. Calculation Example**
Let's consider a proton with a magnetic moment of 1.41 x 10^-26 J/T oriented with a magnetic field strength of 1.50 T.

ΔE = -(1.41 x 10^-26 J/T) * (1.50 T)
ΔE = -2.115 x 10^-26 J

Therefore, the difference in energies of the proton oriented with and against the magnetic field is -2.115 x 10^-26 J.

**In summary**, the difference in energies of protons oriented with and against a magnetic field can be calculated using the formula ΔE = -μ * B, where ΔE is the energy difference, μ is the magnetic moment of the proton, and B is the magnetic field strength. In the example provided, the energy difference was calculated to be -2.115 x 10^-26 J when a proton with a magnetic moment of 1.41 x 10^-26 J/T was oriented with a magnetic field strength of 1.50 T.