# Electric fields with different frequencies, total intensity?

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.

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.

• I do not think this is correct. You can add electric field vectors because they obey the superposition rule, but you cannot add the intensities to get the total intensity. – honeste_vivere May 22 '16 at 19:06
• In this particular case you can add the intensities because the two fields are incoherent (they oscillate at different frequencies). It comes from the fact that when you add the two field vectors and square them to get the intensity, the cross term averages to zero. – Dimitri May 23 '16 at 8:37

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.