A coil having turns is wound tightly in the form of a spiral with inner and outer radii and respectively. When a current passes through the coil, the magnetic field at the center is [2001]

Question

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 center is [2001]

Smart Achievers · Accepted Answer

(3)

Let us consider an element of thickness $a$ of wire. Let it be at a distance $x$ from the centre $O$ .

$Number of turns per unit length = \frac{N}{b - a}$

$∴$ $Number of turns in thickness d x = \frac{N}{b - a} d x$

Magnetic field due to this small element at $O$ ,

$d B = \frac{μ_{0}}{2} \cdot \frac{N I}{(b - a)} \cdot \frac{d x}{x}$

$B = \int_{a}^{b} \frac{μ_{0}}{2} \cdot \frac{N I}{(b - a)} \cdot \frac{d x}{x} = \frac{μ_{0}}{2} \cdot \frac{N I}{(b - a)} \int_{a}^{b} \frac{d x}{x}$

$= \frac{μ_{0}}{2} \cdot \frac{N I}{(b - a)} {[\log_{e} x]}_{a}^{b}$

$∴$ $B = \frac{μ_{0}}{2} \frac{N I}{(b - a)} \ln (\frac{b}{a})$

Solution

Q.
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 center is [2001]

1 $\frac{μ_{0} N I}{b}$

2 $\frac{2 μ_{0} N I}{a}$

3 $\frac{2 μ_{0} N I}{2 (b - a)} \ln (\frac{b}{a})$

4 $\frac{2 μ_{0} N I}{2 (b - a)} \ln (\frac{a}{b})$

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Solution

Q. 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 center is [2001]

1 μ0NIb

2 2μ0NIa

3 2μ0NI2(b-a)ln(ba)