Calculate the magnetic force on an arc 50 millimeters long carrying a current of 400 amperes in a direction perpendicular to a uniform \(\mathrm{B}\) of \(5 \times 10^{-2}\) tesla. High-current circuit hreakers often comprise coils that generate a magnetic field to blow out the arc that forms when the contacts open.

A magnetic field is a vector field that surrounds magnets and electric currents, and affects other magnets and currents within its vicinity. It is represented by the symbol \( B \) and measured in teslas (T). One can visualize it as a field of imaginary lines emanating from north to south poles of a magnet. In our scenario involving a current-carrying wire, the magnetic field is uniform, meaning that its strength and direction are constant across the space in which the wire is located. It’s crucial to understand that the magnetic force experienced by a wire is directly dependent on both the magnitude of this field and the orientation of the wire within it.

When a wire carries an electric current, it creates its own magnetic field around it, and if placed within another magnetic field, a force arises on the wire. This force comes into play when the wire and the external magnetic field interact. The direction of the force is given by the right-hand rule: if you point the thumb of your right hand in the direction of the current, and your fingers in the direction of the magnetic field, the force acts in the direction of your palm. In our exercise, a wire carrying 400 amperes is substantial and underlines why such high currents are functional in scenarios like circuit breakers, where a significant force is needed to blow out an electrical arc.

In physics and most sciences, we use the International System of Units (SI) for consistency and standardization. In this context, lengths are measured in meters (m), force in newtons (N), and magnetic field strength in teslas (T). To accurately calculate the magnetic force, unit conversion is often necessary—like converting the 50 millimeters to 0.05 meters, which ensures all units align within the SI system for proper calculation. Remember, one meter equals one thousand millimeters, and conversions like these are crucial to maintain accuracy in calculations.

The Lorentz force law describes the combined electric and magnetic force on a point charge due to electromagnetic fields. In the context of a current-carrying wire, the magnetic component of the Lorentz force is of interest and it's given by the formula \( F = B \times I \times l \times \text{sin}(\theta) \). To find the magnetic force on a wire, we use this law while considering the magnitude of the current \( I \), the length of the wire \( l \), the magnetic field \( B \), and the angle \( \theta \) between the wire and the magnetic field. In our exercise, the angle is 90 degrees, leading to a sine value of 1. This result simplifies the equation, eliminating the need for incorporating the angle into the force calculations directly, streamlining the process to reach the final force value of 1 newton.