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What happens when electric fields with different frequencies are combined? Is it possible to calculate the intensity of the total electric field?

For a project, I need to simulate a brain treated with microwaves from antennas operating at different frequencies. I would like to calculate the specific absorption rate (SAR), but for that I need the magnitude of the total electric field.

noob
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2 Answers2

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The intensity of the total electric field will just be the sum of the two separate intensities, because the interference term will oscillate at a very fast frequency (namely $\omega_1 - \omega_2$) and its mean value will thus be zero.

See this awnser for more details.

Community
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Dimitri
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Electric fields obey the superposition principle. So let's call the two fields $\mathbf{E}_{1}\left( \omega_{1} \right)$ and $\mathbf{E}_{2}\left( \omega_{2} \right)$. Then the total field, $\mathbf{E}_{t}$, is given by: $$ \mathbf{E}_{t} = \mathbf{E}_{1} + \mathbf{E}_{2} \tag{1} $$

Note that the field intensity or energy density is given by: $$ W = \frac{ \varepsilon_{o} \ \left( \mathbf{E} \cdot \mathbf{E} \right) }{ 2 } \tag{2} $$ where $\varepsilon_{o}$ is the permittivity of free space. You can see that $W_{t} \neq W_{1} + W_{2}$ by noticing that: $$ \begin{align} \left( \mathbf{E}_{t} \cdot \mathbf{E}_{t} \right) & = \left( \mathbf{E}_{1} + \mathbf{E}_{2} \right) \cdot \left( \mathbf{E}_{1} + \mathbf{E}_{2} \right) \tag{3a} \\ & = \left( \mathbf{E}_{1} \cdot \mathbf{E}_{1} \right) + \left( \mathbf{E}_{2} \cdot \mathbf{E}_{2} \right) + 2 \left( \mathbf{E}_{1} \cdot \mathbf{E}_{2} \right) \tag{3b} \\ & \neq \left( \mathbf{E}_{1} \cdot \mathbf{E}_{1} \right) + \left( \mathbf{E}_{2} \cdot \mathbf{E}_{2} \right) \tag{3c} \end{align} $$

Is it possible to calculate the intensity of the total electric field?

Yes, add the two input fields together then calculate the intensity using Equation 2 above.

I would like to calculate the specific absorption rate (SAR), but for that I need the magnitude of the total electric field.

Calculate $\mathbf{E}_{t}$ using Equation 1 above and then find its magnitude, which is the total electric field magnitude.

honeste_vivere
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