Hey, everyone. Let's work through this problem together. Here, we're told that we need to build a transformer that drops the 120 volts of a regular North American outlet down to a much safer 15 volts. Now we already have a solenoid with 50 turns, but we need to make a second solenoid in order to complete our transformer. Now here, we want to determine the least number of turns that this second solenoid could have.
So let's go ahead and get started here. Now we know that the ratio of the voltages in our transformer depends on the ratio of the number of turns in our solenoids. That means that \( \frac{V_2}{V_1} \) is equal to \( \frac{N_2}{N_1} \). Now we have both our input and our output voltages here. So this ratio on the left hand side is going to be \( \frac{15}{120} \).
So this will be equal to \( \frac{N_2}{N_1} \). Now we can reduce that fraction. This ends up giving us \( \frac{1}{8} \) is equal to \( \frac{N_2}{N_1} \). Now in our problem, we're told that one of our solenoids has 50 turns, but we're not told which solenoid. So that means that there are 2 possible outcomes here, one in which \( N_1 \) is equal to 50 and one in which \( N_2 \) is equal to 50.
So we need to test both of these outcomes and see which one results in our other solenoid having the least possible number of turns. So let's go ahead and plug these \( N \) values into our ratio here. Now if \( N_1 \) is equal to 50, this then makes our ratio \( \frac{1}{8} \) is equal to \( \frac{N_2}{50} \). Now this means doing some algebra that \( N_2 \) is equal to \( \frac{50}{8} \) or 6.25 turns. Now if \( N_2 \) is instead equal to 50, then this makes our ratio \( \frac{1}{8} \) is equal to \( \frac{50}{N_1} \).
This means that \( N_1 \) will then be equal to 50 times 8 or 400 turns. Now comparing these two values, 6.25 turns is clearly fewer turns than 400. So this means that if \( N_1 \) is equal to 50, this results in our second solenoid having the least possible number of turns, 6.25. Feel free to let us know if you have any questions here, and let's keep going.