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Mathematical physics

Mathematical physics

Are Faraday cages less effective than previously thought?

15 Sep 2015
What's cooking? A microwave oven is a Faraday cage

Is the Faraday cage in your lab less effective than you think? A new study by applied mathematicians at the University of Oxford suggests that the mesh wire cages may not be as good at shielding electromagnetic radiation as previously thought. The team has identified a flaw in the conventional understanding of how the effectiveness of a Faraday cage changes as the wires are moved closer together. Even though engineers might have realized this effect experimentally, this mathematical discovery could improve the design of these cages.

The physicist Michael Faraday came up with the idea of using a conducting wire-mesh cage to block electromagnetic fields in 1836. A familiar modern example of such a cage is the mesh screen on a microwave oven door, which is designed to keep microwaves in while allowing us to watch our food as it cooks. A solid metal cage offers the best protection from electromagnetic waves, but Faraday and many scientists since have assumed that mesh is a good enough approximation of a solid metal cage. Specifically, many had believed that the electric field within the cage rapidly approaches zero as the wires get closer together. In the 1960s, for example, the Nobel laureate Richard Feynman gave a famous series of physics lectures at Caltech that included a short discussion on Faraday cages that has since been adapted in many textbooks and physics lectures. He assumed constant electric charge, and argued that the electric field approaches zero exponentially within the cage.

It takes a lot of time to grow confidence that the intuition [of many people we talked to] is wrong
Nick Trefethen, University of Oxford

Nick Trefethen of the University of Oxford first became interested in Faraday cages in 2013 when he discovered that a thorough mathematical analysis of the cages was lacking in the scientific literature. Trefethen had been working on the trapezoidal quadrature rule – a numerical method for approximating integrals – and he guessed that its equations should also describe the Faraday cage. Because the trapezoidal rule has a property of rapid convergence, he guessed that the Faraday cages would work the same way.

He joined forces with Oxford’s Jon Chapman and David Hewett to study the problem, but it soon became apparent to the team that this rapid convergence to zero electric field does not happen. “We spent months and months fretting about it,” he told, adding “It takes a lot of time to grow confidence that the intuition [of many people we talked to] is wrong.”

The thick of it

Trefethen and colleagues used a simple 2D numerical model of a Faraday cage that comprised a ring made of individual circular conductors, each of which represents a wire in the mesh. The region outside of the cage contained a source of a static electric field, which applies a constant voltage to the conductors. Their simulations suggested that the strength of the electric field inside the cage is proportional to the logarithm of the radius of the individual wire circles – which corresponds to the thickness of the wires in a Faraday cage. This suggests that Faraday cages made from thick wires are better than those made from thin wires.

They also found that the strength of the internal electric field is proportional to the distance between the wires. Trefethen and colleagues were also able to derive a theorem that gives an upper bound for the internal field strength as a function of separation. This theorem also gives a linear relationship. A key finding of the study is that the drop in field strength is not exponential within a Faraday cage, as suggested by Feynman. Although the simulation was 2D, the team says that the result should also apply to 3D cages.

The team also looked at a continuous model of the mesh cage, which Trefethen says is the main result of the research. These calculations revealed that the Faraday cage does not behave like a normal conductor – rather, it behaves like a surface with a limited capacitance. As a result, it takes energy to push charges onto the wires. This means that a mesh will have a different charge distribution than the ideal Faraday cage, which is a continuous sheet of metal.

Everyday examples

Trefethen points out that there is evidence for the team’s findings in everyday technology. The shielding in microwave oven doors, for example, is usually a solid sheet of metal with holes punched in it – which is essentially a Faraday cage with thick wires. Thinner wires would make it easier to see inside the oven, but are not used by microwave designers – a decision that seems to be in line with the team’s findings. Trefethen also points out that despite the thick wires, microwave ovens are not perfect Faraday cages. That is because if you put your mobile phone – which uses microwaves to communicate – into an oven and call it, there is a good chance that it will ring.

Ali Niknejad is an electrical engineer at the University of California, Berkeley, and acknowledges that there are gaps in our understanding of Faraday cages: “Many of us have done that [phone] test and realized, yes, you know the meshes are not working perfectly.”

Niknejad works with high-frequency electronics including microwave devices, and is interested in shielding computer chips and circuits from interference. He points out that engineers have known for a while that solid metal cages should be used in these applications. He also points out that the Oxford study is an indication that it also happens at low frequencies – specifically in the special case of a static electric field.

Safe emissions

Niknejad also points out that there is no need to worry about emissions from your microwave oven, because engineers will have tested the performance of the mesh cages by making real measurements. Microwave ovens, he adds, have several emission safety requirements.

The research is described in SIAM Review and a preprint of the paper is available from the authors.

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