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Why is a magnetic field outside the toroid zero ?

The magnetic field outside a toroid is effectively zero due to the unique geometry and symmetry of this magnetic structure. A toroid is a doughnut-shaped or ring-like core typically made of ferromagnetic material, around which a coil or winding is wound. The reasons behind the near-zero magnetic field outside the toroid are as follows:

  1. Closed Magnetic Path:
    • The toroidal shape inherently provides a closed magnetic path for the magnetic flux generated by the current flowing through the coil. Unlike other magnetic structures with open paths, the toroid confines the magnetic field within itself, minimizing its external influence.
  2. Cancellation of Magnetic Fields:
    • The toroid’s design involves a series of closely wound turns of wire forming a coil around the toroidal core. The magnetic fields generated by each turn of the coil contribute to the overall magnetic field within the toroid. Due to the toroid’s symmetry, these magnetic fields add up in a way that cancels out the external magnetic field, resulting in minimal flux leakage.
  3. Ampere’s Circuital Law:
    • According to Ampere’s Circuital Law, the integral of the magnetic field along a closed loop is equal to the product of the permeability of free space and the current passing through the loop. In the case of a toroid, the closed loop is within the toroid itself, and the winding of the coil ensures that the magnetic field is concentrated within this closed loop.
  4. Equal and Opposite Magnetic Contributions:
    • The toroidal core has a circular cross-section, and the current passing through the coil produces magnetic fields that circulate around the core. The circular symmetry of the toroid ensures that the magnetic contributions from different points around the toroid are equal and opposite, leading to the cancellation of the magnetic field outside the toroid.

In summary, the combination of the closed magnetic path provided by the toroidal shape, the cancellation of magnetic fields due to the coil winding, and the circular symmetry of the toroid results in an almost negligible magnetic field outside the toroid. This property makes toroids valuable in applications where a confined and controlled magnetic field is desirable, such as in transformers and inductors.

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