Question Video: Describing the Relationship between Magnetic Field Strength- Current- And Distance Away from Straight Wire Physics • Third Year of Secondary School

Video Transcript

Which of the following correctly describes the relationship between 𝐵, 𝑟, and 𝐼, where 𝐵 is the magnetic field strength measured at a perpendicular distance 𝑟 away from a long straight wire carrying a constant current of 𝐼? (A) 𝐵 is proportional to 𝑟 divided by 𝐼. (B) 𝐵 is proportional to 𝑟 divided by 𝐼 squared. (C) 𝐵 is proportional to 𝐼 divided by the square root of 𝑟. (D) 𝐵 is proportional to 𝐼 divided by 𝑟. Or (E) 𝐵 is proportional to 𝐼 divided by 𝑟 squared.

Here, we need to identify the correct mathematical relationship between the current 𝐼 in a long straight wire and the magnetic field strength 𝐵 at a perpendicular distance 𝑟 away from the wire. To do this, let’s recall the formula for determining the strength of a magnetic field some distance away from a current-carrying wire. 𝐵 equals 𝜇 naught times 𝐼 divided by two 𝜋𝑟, where 𝜇 naught is a known constant whose value we actually don’t need to be concerned with right now. What’s important is that we know this equation correctly models the situation we’re given in this question. And because of this, we can use it to find the answer.

Notice though that the answer options are all written as statements of proportionality, not actual equations complete with equal signs. Remember, a proportionality tells us how variables in a formula change with respect to each other. So, we simply ignore the constant unchanging values 𝜇 naught, two, and 𝜋. That’s why those terms don’t even appear in any of the answer options.

Still, the formula has 𝐵 by itself on the left-hand side and so do all of the answer choices. So let’s begin to compare them. First, let’s notice that options (A) and (B) have 𝑟 in the numerator and 𝐼 in the denominator. This doesn’t make sense. For example, holding the current in the wire constant, we know that as we measure the magnetic field farther away from the wire, in which case we’d be increasing the value of 𝑟, the strength of the field decreases, not increases. Therefore, it’s incorrect for 𝑟 to be in the numerator here. It should be in the denominator so that an increase in distance away from the wire corresponds to a decrease in the magnetic field strength.

Further, if we measure the field strength while holding the distance from the wire constant and we increase the current in the wire, the field strength increases as well. Therefore, 𝐼 should not be in the denominator. It should be in the numerator because an increase in 𝐼 corresponds to an increase in 𝐵. For these reasons, we should eliminate options (A) and (B). Now, (C), (D), and (E) all do have 𝐼 in the numerator and 𝑟 in the denominator. What’s different in each one though is the exponent associated with 𝑟. To tell which is correct, let’s simply refer back to this formula.

Remember that in order to turn an equation into a statement of proportionality, we replace the equal sign with this symbol, which indicates that we’re no longer strictly equating the left and right sides, because we ignore all the constant terms in the expression.

So, dropping these constants, we now have that 𝐵 is proportional to 𝐼 divided by 𝑟. Here, the denominator simply includes 𝑟, not the square root of 𝑟 and not 𝑟 squared. So we should eliminate options (C) and (E). This leaves our final answer (D). And thus, we know that the correct relationship between 𝐵, 𝑟, and 𝐼 is that 𝐵 is proportional to 𝐼 divided by 𝑟.