Question

Find the electric field a distance Z above the center of a circular loop of radius r , which carries a uniform line charge λ.

Answer 1

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Answer 2

Answer:

The electric Feild is E = λrz / 2ϵ0 ( z^2 + r^2 )^3 / 2

Explanation:

The radial component of the electric field is offset at each λ by the symmetry of the circle and the fact that the electric field is generated from the linear charge. This leaves the z component of the electric field. This can be calculated by performing the following integral (isn't it  a line integral? )

E = 1/4πϵo ∫ λ/ R^2. z/R dl.

Where R is the hypotenuse of a right triangle formed by the radius distance + the height from the line charge. To get the z component, we need to take the cos of this right triangle so that cos⁡θ is equal to z / r. dl is equal to the circumference of the circle, 2πr.

= λ/4πϵ0 ∫ 2πr/z^2 + r^2 .z/r

which therefore results as:

E = λrz / 2ϵ0 ( z^2 + r^2 )^3 / 2

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