The magnetic field outside a solenoid is typically zero or very weak because the magnetic field lines produced by the current-carrying coils of the solenoid are confined within the interior of the solenoid itself. This confinement occurs because the magnetic field lines generated by each turn of wire in the solenoid loop around the interior of the coil and do not extend beyond the ends of the solenoid to a significant extent. Therefore, outside the solenoid, the magnetic field lines cancel each other out due to their opposite directions, resulting in a net magnetic field that is negligible or zero.
When a solenoid is placed in a uniform magnetic field, the force experienced by the solenoid as a whole can be zero under certain conditions. This occurs when the direction and strength of the external magnetic field are such that the magnetic forces exerted on each side of the solenoid cancel each other out. For example, if the magnetic field is uniform and parallel to the axis of the solenoid, the magnetic forces on the opposite sides of the solenoid may balance each other, resulting in a net force of zero.
Similar to a solenoid, a toroid (a coil wound in the form of a ring or donut shape) also exhibits a near-zero magnetic field outside its structure. The magnetic field lines produced by the current circulating through the toroid’s coil are confined within the core of the toroid due to the circular path of the magnetic field lines around the toroidal core. This confinement prevents the magnetic field from extending significantly outside the toroid, resulting in a magnetic field that is nearly zero in the surrounding space.
Inside a solenoid, the magnetic field is stronger compared to outside because the magnetic field lines generated by the current-carrying coils add up coherently along the axis of the solenoid. Within the interior of the solenoid, the magnetic field lines are closely packed and parallel to the axis of the coil, resulting in a relatively uniform and strong magnetic field. This interior field strength depends on factors such as the number of turns in the solenoid coil, the current flowing through it, and the permeability of the core material if present.
To derive the magnetic field outside a solenoid mathematically, one typically applies Ampère’s law, which relates the magnetic field around a closed loop to the current enclosed by the loop and the permeability of the medium. By considering the geometry and symmetry of the solenoid, one can calculate the magnetic field at different points outside the solenoid using integral calculus methods, assuming an idealized scenario with a long solenoid where end effects are negligible. This derivation provides a quantitative understanding of why the magnetic field is zero or weak outside the solenoid compared to its interior.