Electric current I passes through a cylindrical wire of...

Electric current $I$ passes through a cylindrical wire of sadius $a$. The current density at a point in the wire, proportional to the distance of the point from the'axis of the wire. The magnetic field at the surface of the wire is : $\underline{\mu_{0}} \underline{I}$ (b) $\mu_{0} l$ $\begin{array}{lll}\text { (a) } 2 \pi a & \text { (D) } 4 \pi a\end{array}$ (c) $\frac{2 \mu_{0} I}{\pi a}$ (d) none of these

Electric current $I$ passes through a cylindrical wire of sadius $a$. The current density at a point in the wire, proportional to the distance of the point from the'axis of the wire. The magnetic field at the surface of the wire is :
$\underline{\mu_{0}} \underline{I}$ (b)
$\mu_{0} l$
$\begin{array}{lll}\text { (a) } 2 \pi a & \text { (D) } 4 \pi a\end{array}$
(c) $\frac{2 \mu_{0} I}{\pi a}$
(d) none of these

Key Concepts

Ampere's Law

Ampere's Law states that the line integral of the magnetic field around any closed loop is equal to the permeability times the electric current enclosed by the loop. This principle is particularly useful in systems exhibiting symmetry, such as cylindrical conductors with non-uniform current distributions, to determine the magnetic field at various points.

Current Density

Current density is a measure of the electric current per unit area flowing through a conductor. When the distribution is non-uniform (for example, varying proportionally with distance from a central axis), calculating the effective current in a region requires integration over the area, taking into account the spatial variation of the current density.

Cylindrical Symmetry

Cylindrical symmetry is an important concept in electromagnetism where the physical properties of a system are invariant under rotations about or translations along the axis of a cylinder. This symmetry simplifies the mathematical analysis of the system, particularly when applying laws like Ampere's law to calculate magnetic fields.

Magnetic Field in Current-Carrying Conductors

Calculating the magnetic field produced by a current-carrying conductor involves understanding both the current distribution within the conductor and the geometry of the system. In cases of non-uniform current density, the magnetic field must be determined by integrating the contributions of differential current elements, often using Ampere's Law as a guiding principle.

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