Question 247: Magnetism and Matter

Question

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 $$a/2$$ and $$2a$$ is

Accepted Answer

Here, current is uniformly distributed across the cross-section of the wire, therefore, current enclosed in the ampere-an path formed at a distance $${r_1}\left( { = {a \over 2}} ight)$$
$$ = \left( {{{\pi r_1^2} \over {\pi {a^2}}}} ight) 	imes I,$$ where $$I$$ is total current
$$	herefore$$ Magnetic field at $${P_1}$$ is
$${B_1} = {{{\mu _0} 	imes current\,\,enclosed} \over {path}}$$
$$ \Rightarrow {B_1} = {{{\mu _0} 	imes \left( {{{\pi r_1^2} \over {\pi {a^2}}}} ight) 	imes I} \over {2\pi {r_1}}}$$
$$ = {{{\mu _0} 	imes I{r_1}} \over {2\pi {a^2}}}$$
Now, magnetic field at point $${P_2},$$
$${B_2} = {{{\mu _0}} \over {2\pi }}.{I \over {\left( {2a} ight)}} = {{{\mu _0}I} \over {4\pi a}}$$
$$	herefore$$ Required ratio $$ = {{{B_1}} \over {{B_2}}} = {{{\mu _0}I{r_1}} \over {2\pi {a^2}}} 	imes {{4\pi a} \over {{\mu _0}I}}$$
$$ = {{2{r_1}} \over a} = {{2 	imes {a \over 2}} \over a} = 1.$$

Answer

$$1/2$$

Answer

$$1/4$$

Answer

$$4$$

Answer

$$1$$

Magnetism and Matter

A long straight wire of radius $a carries a steady current i. The current is uniformly distributed across its cross section. The ratio of the magnetic field at a/2 and 2a$ is

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