Video: Finding the Strength of a Uniform Magnetic Field from the Force Experienced by a Current-Carrying Wire

When positioned at 90° to a magnetic field, a wire of length 1 m carrying a current of 4 A experiences a force of 0.2 N. What is the strength of the magnetic field?

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Video Transcript

When positioned at 90 degrees to a magnetic field, a wire of length one meter carrying a current of four amperes experiences a force of 0.2 newtons. What is the strength of the magnetic field?

All right, so in this example, we have a length of wire. This length is one meter long. And we’re also told that the wire carries a current of four amperes. And then, in addition to this, this current-carrying wire is positioned at 90 degrees to a magnetic field. So we could draw magnetic field lines that look like this, where these lines, as we’re told in the problem statement, are perpendicular to our wire. For our purposes, whether the field lines point straight up as we’ve drawn them or if they pointed straight down doesn’t make a difference so long as the field lines are at 90 degrees to our wire.

We’re told that, in this magnetic field, our wire experiences a force, and it has a magnitude of 0.2 newtons. Knowing all this, we want to solve for the strength of the magnetic field that the current-carrying wire is in. We can call that magnetic field strength capital 𝐵. And let’s say that the force the wire experiences in the field at 0.2 newtons is 𝐹 sub 𝐵. So we have all this information, but we need some way to tie it all together. We can do this by recalling a mathematical expression for the magnetic force experienced by a current-carrying wire in a magnetic field. That force is equal to the strength of the magnetic field 𝐵 times the magnitude of the current running through the wire multiplied by the length of wire exposed to the magnetic field.

Now, in our case, it’s not 𝐹 sub 𝐵 we want to solve for, but rather the magnetic field strength 𝐵. So to rearrange and solve for that, let’s divide both sides of the equation by 𝐼 times 𝐿 causing those two terms to cancel out on the right-hand side. So then, the magnetic field is equal to the magnetic force divided by the current times the length of the wire in the field. We can now apply this relationship using the numbers we’ve been given. 𝐵 is equal to 0.2 newtons divided by four amperes times one meter.

And if we consider just the units for a moment, a newton per ampere meter, that’s equal to what’s called a tesla, symbolized capital 𝑇. This is the SI base unit of magnetic field strength. So 𝐵 is equal to 0.2 divided by four times one tesla, which is equal to 0.05 teslas. That’s the strength of the magnetic field that the current-carrying wire is in.

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