An outdoor WiFi unit for a picnic area has a 100-milliwatt output and a range of about 30 metres. What output power would reduce its range to 12 metres for use with the same devices as before? Assume there are no obstacles in the way and that microwaves into the ground are simply absorbed.
We can call the initial power given — 100 milliwatts — 𝑃 sub one. And we can call the range that corresponds to that power 30 metres 𝑟 sub one. We’ll call the range of 12 metres 𝑟 sub two and the power that we want to solve for, which would project the signal at this reduced range, 𝑃 sub two.
As we begin our solution, let’s recall a relationship between signal intensity, power, and area. The intensity of an electromagnetic signal 𝐼 is proportional to the power of that signal divided by the distance squared from the source. In our situation, we can assume that intensity is constant because we want to be able to power the same types of devices as before, even when our range is decreased.
When we write that 𝐼 one is equal to 𝐼 two, this is equivalent to writing 𝑃 sub one, the initial power, divided by 𝑟 sub one squared is equal to 𝑃 sub two, the final power, divided by 𝑟 sub two squared. If we rearrange this equation to solve for 𝑃 two, we see that it’s equal to 𝑃 one times the ratio 𝑟 two over 𝑟 one squared.
We’ve been given 𝑃 one, 𝑟 one, and 𝑟 two in the problem statement and can plug in for those three values now. When we enter these values on our calculator, we find that to two significant figures 𝑃 two is equal to 16 milliwatts. That’s the power the WiFi device will need to be supplied with in order to have a range of 12 metres.