Assume a uniformly charged ring of radius R and charge Q produces an electric fi

Question

Assume a uniformly charged ring of radius R and charge Q produces an electric field Ering at a point P on its axis, at distance x away from the center of the ring as in Figure a. Now the same charge Q is spread uniformly over the circular area the ring encloses, forming a flat disk of charge with the same radius as in Figure b. How does the field Edisk produced by the disk at P compare with the field produced by the ring at the same point?

Assume a uniformly charged ring of radius R and charge Q produces an electric field Ering at a point P on its axis, at distance x away from the center of the ring as in Figure a. Now the same charge Q is spread uniformly over the circular area the ring encloses, forming a flat disk of charge with the same radius as in Figure b. How does the field E disk produced by the disk at P compare with the field produced by the ring at the same point?

Explanation / Answer

Using standard expressions, we get

E_ring = kxQ / ( R^2 + x^2)^(3/2)

similarly,

E_disk = 2kQ / R^2 * [ 1 - x / (x^2 +R^2)^(1/2) ]

Comparing them we get

Er/ Ed = x / ( R^2 + x^2)^(3/2) * R^2 / 2 [ 1 - x / (x^2 +R^2)^(1/2) ]

= x / [ 2R {1 + x^2 / R^2}^1.5 * {1 - x / (x^2 +R^2)^(1/2)} ]

Now comparing the numerator and denominator,

we get

E_ring < E_disk

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