How strong are Wi-Fi signals?

Could someone tell me how strong the signals from the wireless devices are compared to those from lights, and perhaps those from the Sun as well?

At the surface of the Earth, the Sun delivers approximately 1 kW/m$^2$ [Wikipedia]. An average 100 W incandescent light bulb is only about 2.5% efficient so it emits 2.5 W of optical power (the rest is emitted as infrared or ultraviolet as described in the case of the Sun below). Wireless routers emits about 0.1 W of power. A cell phone emits about 1 W.

Let's put all that into a table, but let's make sure we include the frequency of the radiation emitted by each of these sources:

|     Source     | Power (W) | Frequency |
| Sun            | 1000/m^2  | optical   |
| Light bulb     | 2.5       | optical   |
| Cell phone     | 1         | microwave |
| WiFi router    | 0.1       | microwave |
| Microwave oven | 700       | microwave |

The Sun is by far the strongest emitter in our daily lives. That's pretty obvious though if you think about the fact that looking at the Sun is painful and would destroy your eyes while looking at a WiFi router is no problem.

What is the scientific basis for claiming the radiation has or doesn't have effects on the human body?

There are two factors that determine whether radiation is hamful: flux and frequency. Flux means roughly the number of photons flowing through a certain area per time. Frequency means the frequency of each photon. Power $P$ is related to frequency $\omega$ and flux $\Phi$ via $$P = \Phi \hbar \omega \, .$$ However, power is not the only thing that determines harmfulness. It turns out various materials have specific frequencies where they do and do not absorb radiation. For example, glass does not absorb optical radiation, which is why you can see through it.

The Sun emits power over a range of frequencies but the peak is in the optical (i.e. visible) range. That comes as no surprise because of course our eyes are evolved to see the radiation light that exists on Earth. Optical radiation has relatively high energy and because of that it gets readily absorbed by the outer parts of your body (except for clear part of the eyes). However, we're not usually exposed to enough optical radiation flux to do any harm. For example, we don't usually encounter strong enough lights to burn us. A really strong industrial laser would be a counterexample. On the other hand, the part of the solar spectrum at frequencies just above the optical, known as "ultraviolet", has enough energy to damage your body cells, causing sunburn and skin cancer. The part of the solar spectrum at frequencies below the optical, known as "infrared", is commonly called "heat". The infrared is generally too low energy to destroy body cells at the levels coming from the Sun.

Incandescent light bulbs also emit a spectrum of radiation, and the story is relatively similar to the story we told for the Sun.

Now, cell phones, WiFi routers, and microwave ovens all produce microwave radiation, which in the range of 1 GHz frequency. That's about 100,000 times lower frequency than visible light. Microwave radiation penetrates your skin and goes through your body. That's why microwave ovens work; the radiation permeates the food and heats it up. Compare that to putting food right next to the heating element of a broiler in which case the food's outside cooks very quickly before the whole thing is done.

Anyway, the point is that microwave radiation penetrates your body. That might sound scary, but microwave photons are too low in energy to damage your cells the way that ultraviolet does. So even though microwaves heat you up a little bit, they don't give you cancer the way that sunlight does.

Now let's compare power levels in a cell phone and in a WiFi router. Since the phone is right next to your head, about half of the phone's emitted power goes through your brain. On the other hand, if you're $3\text{m}$ away from a WiFi router and we approximate the size of a head cross as $20\text{cm}$, then the fraction of the WiFi router's power going through your head is only about $$\text{WiFi power fraction} = \frac{(20\text{cm})^2}{\underbrace{4 \pi (3\text{m})^2}_\text{surface area of sphere}} = 0.00035 \, .$$ So all together, the ratio of phone energy to WiFi router energy going through your brain is \begin{align} \frac{\text{phone power through brain}}{\text{WiFi power through brain}} &= \frac{\text{phone emitted power}}{\text{WiFi emitted power}} \times \frac{\text{phone power fraction}}{\text{WiFi power fraction}} \\ &= \frac{1\text{W}}{0.1\text{W}} \times \frac{0.5}{0.00035} \\ &\approx 14,000 \, . \end{align} So a cell phone puts about 14,000 times more power through your brain than a WiFi router. If your folks are afraid of the WiFi router, they should be terrified by cell phones.

The highest stated value i've been able to find is that 5GHz routers can transmit at some ~23dBm which corresponds to 200mW. This power is the radiated in a pattern defined by the antenna-setup which means that the energy is typically not very directional (as the router doesn't know where the device will be). The intensity of the received radiation at the receiver (device) would be dependant on distance and receiving area, thus resulting in a $W/m^2$-figure.

If we consider the antenna to be radiation 200mW in an isotropical fashion (equal in all directions) and the receiver is at a distance of 5m, the intensity at the receiver will be 1.27$mW/m^2$. Thsi figure can then be compared to the intesity received by the sun, roughly 1000$W/m^2$ at sea level.

As you can see, these figures are far apart. I wouldn't worry about radiation from wireless comunication. If there's a router plugged in, extra devices won't affect this. Phones for example are always communication (unless airplanemode), either through wifi or dataroaming, and connecting this device to a wifi network won't change this.

The Wifi band does stretch into the spectral region where water absorption may occur, but since wifi routers aren't operated in a resonator chamber, and the power is greatly reduced, i at least wouldn't worry about this. On top of this, we're constantly surrounded by this EM-radiation from cell towers and neighbours' routers and i find it hard to believe that a wifi router would have a greater health risk than what is already present.

Here are pretty recent statements from the World health organization and Canades health department stating that radiation from these bands have not been proven to cause cancer or any other effects. What needs to be mentioned is that it has not been proven to be harmless, but so far, it has not been proven to be harmful either, even though studies have been made into the topic. The links above cite studies.

You only get out what you put in

This aims to be a layperson's answer, rather than specific complex formulae which are mind boggling to comprehend to a layperson, but the easiest way to prove the lightbulbs are more harmful than the wifi, is to consider the input power (as output can never exceed input; due to inefficiencies, it will always be less).

Consider how, after several hours, you can burn your hand on a lightbulb due to the heat it produces, but you cannot really burn your hand on a router.

Most routers only take 5v-12v at perhaps 5 amps; lightbulbs can take, anywhere between 12v to 240v at varying amperages. If you consider how a microwave requires 600 watts of power (110-240 volts depending on system) to take several minutes to boil water, a 5v to 12v device with maybe 5 amps tops (usually 0.8 amps) isn't going to nearly have as much power output as a lightbulb.

You could argue it's the continuous duration of exposure over a long period of time that potentially causes the harm with wifi, but it's worth noting wifi signals will already be in abundance (neighbour's wifi, public wifi, cell phone towers, other people's phones etc). Just short of insulating the entire house, refusing to use microwaves, cellphones, routers etc, they're going to be exposed anyway.