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ON SYMMETRY OPERATIONS

In Physics, we often encounter systems with various symmetries. Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation. Consider, for example an infinite uniform sheet of charge (surface charge density σ) along the y-z plane. This system is unchanged if
(a) translated parallel to the y-z plane in any direction,
(b) rotated about the x-axis through any angle.

As the system is unchanged under such symmetry operation, so must its properties be. In particular, in this example, the electric field E vector must be unchanged.

Translation symmetry along the y-axis shows that the electric field must be the same at a point (0, y1, 0) as at (0, y2, 0). Similarly translational symmetry along the z-axis shows that the electric field at two point (0, 0, z1) and (0, 0, z2) must be the same.
By using rotation symmetry around the x-axis, we can conclude that E vector must be perpendicular to the y-z plane, that is, it must be parallel to the x-direction.

Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of the x-coordinate. It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space. The direction, however, is opposite of each other on either side of the sheet.

Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.
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