Microwave that Brain!

Electricity and Magnetism Level 5

The Microwave Oven works on the principle of Electromagnetism and Dipoles. When the Microwave starts, a Sinusoidal Standing Microwave Radiation is developed inside the chamber. (Consider only the Electric Field)

Now, every food item has a certain amount of water inside it. Every water molecule is a Dipole (Dipole Moment \(\mu\)). This dipole interacts with the Electric Field, and thus, rotates and generates friction in the form of heat, due to which the food cooks.

Consider a Cubical Microwave oven of Length \(\ell\). It generates an Electric Field (Maximum Amplitude \(E_o\)) in the form of standing waves in the 2nd overtone.

Unfortunately, John killed our dear friend Gandalf and decided to test out his new knowledge of Microwaves on his Brain. So, at the Center of the oven, he places the brain, which is surprisingly a Sphere of Volume \(V\). It has a Water Content of \(\eta\) molecules per unit volume. Each Water Molecule (Moment of Inertia \(I\)) generates \(\kappa\) Joules in one Time Period of Oscillation.

If the amount of Heat generated when the Microwave Oven is used for 10 minutes, is \(H\) Joules, then evaluate \(\lfloor H\times 10^8\rfloor+377\)

Details and Assumptions:

\(\bullet V\ll \ell^3\)
\(\bullet\) All water molecules are parallel to the Electric Field initially
\(\bullet\) Molecules perform Simple Harmonic Oscillation
\(\bullet\) Assume the Electric Field to be constant for a Water Molecule
\(\bullet \sqrt{10} \approx \pi\)
\(\bullet I =1.9\times 10^{-47} Kg\cdot m^2\)
\(\bullet \eta = 2.575\times 10^{28}\) molecules/\(m^3\)
\(\bullet \kappa = 10^{-43}\) Joules
\(\bullet V=1260 cm^3\)
\(\bullet E_o = 100 \frac{V}{m}\)
\(\bullet \mu = 1.85\) Debyes
\(\bullet 1\) Debye \(= 3.33\times 10^{-30} C\cdot m\)


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