A long straight wire of radius carries a steady current . The current is uniformly distributed across its cross-section. The ratio of the magnetic field at and from axis of the wire is ______.

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Solution

Q.
A long straight wire of radius $a$ carries a steady current $I$ . The current is uniformly distributed across its cross-section. The ratio of the magnetic field at $\frac{a}{2}$ and $2 a$ from axis of the wire is ______. [2024]

1 1 : 1

2 3 : 4

3 1 : 4

4 4 : 1

Ans.
(1)

Magnetic field inside wire,

$B_{inside} = \frac{μ_{0} J r}{2} = \frac{μ_{0} I r}{2 A} = \frac{μ_{0} I r}{2 π a^{2}}$

$r \to \frac{a}{2} \Rightarrow B_{inside} = \frac{μ_{0} I}{4 π a}$

Magnetic field outside, $B_{2} = \frac{μ_{0} i}{2 π r}$

$B_{2} = \frac{μ_{0} i}{2 π (2 a)} = \frac{μ_{0} i}{4 π a}$

$B_{inside} = \frac{μ_{0} I}{4 π a}$

So $B_{1} = B_{2} \Rightarrow 1 : 1$

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