Question

Consider a long U-shaped conducting tube whose entire inner surface carries a uniform positive charge density. At the bottom of the U, a point P is located at equal distances R from both vertical arms (measured horizontally). What is the electric field at point P?

Answer 1

$\displaystyle \mathbf{E} = \mathbf{0}$

Answer 2

Symmetry argument:
Each element of charged inner surface on the left arm produces an electric field at P that is exactly canceled by a corresponding element on the right arm (since P is equidistant from both arms).
Similarly, contributions from the curved bottom portion have no preferred horizontal direction and sum to zero at the center.
Hence, by superposition, the net electric field at P vanishes.