Tension developed in a charged ring

AI Thread Summary
The discussion centers on calculating the electric field at the circumference of a charged ring, where tension in the ring is balanced by the electric field. The derived equation for tension is T = EQ/(2π), and the field due to an infinitesimal charge is expressed as dE = k(dq)/z². However, the integral to find the net electric field at the circumference does not converge, suggesting an infinite field, which raises questions about the calculations. The user also notes that calculating the field at any point in the plane of the ring proved too complex for computational tools. The divergence of the electric field at a line charge is acknowledged, indicating a deeper physical significance.
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Homework Statement
calculate the tension developed in ring of radius ##R##( of negligible thickness) and charge ##Q##
Relevant Equations
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1622133322336.png

consider a small element that subtends an angle ##2\Delta \theta## at the center of the ring. balancing the forces on this element gives:
(let the field due to the ring be at its circumference be ##E##).
$$2T\Delta \theta = E(dq) = E (\frac{Q}{2\pi})(2\Delta \theta)$$
$$T = \frac{EQ}{2\pi}$$
now the problem is reduced to finding the field due to the charged ring at its circumference:
1622133693717.png

let the distance from an infinitesimal charge ##(dq)## to the required point on the circumference be ##z##
$$z^2 = (Rsin\theta + R)^2 + (Rcos\theta)^2 = 2R^2(1+sin\theta)$$
$$(dE) = k(dq)/z^2 = \frac {k(dq)}{2R^2(1+sin\theta)}$$
by symmetry arguments the net field at the circumference will be only in the y direction:
$$(dE_{net}) = (dE)(cos\alpha)$$
$$\alpha = \pi/4 - \theta/2$$
$$(dq) = (Q/2\pi) d\theta$$
$$dE_{net} = k(dq)\frac{cos(\pi/4 - \theta/2)}{1+sin\theta} = \frac{kQ}{4\pi R^2} \frac{cos(\pi/4 - \theta/2)}{1+sin\theta} (d\theta)$$
$$E_{net} = \frac{kQ}{4\pi R^2} \int_0^{2\pi} \frac{cos(\pi/4 - \theta/2)}{1+sin\theta} d\theta$$

I put the integral into wolframalpha and it does not converge!
that would mean the field is infinite which obviously can't be true.
is there any physical significance to this or have i made a mistake in calculating the field at the circumference of the ring.

I had also tried to find the field at any general point in the plane of the ring and then wanted to find the field at circumference but the integral proved to be very complex (wolframalpha exceeded standard computation timeo_O) hence i didn't go down that path.
 
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