A long straight wire of radius carries a steady current . The current is uniformly distributed over its cross-section. The ratio of the magnetic fields and , at radial distances and respectively, from the axis of the wire is [2016]

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Solution

Q.
A long straight wire of radius $a$ carries a steady current $I$ . The current is uniformly distributed over its cross-section. The ratio of the magnetic fields $B$ and $B'$ , at radial distances $a / 2$ and $2 a$ respectively, from the axis of the wire is [2016]

1 1

2 4

3 $\frac{1}{4}$

4 $\frac{1}{2}$

Ans.
(1)

Magnetic field at a point inside the wire at distance $r = (\frac{a}{2})$ from the axis of wire is

$B = \frac{μ_{0} I}{2 π a^{2}} r = \frac{μ_{0} I}{2 π a^{2}} \times \frac{a}{2} = \frac{μ_{0} I}{4 π a}$

Magnetic field at a point outside the wire at a distance $r = 2 a$ from the axis of wire is

$B' = \frac{μ_{0} I}{2 π r} = \frac{μ_{0} I}{2 π} \times \frac{1}{2 a} = \frac{μ_{0} I}{4 π a}$ $∴$ $\frac{B}{B'} = 1$

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