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Chapter 26: Problem 60

You and a friend get lost while hiking, so your friend pulls out a magnetic compass to get re-oriented. However, you're standing right under a power line carrying \(1.5 \mathrm{kA}\) toward magnetic north; it's \(10 \mathrm{m}\) above the compass. The horizontal component of Earth's magnetic field at your latitude points northward and has magnitude 0.24 G. Will the compass help you find your way?

Short Answer

Expert verified
After evaluating the above steps, a calculation gives \(B_{line} \approx 3.0 \times 10^{-5} T\). Comparing this value \(B_{line}\) with \(B_{earth} = 0.24 \times 10^{-4} T\), one can see \(B_{line}\) is less than \(B_{earth}\). Therefore, the compass will still be more influenced by the Earth's magnetic field and can help to find the way.

Step by step solution

01

Identify known and unknown quantities

In this Setup, the current in the power line \(I = 1.5 kA = 1500 A\), the distance from the power line \(r = 10 m\), the Earth's magnetic field \(B_{earth} = 0.24 G = 0.24 \times 10^{-4} T\). We want to find if the compass will work, which requires finding out the value of the magnetic field produced by the power line, say \(B_{line}\).
02

Calculate Magnetic Field from Power Line

We have to find the magnetic field caused by the power line. The formula for the magnetic field \(B\) at a distance \(r\) from a long, straight conductor carrying a current \(I\) is given by Ampere’s law as \(B = \mu_{0}I / 2\pi r\), where \(\mu_{0}\) is the permeability of free space and has a value of \(4\Pi \times 10^{-7} Tm/A\). Using this formula, the magnetic field \(B_{line}\) produced by the power line is given by \(B = \mu_{0}I / 2\pi r = (4\Pi \times 10^{-7} Tm/A) * (1500 A) / (2\pi *10 m)\)
03

Evaluate and Compare With Earth's Magnetic Field

Evaluate the above expression to find \(B_{line}\). If \(B_{line}\) is greater than \(B_{earth}\), then the compass will be more influenced by the magnetic field of the power line than by the Earth's magnetic field, and so it would not function properly. If \(B_{line}\) is less than \(B_{earth}\), then the compass will still point towards the Earth's magnetic north.

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