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 4๐œ‹ ร— 10โปโท Tโ‹…m/A for the value of ๐œ‡โ‚€. 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 ๐œ‹ times 10 to the negative seventh tesla meters per ampere for the value of ๐œ‡ 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 weโ€™ve drawn here. And weโ€™re told that this wire carries a current ๐ผ of 100 amperes. Thatโ€™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 weโ€™ll call that distance ๐‘‘, then we want to solve for the strength of the magnetic field at that distance. We can refer to that magnetic field as capital ๐ต.

And we can recall a mathematical relationship for the strength of the magnetic field a perpendicular distance ๐‘‘ from a current-carrying wire. Itโ€™s equal to a constant ๐œ‡ naught, the permeability of free space, multiplied by the current in the wire divided by two times ๐œ‹ times the distance from the wire ๐‘‘. Since our problem statement tells us ๐ผ and ๐‘‘ as well as the value to use for ๐œ‡ naught, we can substitute those given values into this equation.

Here, weโ€™ve used four ๐œ‹ times 10 to the negative seventh tesla meters per ampere for ๐œ‡ naught, 100 amperes for the current ๐ผ, and 0.06 meters for the distance ๐‘‘. All the units in this expression are already in the form weโ€™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. Weโ€™ll be left with an answer in units of teslas, which is good because weโ€™re calculating a magnetic field strength. When we compute ๐ต and we give our answer in scientific notation, keeping two decimal places, we find that itโ€™s equal to 3.33 times 10 to the negative fourth teslas. Thatโ€™s the strength of the magnetic field this perpendicular distance away from the wire.

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