The microwaves will travel and penetrate into the potato. As they propagate through, they will give up their energy into the potato. Classically, this would be by generating currents in the potato that work to oppose the incident electromagnetic wave. These currents will generate heat due to the finite conductivity of the potato and heat up the food. In a more modern sense, polar molecules, particularly water, will oscillate as they rotate their dipole moment back and forth in accordance with the induced oscillating currents. The movement causes thermal vibrations, giving up heat.
Either mechanism though can be modeled by looking at the absorption of the microwave in the dielectric using classical models. For example, it is very common to look at the absorption of radiation from a cell phone antenna by using FDTD of the cell phone against a model head. MRI simulations are also similarly done using FDTD and a model of the radiating antennas and head.
Your lecturer has suggested using a Beer Lambert model. Ideally, what I would suggest you do is a FDTD or FEM model of the potato in a resonant cavity and observe the absorption. However, the assumption that you state for the Beer Lambert model sounds like it will be a sufficient approximation. I already posted some calculations for the penetration of microwaves into water in the GPS thread. Now I do not know the conductivity (or permittivity) of a potato, however I would assume that it would be on the order of one Siemen, similar to salt water. For 5 Siemens, the skin depth is 6 mm for 1.2 GHZ wave in free space. I think we can assume this high of a conductivity due to the fact that a potato is often used as a classroom example of a battery. The potato is used as the electolytic medium that separates an anode and cathode but still has high enough conductivity to allow the ready exchange of ions between the two plates. The permittivity and permeability of the potato would shorten this distance as the wavelength would be contracted. Either way, it is probably safe to assume that an incident microwave on a potato would probably be completely absorbed. In fact, this is often the problem with microwaving foods. A microwave will often fail to penetrate fully into a product, thus only heating the exterior (the bane of my microwave burrito attempts). This is countered by moving the frequency of the microwave away from the actual resonant frequency of water (so that we do not over heat the exterior), thus slowing down the cooking process to allow the heat to conduct from the exterior to the inside.
This latter problem seems to me the object of your modeling problem.
EDIT:
Taking a look at the Beer-Lambert law on Wiki suggests that this is exactly the model that I would suggest. It is analogous to the equations that define the propagation of an electromagnetic wave through a lossy medium. It does not take into account however, the angle of incidence and the reflections off of the air to potato boundary and all of the reflections and transmissions within the potato due to the potato-air interface. However, since the microwaves will pretty much be completely absorbed before propagating to any appreciable depth, the latter effects are minute. The transmittance equation is completely analogous to the the power loss of an electromagnetic wave in a conductive medium.
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Page 7, or slide 95, shows the average power flow density of a wave inside a homogeneous medium. Essentially, we only care about the exp{-2\alpha z} factor as that is the only term that will vary with the distance travelled. Here, alpha is called the skin depth of the medium and is frequency dependent. the -2\alpha coefficient is similar to the absorption and concentration coefficients in the transmittance equation from Beer-Lambert.
So yes, I agree that the Beer-Lambert law will be sufficient. It should be a good approximation for the power absorption of the microwaves as they travel through the potato due to the fact that the waves will most certainly be completely absorbed before reaching the potato-air interface and be partially reflected back inside and transmitted out. If this wasn't the case, then you would probably have to do a full FDTD or FEM simulation to obtain the absorption of the waves in your target.
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