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Two identical circular loops 1 and 2 of radius R each have linear charge densities −λ and +λ C/m respectively. The loops are placed coaxially with their centres `Rsqrt3` distance apart. Find the magnitude and direction of the net electric field at the centre of loop 1.

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#### Solution

Magnitude of electric field at any point on the axis of a uniformly charged loop is given by,

`E=λ/(2∈_0) (rR)/(r^2+R^2)^(3/2) .....(i)`

where

R = Radius of the loop

r = Position of the point from the centre of the loop

λ = Linear charge density of the loop

Electric field at the centre of loop 1 due to charge present on it is zero. [ From (i), when r = 0 ]

`| vecE_1 | = 0 (As Z = 0)`

Electric field at a point outside the loop 2 on the axis passing normally is

`|vecE_2| = [lambdaR]/(2ε_0) . [Z]/[R^2 + Z^2]^(3/2)`

Since, Z = `Rsqrt3`

= `[lambdaR]/(2ε_0) . [Rsqrt3]/[(R^2 + 3R^2)^(3/2)]`

= `[lambdasqrt3]/[16ε_0R]` twowards right (As λ is positive)

So, net electric field at the centre of loop 1

`vecE = vecE_1 + vecE_2`

= 0 + `[lambdasqrt3]/[16ε_0R] = [lambdasqrt3]/[16ε_0R]`

This is the net electric field at the centre of loop 1 due to the charge on both the loops. The direction of this net field is from loop 2 to loop 1 as shown in the above figure.

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