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bob012345

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In a previous thread* the field in a charged ring was discussed and it was shown to be not zero except at the center. In *post #45 a video is referenced that says the field diverges as one gets close to the ring and it was argued that at very close distances the field looks like an infinite line charge which diverges so this diverges.

The video asserts that as ##a=\frac{r}{R}→1## the field diverges. It is based on looking at the integral and making arguments it looks like an infinite line charge as you get close but the results of the integration eliminate the divergence. It is true K(k) the complete elliptical integral of the first kind approaches infinity as k→1 but the term in front of it (1−a) approaches zero faster. This seems right and numerically works as far as I can take it which was closer than one part in ##10^{−18}## of R from the ring but I'm not sure how to prove it works in the limit or if there is a surprise where K(k) starts to grow faster and the field does diverge. The exact field is here;

$$E_r = \frac{ k_e Q}{ \pi R^2} \left[\frac{1}{a} \frac{(1-a)}{(1+a)^3} \left((1-a)K(k) - (1+a)E(k)\right)\right],~~~a>0$$

This plot shows the exact field as points and the blue curve is a series approximation based on the exact form with many terms. The black curve is a point charge of the same total charge for reference.

*https://www.physicsforums.com/threa...-point-within-a-charged-circular-ring.1006381

The video asserts that as ##a=\frac{r}{R}→1## the field diverges. It is based on looking at the integral and making arguments it looks like an infinite line charge as you get close but the results of the integration eliminate the divergence. It is true K(k) the complete elliptical integral of the first kind approaches infinity as k→1 but the term in front of it (1−a) approaches zero faster. This seems right and numerically works as far as I can take it which was closer than one part in ##10^{−18}## of R from the ring but I'm not sure how to prove it works in the limit or if there is a surprise where K(k) starts to grow faster and the field does diverge. The exact field is here;

$$E_r = \frac{ k_e Q}{ \pi R^2} \left[\frac{1}{a} \frac{(1-a)}{(1+a)^3} \left((1-a)K(k) - (1+a)E(k)\right)\right],~~~a>0$$

**Edit: This is wrong due to an algebra mistake! There is a divergence after all and this video is correct! See below.**

These two plots are the wrong. Corrected plot is at the bottom of the thread.These two plots are the wrong. Corrected plot is at the bottom of the thread.

This plot shows the exact field as points and the blue curve is a series approximation based on the exact form with many terms. The black curve is a point charge of the same total charge for reference.

*https://www.physicsforums.com/threa...-point-within-a-charged-circular-ring.1006381

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