A long wire is known to have a radius greater than 4.0 mm and to carry a current that is uniform

step 1 For the given problem, we use a two equation, that is B1 is equal to mu nod i divide by 2 pi r s

A long wire is known to have a radius greater than 4.0 mm...

A long wire is known to have a radius greater than 4.0 $\mathrm{mm}$ and to carry a current that is uniformly distributed over its cross section. The magnitude of the magnetic field due to that current is 0.28 $\mathrm{mT}$ at a point 4.0 $\mathrm{mm}$ from the axis of the wire, and 0.20 $\mathrm{mT}$ at a point 10 $\mathrm{mm}$ from the axis of the wire. What is the radius of the wire?

A long wire is known to have a radius greater than 4.0 $\mathrm{mm}$ and to carry a current that is uniformly distributed over its cross section.
The magnitude of the magnetic field due to that current is 0.28 $\mathrm{mT}$
at a point 4.0 $\mathrm{mm}$ from the axis of the wire, and 0.20 $\mathrm{mT}$ at a point 10
$\mathrm{mm}$ from the axis of the wire. What is the radius of the wire?

Key Concepts

Ampere's Law

Ampere's Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. In the context of current-carrying wires, it is used to derive expressions for the magnetic field both inside and outside the cylindrical wire, making it a foundational principle for solving problems involving magnetic fields around conductors.

Uniform Current Distribution

Uniform current distribution means that the electric current is spread evenly across the cross-sectional area of the conductor. This assumption allows one to calculate the current density and subsequently determine the magnetic field at any point inside the conductor using symmetry arguments and Ampere’s Law.

Magnetic Field Inside a Conductor

Inside a cylindrical conductor with a uniform current distribution, the magnetic field increases linearly with the distance from the center. This behavior is derived using Ampere’s Law by considering an Amperian loop of radius smaller than the conductor’s radius, leading to an expression that incorporates the current density and the ratio of the enclosed current to the total current.

Magnetic Field Outside a Conductor

For points outside a current-carrying conductor, the entire current is enclosed by the Amperian loop, and the magnetic field follows an inverse relationship with distance from the wire. Ampere’s Law in this region yields the familiar expression for the magnetic field that decays as one over the distance from the conductor’s axis.

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