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Chapter 26: Problem 7

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 is not possible to construct a complete current-carrying circuit whose magnetic field at any point decreases as \(1 / r^{2}\) with distance from the circuit.

Step by step solution

01

Understanding The Biot-Savart law

The Biot-Savart law gives the magnetic field produced by a current element in a current carrying conductor. The magnetic field magnitude due to a differential current element varies inversely with the square of the distance from the element.
02

Analyzing the Requirement

The task is to construct a complete circuit that exhibits a magnetic field that decreases at the rate of \(1 / r^{2}\) as the distance 'r' from the current element increases. Since the question asks for a complete circuit, it means looking for a situation where the magnetic field at any point due to the entire circuit decreases with \(1 / r^{2}\).
03

Conclusion of the Analysis

The decrease in magnetic field with \(1 / r^{2}\) is true only for small differential current element. However, for a complete current carrying conductor or circuit, the magnetic field does not decrease as \(1 / r^{2}\). The decrease in magnetic field depends on the geometry of the circuit. Therefore, it's not possible to put together a complete circuit whose magnetic field exhibits this decrease as \(1 / r^{2}\). The superposition principle is applied for whole circuits and this involves integration over the circuit which, depending on the geometry of the circuit, does not result in a \(1 / r^{2}\) relationship.

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