Final answer:
To calculate the magnetic force on a segment of a current-carrying wire within a magnetic field, one must use the formula F = IℓB sinθ and perform an integration of the magnetic field over the length of the wire while considering the current carried by the wire.
Step-by-step explanation:
The question is asking to calculate the magnetic force on a segment of a current-carrying wire that is placed in a magnetic field. The wire carries a current I and the magnetic field B is given by B = 2.01 + 5.0x²J, with x in meters and B in millitesla. To find the force, we can use the formula F = IℓB sinθ, where θ is the angle between the direction of the current and the magnetic field.
In this case, since the wire lies along the x-axis and the direction of the magnetic field is perpendicular to the wire, the sinθ is equal to 1 (as θ=90°). Therefore, the force can be calculated by integrating the magnetic field over the length of the wire from x = 2.0 m to x = 4.0 m. The integral of the magnetic field with respect to x within those bounds, multiplied by the current I, will give the total force on the wire segment. Since the wire is carrying a current of 2.5 A, we can plug this into the calculation.
To solve the mathematical problem completely, one would perform the following integral:
∫2.0∫4.0 (2.01 + 5.0x²) dx, and then multiply the result by the current I to find the total force F.