A long vertical wire carries an unknown current. Coaxial...

A long vertical wire carries an unknown current. Coaxial with the wire is a long, thin, cylindrical conducting surface that carries a current of 30 mA upward. The cylindrical surface has a radius of $3.0 \mathrm{~mm} .$ If the magnitude of the magnetic field at a point $5.0 \mathrm{~mm}$ from the wire is $1.0 \mu \mathrm{T},$ what are the (a) size and (b) direction of the current in the wire?

Key Concepts

Ampère’s Law

Ampère’s law relates the magnetic field around a closed loop to the net current that passes through the loop. It is expressed as the line integral of the magnetic field along a closed path being equal to ?? times the enclosed current. This principle is crucial for deriving the magnetic field of symmetric current distributions, such as a long straight wire, and is instrumental in solving problems where the current distribution is known or needs to be determined.

Magnetic Field of a Long Straight Current-Carrying Wire

The magnetic field produced by a long, straight current-carrying wire can be calculated using the formula B = (??I)/(2?d), where I is the current and d is the distance from the wire. This concept underlines how the magnetic field decreases with distance and is derived from Ampère’s law. It is a central idea when analyzing problems involving wire currents and magnetic field measurements.

Superposition Principle

The superposition principle in magnetism states that the net magnetic field at a given point is the vector sum of the magnetic fields produced by each individual current element. This concept is essential when dealing with multiple current sources, such as a wire and an additional current-carrying conductor, as it allows the individual contributions to be calculated and then combined to determine the overall magnetic effect.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. By pointing the thumb of the right hand in the direction of the conventional current, the curl of the fingers indicates the direction in which the magnetic field circulates. This tool is fundamental in deducing the orientation of magnetic fields relative to the current direction in various electromagnetic problems.

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