# Video: Calculating the Strength of the Magnetic Field at a Distance from a Straight Cable

A long, straight cable in an industrial power plant carries a direct current of 100 A. Calculate the strength of the resulting magnetic field at a perpendicular distance of 0.06 m from this cable. Use 4𝜋 × 10⁻⁷ T⋅m/A for the value of 𝜇₀. Give your answer in scientific notation to two decimal places.

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### Video Transcript

A long, straight cable in an industrial power plant carries a direct current of 100 amperes. Calculate the strength of the resulting magnetic field at a perpendicular distance of 0.06 meters from this cable. Use four 𝜋 times 10 to the negative seventh tesla meters per ampere for the value of 𝜇 naught. Give your answer in scientific notation to two decimal places.

Alright, so in this example, we have a long, straight wire. We can imagine it going on even beyond the length that we’ve drawn here. And we’re told that this wire carries a current 𝐼 of 100 amperes. That’s a lot of current, but then again this is an industrial power plant. Our problem statement tells us that if we go a perpendicular distance of 0.06 meters from the wire, and we’ll call that distance 𝑑, then we want to solve for the strength of the magnetic field at that distance. We can refer to that magnetic field as capital 𝐵.

And we can recall a mathematical relationship for the strength of the magnetic field a perpendicular distance 𝑑 from a current-carrying wire. It’s equal to a constant 𝜇 naught, the permeability of free space, multiplied by the current in the wire divided by two times 𝜋 times the distance from the wire 𝑑. Since our problem statement tells us 𝐼 and 𝑑 as well as the value to use for 𝜇 naught, we can substitute those given values into this equation.

Here, we’ve used four 𝜋 times 10 to the negative seventh tesla meters per ampere for 𝜇 naught, 100 amperes for the current 𝐼, and 0.06 meters for the distance 𝑑. All the units in this expression are already in the form we’d like them. And we can see that when we calculate this fraction, the units of meters in numerator and denominator will cancel out, as will the units of amperes. We’ll be left with an answer in units of teslas, which is good because we’re calculating a magnetic field strength. When we compute 𝐵 and we give our answer in scientific notation, keeping two decimal places, we find that it’s equal to 3.33 times 10 to the negative fourth teslas. That’s the strength of the magnetic field this perpendicular distance away from the wire.