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Question

In figure, infinite conducting rings each having current $$i$$ in the direction shown are placed concentrically in the same plane as shown in the figure. The radii of rings are $$r,2r,{2}^{2}r,{2}^{3}r,.....\infty$$. The magnetic field at the center of the rings will be :

167093_976e387389a14323a7a2b37ae5a9b79e.png


A
zero
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B
μ0ir
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C
μ0i2r
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D
μ0i3r
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Solution

The correct option is C $$\cfrac { { \mu }_{ 0 }i }{ 3r } $$
$$B=\cfrac { { \mu  }_{ 0 }i }{ 2r } -\cfrac { { \mu  }_{ 0 }i }{ 2(2r) }
+\cfrac { { \mu  }_{ 0 }i }{ 2({ 2 }^{ 2 }r) } -......\infty $$
or, $$B=\cfrac
{ { \mu  }_{ 0 } i}{ 2r } \left[ 1-\cfrac { 1 }{ 2 } +\cfrac { 1 }{ { 2
}^{ 2 } } -\cfrac { 1 }{ { 2 }^{ 3 } } +........\infty  \right] $$
or, $$B=\cfrac
{ { \mu  }_{ 0 }i }{ 2r } \left[ \cfrac { 1 }{ 1-\left( -\cfrac { 1 }{ 2
}  \right)  }  \right] =\cfrac { { \mu  }_{ 0 } i}{ 3r } $$

Physics
NCERT
Standard XII

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