(a) Most microwave ovens operate at \(2.45 \mathrm{GHz}\). Assume that \(\sigma=1.2 \times\) \(10^{6} \mathrm{~S} / \mathrm{m}\) and \(\mu_{r}=500\) for the stainless steel interior, and find the depth of penetration. \((b)\) Let \(E_{s}=50 \angle 0^{\circ} \mathrm{V} / \mathrm{m}\) at the surface of the conductor, and plot a curve of the amplitude of \(E_{s}\) versus the angle of \(E_{s}\) as the field propagates into the stainless steel.

Microwave ovens operate on a specific frequency to heat food efficiently. Most microwave ovens use a frequency of 2.45 gigahertz (GHz), which translates to a wavelength that water molecules, present in most foods, can absorb effectively. This absorption excites the water molecules, creating heat through friction, ultimately cooking the food.

When engineers design microwave ovens, they must consider how the frequency interacts with the interior materials of the oven, like the stainless steel walls. The operating frequency not only determines how the food is heated, but also how deeply the microwaves penetrate into the oven's internal surfaces. Precise calculations are important to ensure safety and efficiency, as well as to avoid interference with other electronic devices operating on nearby frequencies.

The depth of penetration, often referred to as skin depth, is a measure of how far an electromagnetic field can penetrate into a conductor before it decays significantly. This concept is pivotal in understanding how microwave ovens work and how the microwaves interact with the materials inside them.

The skin depth is influenced by the frequency of the electromagnetic field, the electrical conductivity, and the magnetic permeability of the material. Calculating the skin depth involves these variables, and the depth can be very shallow for high frequencies or for materials with high electrical conductivity and magnetic permeability. This is particularly relevant in microwave ovens, where the penetration depth determines how the microwaves interact with the oven's stainless steel interior, affecting design considerations to ensure the safe containment of microwaves within the oven.

The electric field amplitude is a quantitative expression of the magnitude of the electric field at a given point in space. For a microwave oven, the electric field amplitude at the surface of the stainless steel interior is an important parameter. It determines the initial intensity of the microwaves as they begin to penetrate the material of the oven's walls.

The amplitude of the electric field decreases exponentially with distance from the surface due to the conductive nature of the metal, following the skin depth principle. This measurement is essential for plotting the expected behavior of the field as it moves through the material, and ultimately for ensuring that the design of the oven directs the microwaves appropriately to heat food without causing damage to the oven itself or to its surroundings.