Single Choice

A circular current carrying coil has a radius $$R$$. The distance from the centre of the coil on the axis, where the magnetic induction will be $$\dfrac{1}{8}$$th to its value at the centre of the coil is:

A$$\dfrac { R }{ \sqrt { 3 } } $$
B$$R\sqrt { 3 } $$
Correct Answer
C$$2\sqrt { 3 } R$$
D$$\dfrac { 2 }{ \sqrt { 3 } } R$$

Solution

Here, the ratio, $$\dfrac { { B }_{ centre } }{ { B }_{ axis } } ={ \left( 1+\dfrac { { x }^{ 2 } }{ { R }^{ 2 } } \right) }^{ { 3 }/{ 2 } }$$
Also, $${ B }_{ axis }=\dfrac { 1 }{ 8 } { B }_{ centre }$$
$$\dfrac { 8 }{ 1 } ={ \left( 1+\dfrac { { x }^{ 2 } }{ { R }^{ 2 } } \right) }^{ { 3 }/{ 2 } }\Rightarrow 4=1+\dfrac { { x }^{ 2 } }{ { R }^{ 2 } } $$
$$3=\dfrac { { x }^{ 2 } }{ { R }^{ 2 } } \Rightarrow { x }^{ 2 }=3{ R }^{ 2 }$$
or, $$x=\sqrt { 3 } R$$

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