Question

You are standing at a spot where the magnetic field of the Earth is horizontal, points due northward, and has magnitude \(40.0 \mu \mathrm{T}\). Directly above your head, at a height of \(12.0 \mathrm{~m},\) a long, horizontal cable carries a steady direct current of \(500 .\) A due northward. Calculate the angle \(\theta\) by which your magnetic compass needle is deflected from true magnetic north by the effect of the cable. Don't forget the sign of \(\theta\) - is the deflection eastward or westward?

Short answer

#tag_title# Calculating the magnetic field due to the cable #tag_content# To calculate the magnetic field due to the cable (B_cable), we use the formula: \(B_{cable} = \frac{\mu_0*I}{2*\pi*r}\) Substituting the given values: \(B_{cable} = \frac{(4 \pi × 10^{-7} \frac{Tm}{A}) * (500 A)}{2 * \pi * (12 m)}\) Now we simplify the expression: \(B_{cable} = \frac{2 \times 10^{-4} T}{12 m}\) \(B_{cable} = \frac{1}{6} × 10^{-4} T\) So, the magnetic field due to the cable is \(1.67 × 10^{-4} T\).

Step-by-step solution

Calculate magnetic field due to the cable

To calculate the magnetic field created by the cable, let's first consider the distance from the cable to the spot where you are standing (which is 12 meters below) and the magnitude of the current carried by the cable (which is 500 A). Using Biot-Savart Law, the magnetic field at a distance r from a long, straight wire carrying current I can be expressed as: \(B_{cable} = \frac{\mu_0*I}{2*\pi*r}\) Where, \(B_{cable}\) is the magnetic field due to the cable, \(\mu_0\) is the permeability of free space (\(4 \pi × 10^{-7} \frac{Tm}{A}\)), \(I\) is the current (500 A), and \(r\) is the distance from the cable (12 m). Now, calculate the magnetic field due to the cable.

Key concepts

Understanding Biot-Savart Law

The Biot-Savart Law is a critical concept in electromagnetism, providing a mathematical model for calculating the magnetic field produced by a steady electric current. In essence, it relates the magnetic field at a point in space to the current distribution that produces it.

For a long, straight conductor carrying a current, the Biot-Savart Law simplifies to a formula for the magnetic field \(B_{cable}\) at a distance \(r\) from the wire, which you can express as:
\[B_{cable} = \frac{\mu_0 * I}{2 * \pi * r}\]
Here, \(I\) represents the current through the conductor, \(\mu_0\) is the magnetic constant (also known as the permeability of free space), and \(\pi\) is the mathematical constant Pi — an intrinsic part of the circular geometry involved.

It's important to understand that this formula assumes an infinitely long wire and a vacuum surrounding it. While this may not reflect real-world situations perfectly, it's a close approximation that works well for problems like the textbook exercise.

Exploring Earth's Magnetic Field

Earth's magnetic field is like a giant bar magnet embedded within the planet; although it's far more complex due to the fluid nature of Earth’s outer core. At ground level, we measure the Earth's magnetic field in microteslas (µT), and it plays a vital role in navigation and animal migration.

Interestingly, at the location described in the exercise, Earth's magnetic field is horizontal and pointing due north with a magnitude of \(40.0 \mu \mathrm{T}\). The field direction gives us what we call 'magnetic north,' different from 'true north' (geographic North Pole) because Earth's magnetic poles do not align exactly with its axis of rotation.

The Earth's magnetic field, combined with other local magnetic influences, affects magnetic compasses. When doing field calculations, it’s essential to consider both the Earth’s inherent magnetic field and any additional fields, ensuring accurate measurements and orientations in navigation.

Correcting Magnetic Compass Deviation

The deviation of a magnetic compass refers to the error between the actual magnetic north and the indicated north on the compass. This deviation can be caused by local magnetic fields, such as the field generated by a current-carrying cable in the exercise.

Compass deviation adjustment is crucial for accurate navigation. To calculate the deflection caused by the cable's magnetic field, we assess the combined effect of Earth's magnetic field and the additional field due to the cable. It is computed using vector addition since magnetic fields are vector quantities that have both magnitude and direction.

Once the magnetic field due to the cable is calculated using the simplified Biot-Savart equation, the total magnetic field at the compass's location is found by vector addition of Earth's field and the cable's field. The angle \(\theta\) of compass needle deflection is then determined from the resultant field vector. Depending on the direction of the current and the positioning of the cable, the deflection \(\theta\) could be eastward or westward, aiding in correcting the compass reading for precise navigation.