Question Video: Calculating the Magnetic Field at a Point around a Constant Electric Current

A typical current in a lightning bolt is 4.0 × 10⁴ A. Estimate the magnetic field 5.0 m from the bolt.

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Video Transcript

A typical current in a lightning bolt is 4.0 times 10 to the fourth amperes. Estimate the magnetic field 5.0 meters from the bolt.

Say then that we have a lightning bolt moving through the sky. And this bolt is essentially a gigantic electric current, movement of charge from the sky to the Earth. Since the bolt is an electric current, that means it creates a magnetic field around it.

With that field moving in circles around the axis of this lightning bolt, the question asks if we were to move a distance of five meters away from the axis of this lightning bolt, what would the magnetic field at that location be?

In particular, we’re asked to estimate the value of the field at this location. That’s helpful because if we were to treat this lightning bolt as though it went on forever and ever, that is, it was a line that kept going on and on, then under that assumption, we can recall an equation which will tell us exactly what the magnetic field is a certain distance away from the axis of the lightning bolt.

For an infinitely long straight wire carrying a current 𝐼, the magnetic field a given distance 𝑟 away from the axis of that wire is equal to the permeability of free space 𝜇 naught times the current, all divided by two 𝜋 times the distance 𝑟.

The constant 𝜇 naught the permeability of free space we can treat as exactly 1.26 times 10 to the negative sixth tesla meters per ampere. Knowing that as well as the effective current of this lightning bolt and the distance from the bolt at which we want to solve for the magnetic field, we have everything we need to plug in and solve for that magnetic field 𝐵.

With those values in, notice that the units of meters as well as the units of amperes cancel out, leaving us with units of tesla, the units of magnetic field.

This fraction is equal to 1.6 times 10 to the negative third tesla. That’s the approximate magnetic field’s strength at distance of 5.0 meters from this lightning bolt.

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