Electric field on the axis of a ring-shaped charged conductor

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SUMMARY

The discussion centers on the calculation of the electric field along the axis of a ring-shaped charged conductor. A participant questions the omission of multiplying the cosine component by 2, suggesting that both horizontal components should be considered due to symmetry. However, it is clarified that the integration bounds from 0 to 2π inherently account for this symmetry, negating the need for an additional factor of 2. The correct approach ensures that the contributions from both halves of the ring are included in the final result.

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yashboi123
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Homework Statement
A ring-shaped conductor with radius a = 2.20 cm has a total positive charge Q = 0.130 nC uniformly distributed around it
Relevant Equations
E = F/q
1706663702409.png

Hello. I was wondering why do we not multiply cos(alpha) by 2. I believe we should do this since the y-components of the electric field cancel out, meaning there would be 2 x-components of the electric field(at least I think so). Currently, this derivation/answer only considers one horizontal component, not the other half.
 
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How does it only consider one horizontal component when the integration is from ##0## to ##2 \pi##?

The integration bounds take care of this “doubling up effect”.

If you did it from ##0## to ##\pi## you would certainly have to multiply by 2 at the end.
 
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