A coil having N turns is wound tightly in the form of a...

A coil having N turns is wound tightly in the form of a spiral with inner and outer radii "a" and "b" respectively. When a current I passes through the coil, the magnetic field at the centre is (a) $\left[\left(\mu_{0} \mathrm{NI}\right) / \mathrm{b}\right]$ (b) $\left[\left(2 \mu_{\circ} \mathrm{NI}\right) / \mathrm{a}\right]$ (c) $\left[\left(\mu_{0} \mathrm{NI}\right) /\{2(\mathrm{~b}-\mathrm{a})\}\right] \ln (\mathrm{ab})$ (d) $\left[\mu_{0} /\{2 \mathrm{NI}(\mathrm{b}-\mathrm{a})\}\right] \ln (\mathrm{ab})$

A coil having N turns is wound tightly in the form of a spiral with inner and outer radii "a" and "b" respectively. When a current I passes through the coil, the magnetic field at the centre is
(a) $\left[\left(\mu_{0} \mathrm{NI}\right) / \mathrm{b}\right]$
(b) $\left[\left(2 \mu_{\circ} \mathrm{NI}\right) / \mathrm{a}\right]$
(c) $\left[\left(\mu_{0} \mathrm{NI}\right) /\{2(\mathrm{~b}-\mathrm{a})\}\right] \ln (\mathrm{ab})$
(d) $\left[\mu_{0} /\{2 \mathrm{NI}(\mathrm{b}-\mathrm{a})\}\right] \ln (\mathrm{ab})$

Key Concepts

Spiral Coil Geometry

Spiral coil geometry refers to a coil where current-carrying turns are arranged in a spiral form, with the radius of each turn changing continuously from an inner limit to an outer limit. This geometry requires careful consideration of how the magnetic field contributions vary with radius, leading to integrals that capture the overall effect from the entire spiral.

Integration over a Continuous Distribution

In many practical situations, current-carrying elements are not discrete but continuously distributed, as in the case of a spiral coil with changing radius. Integration is used to accumulate the contributions from each infinitesimal segment of the coil to obtain the overall magnetic field, often resulting in expressions that involve logarithmic functions.

Biot–Savart Law

The Biot–Savart Law is instrumental in calculating the magnetic field produced by a small element of current. By summing (integrating) contributions from each such differential element, it provides the theoretical basis for determining the magnetic field due to a current distribution, including those in loops and coils.

Magnetic Field of a Circular Current Loop

This concept involves the result from electromagnetism that a circular loop carrying a steady current produces a magnetic field at its center given by the formula B = ??I/(2r), where r is the radius of the loop. It is fundamental when analyzing magnetic fields in systems containing current loops and is often the starting point for more complex geometries.

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