You are standing at a spot where the magnetic field of the Earth is horizontal, points due northward, and has magnitude 36.9 μT. Directly above your head, at a height of 12.3 m, a long, horizontal cable carries a steady DC current of 381 A due northward. Calculate the angle θ by which your magnetic compass needle is deflected from true magnetic north by the effect of the cable.
To calculate the angle by which your magnetic compass needle is deflected, we need to use the formula for the magnetic field produced by a straight current-carrying wire.
The formula is given as:
B = (μ₀ * I) / (2π * r)
Where:
- B is the magnetic field produced by the wire,
- μ₀ is the permeability of free space (4π × 10^-7 T·m/A),
- I is the current flowing through the wire,
- r is the distance between the wire and the point where we want to calculate the magnetic field.
In this case, the distance from the wire to the point where your compass needle is located is 12.3 m.
Plugging in the known values:
B = (4π × 10^-7 T·m/A * 381 A) / (2π * 12.3 m)
Simplifying the equation:
B = (4π × 10^-7 * 381) / 24.6 T
B ≈ 2π × 10^-5 T
Now, we can use the trigonometric relationship between the angle of deflection (θ) and the magnetic field strengths (B) as follows:
tan(θ) = (Bcompass / Bwire)
where Bcompass is the magnetic field at your location of 36.9 μT (microtesla).
Let's calculate θ:
θ = arctan(Bcompass / Bwire)
Converting Bcompass to Tesla:
Bcompass = 36.9 μT = 36.9 × 10^-6 T
Plugging in the values:
θ = arctan((36.9 × 10^-6 T) / (2π × 10^-5 T))
θ ≈ arctan(0.0185)
Finally, using a calculator or function, we can find:
θ ≈ 1.06 degrees
Therefore, your magnetic compass needle is deflected approximately 1.06 degrees from true magnetic north by the effect of the cable.