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vcsharp2003

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- Homework Statement
- A disk that is very thin and uniformly charged has a non-zero electric field at its center. Explain why this statement is true or false.

- Relevant Equations
- ##E = \dfrac {\sigma} {2 \epsilon_{0}} [1 - \dfrac {x} {(R^2 + x^2)^{\frac {1}{2}}} ]##, where

##E## is electric field strength at a point P on the axis of the disk

##x## is distance of point P from the center of the sphere ( axis is perpendicular to the disk and passing through its center)

##R## is radius of disk

##\sigma## is the uniform charge density per unit area of the thin disk

##\epsilon_{0}## is electrical permittivity of vacuum

The electric field strength at the center of a uniformly charged disk should be zero according to symmetry of concentric rings about the center, where each ring is contributing to the electric field at the center of the disk.

For a thin ring of uniform charge distribution the formula is ##E = \dfrac {1} {4 \pi \epsilon_{0}} \dfrac {Qx} {(R^2 + x^2)^{\frac {3}{2}}}##, where the electric field ##E## is at a point P that is a distance ##x## from the center of the ring and along the ring's axis. When we consider the center of the ring, then ##x =0## which gives us ##E = 0## at the center of the ring.

Thus, each concentric ring will contribute ##0## to the electric field at the center of the thin disk. Consequently, the electric field at the center of the thin disk must be ##0##.

However, I do see from the formula for a thin charged disk as given under the relevant equations, that the electric field at the center of a disk is found to be ##E = \dfrac {\sigma} {2 \epsilon_{0}}## when we substitute ##x =0## in the mentioned formula.

I am unable to understand the flaw in my logic.

For a thin ring of uniform charge distribution the formula is ##E = \dfrac {1} {4 \pi \epsilon_{0}} \dfrac {Qx} {(R^2 + x^2)^{\frac {3}{2}}}##, where the electric field ##E## is at a point P that is a distance ##x## from the center of the ring and along the ring's axis. When we consider the center of the ring, then ##x =0## which gives us ##E = 0## at the center of the ring.

Thus, each concentric ring will contribute ##0## to the electric field at the center of the thin disk. Consequently, the electric field at the center of the thin disk must be ##0##.

However, I do see from the formula for a thin charged disk as given under the relevant equations, that the electric field at the center of a disk is found to be ##E = \dfrac {\sigma} {2 \epsilon_{0}}## when we substitute ##x =0## in the mentioned formula.

I am unable to understand the flaw in my logic.

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