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 due to the effect of the cable, you can use the Biot-Savart Law. This law determines the magnetic field produced by a current-carrying wire at a given distance.

The formula for the magnetic field produced by a long straight wire at a perpendicular distance is given by:

B = (μ₀ * I) / (2 * π * 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 from the wire.

In this case, you need to find the magnetic field produced by the cable at a height of 12.3 m.

Using the given information:
I = 381 A (current flowing through the wire),
r = 12.3 m (distance from the wire),
μ₀ = 4π × 10^(-7) T*m/A (permeability of free space).

Plugging in these values into the formula, we get:

B = (4π × 10^(-7) T*m/A * 381 A) / (2π * 12.3 m)

Simplifying this equation, we find:

B ≈ 0.0617 T

Now, the angle θ by which your compass needle is deflected can be determined using the magnitude and directions of the Earth's magnetic field and the magnetic field of the cable.

Given that the Earth's magnetic field is horizontal, points directly northward, and has a magnitude of 36.9 μT, we need to calculate the angle θ.

Using the tangent function:

tan(θ) = (B_cable / B_earth)

Plugging in the values:

tan(θ) = (0.0617 T / 36.9 × 10^(-6) T)

Now, calculate the angle θ by taking the arctangent of both sides of the equation:

θ = arctan(0.0617 T / 36.9 × 10^(-6) T)

Using a calculator, the angle θ is found to be approximately:

θ ≈ 0.1006 radians

Converting this to degrees:

θ ≈ 5.77 degrees

Therefore, your magnetic compass needle would be deflected by approximately 5.77 degrees from true magnetic north due to the effect of the cable's magnetic field.