Plane Wave Shielding: Impact of the Conducting Shield on the Electric and Magnetic Fields

This article discusses the impact of a solid conducting shield on the electric and magnetic fields when a normally incident uniform plane wave penetrates it. The shield is made of a good conductor with no aperture and a thickness much greater than the skin depth at the frequency of interest. A good conductor is defined as a medium in which the conduction current density is much greater than the displacement current density, or equivalently, the loss tangent of a medium (σ ⁄ ω >>⊇1). [1] Plane wave shielding is achieved with enclosures that are located far enough from the source to be in the far field where the ratio of the electric and magnetic field amplitudes approximately equals to the intrinsic impedance of free space (η₀ ≅ 377Ω). Examples of this type of shielding may include fixed or mobile installations of electronics protected from airborne or terrestrial sources. A specific example of this shielding approach is when lighting electronics kept at the top of a stadium light pole are enclosed in a shielded box.  This shielding box is located far from a source, such as Wi-Fi used in the stadium, and is therefore considered to be in the far field.

Uniform Plane Wave Incident on a Shield

Let us consider a conducting shield of thickness t, conductivity σ, permittivity ε, and permeability µ, surrounded on both sides by air (free space, and thus a perfect dielectric), as shown in Figure 1.

A uniform plane wave is normally incident on its left interface. Uniformity assumption together with normal incidence implies that the shield is in the far field of the radiation source.

The incident wave, upon arrival at the leftmost interface , will be partially reflected and partially transmitted through the left interface of the shield. Subsequently, the transmitted wave will be partially reflected and partially transmitted through the right interface of the shield [2, 3].

Let us make a few comments about the magnitudes and directions of the fields and the directions of the wave propagation, shown in Figure 1. The direction of a wave propagation is the direction of the Poynting vector (power density vector ). This direction, a_k, is determined by the directions of the electric field, a_E, and the magnetic field, a_H, according to the cross product as

(1.1)

The magnitudes of the fields in Figure 1 are drawn not to scale.

Impact of the Shield on the Electric Field

At the left interface, the magnitude of the reflected electric field _r is related to the magnitude of the incident electric field _i by

(2.1)

where the reflection coefficient at the left interface is

(2.2)

with the intrinsic impedance of free space given by

(2.3)

and the intrinsic impedance of the shield being

(2.4)

If the shield is made of a good conductor, then << η₀ and the reflection coefficient at the left interface approximately equals

(2.5)

The reflected electric field is equal in magnitude to that of the incident electric field, but the directions are opposite, as shown in Figure 1.

The total electric field at the left interface is

(2.6)

Note that the reflection coefficient at the left interface is not equal to -1; it is approximately equal to -1. It follows that the total field at the left interface is not equal to 0; it is approximately equal to 0.

Very little of the electric field penetrates the left interface of the shield. Effectively, a shield consisting of a good conductor almost entirely “shortens out” the electric field. Still, a small portion of the electric field, ₁ , penetrates the shield and travels through it to reach the right interface.

We will discuss the shield impact and the right interface impact on ₁ shortly. For now, let’s look at the transmission coefficient at the left interface.

The magnitude of the transmitted electric field ₁ is related to the magnitude of the incident electric field ₁ by

(2.7)

where the transmission coefficient at the left interface is

(2.8)

This is consistent with the earlier observation that very little of the electric field penetrates the left interface of the shield (the magnitude of the transmitted electric field is drawn not to scale).

The transmitted electric field ₁ travels through the shield towards the right interface. Upon reaching the right interface, it gets reflected ₂ and transmitted _t. If the shield is several skin-depths thick at the frequency of interest (t >> δ), then the field ₁ is greatly attenuated when it reaches the right interface.

The reflection coefficient at the right interface is

(2.9)

Thus, at the right interface, the reflected field ₂ is approximately equal to the incident field ₁. As the reflected field travels back to the left interface, it gets attenuated again. Since the ₁ field was greatly attenuated when it reached the right interface, the ₂ field will again be greatly attenuated upon reaching the left interface and thus can be disregarded (under the thick shield assumption).

The total field at the right interface is

(2.10)

Thus, the field doubles at the right interface, but this is of little consequence as the ₁ field is greatly attenuated when it arrives there.

Continuity condition at the right interface requires

(2.11)

and thus

(2.12)

This result means that a very small portion of the original wave that is incident on the left interface of the shield ₁ is doubled in magnitude when it is transmitted through the right interface of the shield. Even though ₁ doubles in magnitude, it is of little consequence since it has been greatly attenuated when traveling through the shield.

