Question Video: Calculating the Current in a Straight Wire given the Magnetic Field Strength Physics • Third Year of Secondary School

Video Transcript

A straight wire in an electrical circuit carries a direct current of 𝐼 amperes. The resulting magnetic field at a perpendicular distance of 18 millimeters from this wire is measured to be 1.2 times 10 to the negative four teslas. Calculate 𝐼 to the nearest ampere. Use four 𝜋 times 10 to the negative seven tesla meters per ampere for the value of 𝜇 naught.

Let’s begin by drawing a diagram. Here’s a section of the wire, which we know is carrying a current of 𝐼 amperes. This produces a magnetic field of strength 𝐵 measured at a perpendicular distance 𝑑 away from the wire. In this question, the magnetic field was measured at a distance of 18 millimeters from the wire. So this is our value of 𝑑. We also know that the strength of the magnetic field 𝐵 is measured as 1.2 times 10 to the negative four teslas. Of course, we don’t yet know the value of current in the wire, but we can recall a formula that relates 𝐼 to these values that we already do know. It’s the formula for the strength of the magnetic field due to a current in a straight wire, which reads 𝐵 equals 𝜇 naught times 𝐼 divided by two 𝜋𝑑.

We can use this formula to solve for the value of current in the wire. But first, we’ll have to rearrange it to make 𝐼 the subject. Let’s start by copying the formula. And then we’ll multiply both sides by two 𝜋𝑑 divided by 𝜇 naught so that all those terms can cancel from the right-hand side of the expression, leaving 𝐼 by itself. Now flipping this the other way and writing it a bit more neatly, we have 𝐼 equals two 𝜋 times 𝑑 times 𝐵 divided by 𝜇 naught. And because we’ve already been told the value of 𝜇 naught as well as 𝑑 and 𝐵, let’s go ahead and substitute them into the equation.

Okay, we have everything plugged in. But before we calculate, let’s take a moment to think about the units here. Notice that the values for 𝜇 naught and 𝐵 are expressed entirely in base SI units, but the value for distance is written in millimeters. So let’s convert it to plain meters. To do this, we should recall that the prefix milli- means 10 to the negative three. So we can essentially undo this prefix by moving the decimal point of the millimeter value three places to the left. So 18 millimeters becomes 0.018 meters.

Now it’s easier to see that units of meters and teslas in the numerator will cancel with meters and teslas in the denominator so that the only unit associated with this expression is inverse amperes in the denominator or just plain amperes in the numerator, which is a good sign because we are solving for a value of current after all.

Finally, let’s go ahead and plug this into a calculator, giving a result of 10.8 amperes. Now rounding to the nearest whole number, we found that the current in the wire is 11 amperes. To reach our final answer to this question, recall that we were told that the wire carries a current of 𝐼 amperes. We should be careful not to confuse this 𝐼 with the 𝐼 we encountered in the formula for the magnetic field. The goal of this question was to calculate 𝐼. But it’s important to understand that here 𝐼 is acting as a number placeholder in the value 𝐼 amperes. So because we found that 𝐼 amperes is 11 amperes, we found that 𝐼 is 11. This is our final answer.