Final answer:
The expression for the magnetic field on the axis of a coil is justified by Ampere's law and the geometry of the setup. When the magnetic field sensor is moved along the axis of the coils, the magnitude of the magnetic field is expected to decrease as the distance from the coils increases.
Step-by-step explanation:
The expression for the magnetic field on the axis of a coil, Bₓ = ∐₀I / 2 Na² / (a² + x²)³/², is justified by Ampere's law and the geometry of the setup. Ampere's law states that the magnetic field created by a closed loop is proportional to the current enclosed by the loop and inversely proportional to the distance from the loop. The equation for Bₓ takes into account the parameters of the coil, such as the number of turns (N), current (I), and radius (a), as well as the distance from the axis (x).
When the magnetic field sensor is moved along the axis of the coils, the magnitude of Bₓ is expected to decrease as the distance from the coils increases. This is because the inverse square relationship between the magnetic field and the distance from the loop causes the field to weaken as the sensor moves farther away from the coils. The term Bₓ = 4∐₀ / a5√5 N + 0 (y) does not provide information about the magnitude of Bₓ, but rather describes the direction of the magnetic field in terms of its components along the x and y axes.