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.
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.
