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 from true magnetic north due to the current-carrying cable, you can use the right-hand rule for the magnetic field produced by a straight wire.
First, determine the magnetic field created by the current-carrying cable at your location. The magnetic field produced by a long straight wire is given by the equation:
B = μ₀ * I / (2πd)
Where B is the magnetic field, μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A), I is the current, and d is the perpendicular distance from the wire.
In this case, the current (I) is 381 A and the distance (d) is the height above the cable, which is 12.3 m. Plugging these values into the equation, we get:
B = (4π × 10⁻⁷ T·m/A) * (381 A) / (2π * 12.3 m)
Simplifying the equation, we have:
B = 1.23 × 10⁻⁵ T
So, the magnetic field produced by the cable at your location is 1.23 × 10⁻⁵ T.
Now, we can calculate the angle (θ) by which your magnetic compass needle is deflected. The deflection angle (θ) can be determined using the tangent of the deflection angle:
tan(θ) = B_earth / B_cable
Where B_earth is the magnitude of the Earth's magnetic field (36.9 μT or 36.9 × 10⁻⁶ T) and B_cable is the magnetic field produced by the cable (1.23 × 10⁻⁵ T).
Plugging in the values, we have:
tan(θ) = (36.9 × 10⁻⁶ T) / (1.23 × 10⁻⁵ T)
tan(θ) ≈ 0.3
To find the angle (θ), we can take the arctan of both sides:
θ ≈ arctan(0.3)
Using a calculator or math software, we find:
θ ≈ 16.7°
Therefore, the angle by which your magnetic compass needle is deflected from true magnetic north due to the effect of the cable is approximately 16.7°.