Question 224: Magnetism and Matter

Question

Two identical wires $$A$$ and $$B,$$ each of length $$'l'$$, carry the same current $$I$$. Wire $$A$$ is bent into a circle of radius $$R$$ and wire $$B$$ is bent to form a square of side $$'a'$$. If $${B_A}$$ and $${B_B}$$ are the values of magnetic fields at the centres of the circle and square respectively, then the ratio $${{{B_A}} \over {{B_B}}}$$ is:

Accepted Answer

Case (a) :
$${B_A} = {{{\mu _0}} \over {4\pi }}{I \over R} 	imes 2\pi $$
$$ = {{{\mu _0}} \over {4\pi }}{I \over {\ell /2\pi }} 	imes 2\pi $$  $$\,\,\,\,\,\,\,\,\,$$ $$\left( {2\pi R = \ell } ight)$$
$$ = {{{\mu _0}} \over {4\pi }}{I \over \ell } 	imes {\left( {2\pi } ight)^2}$$
Case (b) :
$${B_B} = 4 	imes {{{\mu _0}} \over {4\pi }}{I \over {a/2}}\,\,\,$$ $$\left[ {\sin \,\,{{45}^ \circ } + \sin \,\,{{45}^ \circ }} ight]$$
$$ = 4 	imes {{{\mu _0}} \over {4\pi }} 	imes {I \over {\ell /8}} 	imes {2 \over {\sqrt 2 }}$$
$$ = {{{\mu _0}I} \over {4\pi \,\ell }} 	imes oot {32} \of 2 \,\,\,\,\,\,\left[ {4a = 1} ight]$$

Answer

$${{{\pi ^2}} \over {16}}$$

Answer

$${{{\pi ^2}} \over {8\sqrt 2 }}$$

Answer

$${{{\pi ^2}} \over {8}}$$

Answer

$${{{\pi ^2}} \over {16\sqrt 2 }}$$

Magnetism and Matter

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