Electromagnetic Shielding: How a Faraday Cage Cancels a Field

Cancelling a field with free charges

A conductor placed in an external electric field responds instantly, on the scale of a circuit: its free electrons drift until the field they create inside exactly cancels the applied one. The result is electrostatic equilibrium — zero net field anywhere inside the conducting shell, no matter how strong the field is outside. This is the whole idea behind a Faraday cage, named after Michael Faraday, who demonstrated it in 1836 by building a room-sized wire cage, charging it to high voltage, and stepping inside with an electroscope that read exactly zero.

For a conducting sphere of radius R in a uniform external field E0, the induced surface charge density is sinusoidal in the polar angle θ measured from the field direction, and the interior is field-free by superposition of the applied field and the field of that induced charge layer:

sigma_induced(theta) = 3 * eps0 * E0 * cos(theta)
E_inside = 0                      (exact, for an ideal conductor)
E_outside = E0 * (1 + 2*(R/r)^3 * cos(theta) * r_hat_component)

The AC case: skin depth

A static field is cancelled by charge redistribution; a time-varying field is cancelled by induced eddy currents that flow in the conductor's surface layer and generate an opposing field. Those currents cannot penetrate deep into a good conductor — they decay exponentially with depth, characterised by the skin depth δ, the distance over which the field falls to 1/e of its surface value:

delta = sqrt(2 / (omega * mu * sigma))
   omega = 2*pi*f      (angular frequency)
   mu    = permeability of the shield material
   sigma = electrical conductivity of the shield material

Higher frequency, higher conductivity, or higher permeability all shrink δ, which is why a thin sheet of copper stops a gigahertz signal completely but barely dents a 50 Hz field — at 1 MHz copper's skin depth is about 66 micrometres, at 50 Hz it swells to roughly 9 millimetres. Total shielding effectiveness in decibels is normally split, following Schelkunoff's classical treatment, into an absorption term A that grows linearly with thickness over δ, a reflection term R set by the impedance mismatch at the boundary, and a smaller multiple-reflection correction B that only matters for very thin foils.

Why the holes matter more than the metal

Real enclosures are not solid — they have seams, doors, ventilation grilles and cable entries, and a single gap leaks far more field than an equivalent area of solid wall. A small aperture behaves like a waveguide operated below cutoff: it strongly attenuates any wavelength much larger than the hole, which is precisely the principle used in a microwave oven door, where a metal mesh with holes a few millimetres across is transparent to visible light but opaque to the oven's 12 cm microwaves. The rule of thumb engineers use is that the largest linear dimension of an aperture should stay below roughly a tenth of the wavelength you need to block; above that, leakage rises quickly.

Static magnetic fields need a different trick

A subtlety that trips people up: a copper Faraday cage is excellent against electric fields and against time-varying magnetic fields (because those induce the eddy currents that do the cancelling), but it is nearly useless against a static magnetic field, since a constant field induces no current at all. Shielding DC magnetic fields — inside an MRI room, for instance — instead requires high-permeability material such as mu-metal, which does not cancel the field but reroutes it, providing a low-reluctance path that pulls flux lines around the protected volume rather than through it.

Where the physics shows up

MRI suites are lined with continuous copper sheeting to keep the scanner's own radio pulses from leaking out and hospital equipment's radio noise from leaking in. Car bodies act as rough Faraday cages during a lightning strike, which is one reason (not the only one) that a metal car is a relatively safe place to be in a storm. Coaxial cable shielding, anechoic chamber walls, and the tin-plated steel can around a sensitive op-amp are all the same idea at different scales.

Frequently asked questions

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