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Chapter 26: Problem 8

The Biot-Savart law shows that the magnetic field of a current element decreases as \(1 / r^{2} .\) Could you put together a complete circuit whose field exhibits this decrease? Why or why not?

Short Answer

Expert verified
No, it's not possible to put together a complete circuit whose magnetic field decreases as \(1 / r^{2} \) due to the vector nature of the magnetic field.

Step by step solution

01

Understand the Biot-Savart Law

The Biot-Savart law describes how the magnetic field decreases with distance from a current carrying element. The magnetic field \(B\) created by a current element \(I\) at a distance \(r\) from the element is inversely proportional to the square of the distance, hence \(B \propto 1/ r^{2}\).
02

Consider a Current in a Loop

Imagine a current \(I\) flowing in a loop. The magnetic field at the center of the loop can be calculated using the Biot-Savart law. In any point within the loop, the magnetic field decreases with the square of the distance from the current.
03

Consider the Magnetic Field of a Complete Circuit

For an entire circuit, things change because the magnetic field is a vector quantity and depends on the direction as well as the magnitude, not just the distance from the current. In a complete circuit, different current elements contribute to the magnetic field at a given point in space but from different directions, therefore a simple inverse square relationship for the whole circuit would not hold up.
04

Conclude

So, for a complete circuit, the magnetic field does not necessarily decrease as \(\ 1 / r^{2} \), as the magnetic fields resulting from different parts of the circuit can interfere constructively or destructively, depending on their directions and the location of the point in space. Thus, it's not possible to put together a complete circuit whose field exhibits only this decrease.

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Most popular questions from this chapter

A simple electric motor consists of a 220 -turn coil, \(4.2 \mathrm{cm}\) in diameter, mounted between the poles of a magnet that produces a 95 -mT field. When a 15 -A current flows in the coil, what are (a) the coil's magnetic dipole moment and (b) the motor's maximum torque?

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