Final answer:
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).