Question

At 2 GHz, the conductivity of meat is on the order of 1 (S/m). When a material is placed inside a microwave oven and the field is activated, the presence of the electromagnetic fields in the conducting material causes energy dissipation in the material in the form of heat. Develop an expression for the time-average power per mm³ dissipated in a material of conductivity o if the peak electric field in the material is Eo.

Answer 1

The expression for the time-average power dissipated per mm³ in a given material with conductivity σ and peak electric field Eo is p = ½ σ Eo². The factor of ½ accounts for the time averaging of the oscillating electric field in the material when exposed to microwaves, such as in a microwave oven.

Step-by-step explanation:

The question is asking to develop an expression for the time-average power dissipated per unit volume in a material with a given electrical conductivity (o) when subjected to a peak electric field (Eo). In electromagnetic theory, the power dissipation per unit volume (p) in a conductive material due to an electric field can be given by the formula p = ½ σ Eo², where σ represents the conductivity of the material, and Eo is the peak electric field amplitude. Since power is energy per time, and the question specifies per mm³, we should ensure to express our final answer with appropriate units.

The time-average power is given by the fact that the electric field E in the material oscillates sinusoidally, which leads to an oscillating current and alternating power dissipation. We use the time-average value since the microwave operates at high frequency. The conductivity σ, in this case, is given as approximately 1 (S/m) at 2 GHz. The expression p = ½ σ Eo² is derived from the formula where instantaneous power is given by σ E², and the factor of ½ comes from averaging the sinusoidally varying electric field over time. To express this power dissipation in terms of mm³, we convert S/m to S/mm by dividing by 1000 (since 1 m = 1000 mm).