Three-Phase AC Magnetic Field

Three conductors carry the following AC currents:

I A = I s i n ( ω t ) I B = I s i n ( ω t 2 π / 3 ) I C = I s i n ( ω t + 2 π / 3 ) I_A = I \, sin(\omega t) \\ I_B = I \, sin(\omega t - 2 \pi / 3) \\ I_C = I \, sin(\omega t + 2 \pi / 3)

The currents flow into the page, and the conductor geometries are shown in the diagram. There is a test point to the right of the conductors.

The peak value of the magnetic flux density ( B ) (B) at the test point can be expressed as:

B m a x = α μ 0 I 2 π \large{B_{max} = \alpha \, \frac{\mu_0 \, I}{2 \pi }}

What is the value of α \alpha ?

Note: Neglect the thickness of the conductors


The answer is 0.220479.

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