Let’s look at the transmission coefficient at the right interface. The magnitude of the transmitted electric field _t is related to the magnitude of the incident electric field ₁ by

(2.13)

where the transmission coefficient at the right interface is

(2.14)

which is consistent with equation 2.12.

Conclusions

Most of the electric field is reflected at the conducting shield interface, regardless of its thickness. Thus, the primary requirement is that the shield is made of a good conductor.

The secondary requirement is that the shield is much thicker than the skin depth at the frequency of interest.

Impact of the Shield on the Magnetic Field

At the left interface, the magnitude of the transmitted magnetic field ₁ is related to the magnitude of the incident magnetic field _i by

(3.1)

where the transmission coefficient at the left interface is

(3.2)

Since the shield is in the far field of the radiation source, the magnitudes of the fields are related by the intrinsic impedances of the respective medium [2]. Thus,

(3.3)

(3.4)

Thus, the transmission coefficient at the left interface becomes

(3.5)

From Eq. 2.8, we obtain

(3.6)

Utilizing Eq. (3.6) in Eq. (3.5) we get

(3.7)

or

(3.8)

Thus, the primary transmission of the magnetic field occurs at the left interface, indicating that a thick shield to attenuate it is of primary importance (this was not the case for the electric field).

Continuity condition at the left interface requires

(3.9)

From Eq. 3.8, we get

(3.10)

Thus, at the left interface, the magnitude of the incident magnetic field doubles as the field penetrates the left interface. This is of great concern, once again underscoring the need for a thick shield to attenuate the magnetic field.

Substituting this result into Eq. 3.9, we obtain

(3.11)

or

(3.12)

and hence

(3.13)

Thus, at the left interface, the magnitude of the reflected magnetic field equals the magnitude of the incident magnetic field. The direction of the reflected magnetic field is the same as the direction of the incident magnetic field, as shown in Figure 1.

Let’s return to the ₁ field. As this field travels through the shield towards the right interface, it is attenuated. Upon reaching the right interface, it gets reflected ₂ and transmitted _t .

The transmission coefficient at the right boundary is

(3.14)

From Eq. 2.14, we obtain

(3.15)

Utilizing Eq. 3.15 in Eq. 3.14, we get

(3.16)

or

(3.17)

Thus, little transmission of the magnetic field occurs at the right interface. Let’s confirm this by looking at the reflection coefficient at the right interface.

The continuity condition at the right interface requires

(3.18)

From Eq. 3.17, we get

(3.19)

Utilizing Eq. 3.19 in Eq. 3.18, we obtain

(3.20)

or

(3.21)

and hence

(3.22)

Confirming that most of the magnetic field incident on the right interface gets reflected, and thus little transmission of the magnetic field occurs at the right interface.

Conclusions

Little of the magnetic field is reflected at the left shield interface. Most of it gets doubled and transmitted through the left interface. Thus, the primary requirement is that the shield is much thicker than the skin depth at the frequency of interest.

The secondary requirement is that the shield is made of a good conductor.

Summary

Plane wave shielding against electric and magnetic fields places different—and in some ways opposite—demands on shield design.

For the electric field, material conductivity is of greater importance than the thickness of the shield. Shield material does not have to be perfectly (or almost perfectly) conducting as long as the charges can redistribute themselves fast enough to create a reflection that cancels the incident wave, [4]. Most of the field is reflected at the left interface regardless of shield thickness, with thickness playing a secondary role in attenuating whatever small portion penetrates. For the magnetic field, the priorities reverse. Because the field largely passes through the left interface—doubling in magnitude as it does—thickness is the primary requirement, with conductivity in a supporting role.

A shield that is both a good conductor and substantially thicker than the skin depth at the frequency of interest satisfies the dominant requirement for each field component, providing reliable protection against a normally incident uniform plane wave.

References

1. Bogdan Adamczyk, Peter Reiser, and Scott Mee, “Shielding Effectiveness – The Impact of Loss Tangent on the Reflection and Absorption Losses,” In Compliance Magazine, June 2026
2. Clayton R. Paul, Introduction to Electromagnetic Compatibility, Wiley, 2006.
3. Bogdan Adamczyk, Principles of Electromagnetic Compatibility – Laboratory Exercises and Lectures, Wiley, 2023.
4. Todd H. Hubing, Electromagnetic Compatibility Course Notes, LearnEMC, 2026.