A long, straight wire is carrying a direct current, which produces a magnetic field of strength 𝐵 one teslas at a perpendicular distance of 𝑑 centimeters from the wire. Assuming the system does not change, what is the relationship between 𝐵 one and the strength of the magnetic field 𝐵 two at a perpendicular distance of two 𝑑 centimeters from the wire? Assume 𝐵 one and 𝐵 two are much greater than the strength of Earth’s magnetic field.
Alright, in this question, we want to write a relationship between two different measurements of the magnetic field produced by a current-carrying wire. And since we’re working under a couple of assumptions here, let’s start by briefly clarifying what they are.
First, we’re assuming that the system does not change as we compare 𝐵 one and 𝐵 two, which really just means that the current in the wire remains constant. All that differs between the two measurements of the magnetic field is how far away from the wire the measurements are made and thus the strength of the measurements themselves. Nothing else changes. The other assumption is that the measurements 𝐵 one and 𝐵 two are much greater than the strength of Earth’s magnetic field. This just tells us that we should ignore any magnetic field that isn’t produced by the current-carrying wire in this question.
Alright, now let’s get on to actually solving the question. Clearing some room on screen, let’s draw a diagram to show what’s happening. We have a wire that’s carrying some current 𝐼, which produces a magnetic field composed of closed concentric circles represented here by the pink and orange loops or field lines. We measure the strength of the field at two different distances away from the wire and get a result of 𝐵 one at a distance 𝑑 and 𝐵 two at a distance of two 𝑑.
Now is a good time to recall the formula for finding the strength of a magnetic field due to a straight current-carrying wire, which states that 𝐵 equals 𝜇 naught times 𝐼 divided by two 𝜋𝑑, where 𝐵 is the strength of the magnetic field some distance 𝑑 away from a wire carrying a current 𝐼. And 𝜇 naught is a constant whose value we actually don’t need to recall in order to answer this question, and we’ll see why in just a moment.
Our job is to find a relationship between 𝐵 one and 𝐵 two, and we’ll use this formula to do so. Now, we were given values in units of centimeters and teslas, but really, 𝑑, two 𝑑, 𝐵 one, and 𝐵 two are so general that it will be useful to treat them as typical variables in the formula, just plain 𝑑 and 𝐵. And what we really want to know is this. When measuring the magnetic field produced by the wire, how does doubling the distance from the wire 𝑑 affect the strength of the field 𝐵?
Well, whenever we wanna know how variables in a formula change with respect to each other, it’s helpful to devise a statement of proportionality. So let’s copy the formula down here and replace the equal sign with this symbol, which means is proportional to. This tells us that we’re no longer strictly equating the left and right sides of this expression. Remember, a proportion tells how variables relate to each other. So we simply ignore all the constant unchanging values here. This includes two, 𝜋, 𝜇 naught, and the current 𝐼, since we’re working under the assumption that the current in the wire remains constant.
Now, let’s set all four of these values equal to one just to hold their place in the expression. And like that, we have our proportionality. It reads that 𝐵 is proportional to one divided by 𝑑. And another way to say this is that 𝐵 is inversely proportional to 𝑑 because 𝐵 is in the numerator of the left-hand side, while 𝑑 is in the denominator of the right-hand side. This means that as one quantity increases, the other must decrease and vice versa.
Now, to tell exactly how much the quantity is changed by, we can use the proportionality sort of like a formula by plugging in a factor that we know one of them changes by. Since we’re concerned with doubling the distance away from the wire, we’ll be multiplying 𝑑 by a factor of two. Now, with this factor of two in the denominator, notice that we can rewrite this as one divided by 𝑑 all times one-half. And to keep this new expression congruent with the general expression we devised up here, the left-hand side also needs to be multiplied by one-half.
Remember, increasing the denominator means we’re dividing by a bigger number, which corresponds to a decrease in overall value. This is what we mean by inversely proportional. Increasing 𝑑 causes a decrease in 𝐵. All this is to say that if we increase the distance from the wire by a factor of two, the strength of the magnetic field must decrease by a factor of two. So to answer this question, doubling the value of 𝑑 halves the value of 𝐵.
Finally, we’re ready to write out a relationship between 𝐵 one and 𝐵 two. Compared to 𝐵 one, 𝐵 two was measured at twice the distance away from the wire. And using this formula to devise this proportionality, we were able to find that the strength of the magnetic field 𝐵 two is half that of 𝐵 one, or 𝐵 two equals one-half times 𝐵 one.