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I had a question on symmetry operations that exactly resembles this post. The selected answer there mentions the required symmetry operation to be scale symmetry, and says:

An infinite plate looks the same no matter how far away from it you are.

However, an infinite linear charge configuration would also look the same no matter how far away from it we are, but electric field falls off as $1/r$ and does not remain constant. So, is the answer valid? If yes, why is it not applicable for the linear charge configuration? If no, what is the correct symmetry operation for thinking about the non-dependence on $r$ of the electric field due to an infinite plate of uniform surface charge density?

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The statement

An infinite plate looks the same no matter how far away from it you are.

is not sufficient to conclude that the field is constant. This argument is trying to handwavily get you to accept that the physical situation is scale invariant; in other words, if you send all the lengths $x\rightarrow \lambda x$, that the field will not change. This is not correct for the line of charge or the point charge. You know it can't be, since in three dimensions the force between two point charges depends on their distance.

Really, we should use Gauss's law to determine the falloff of a symmetric charge distribution.

A sheet of charge is like point particle in one spatial dimension, whereas a line of charge is like a point particle in two spatial dimensions, and a point charge is, well, a point particle in three spatial dimensions. In all of these cases, applying Gauss's law we get

$$ E A_d = \frac{Q}{\epsilon_0} $$ where $A_d$ is the area of a $d$-dimensional sphere \begin{eqnarray} A_d &=& 2 \ \ \ \ \ \ \ \ \ \ (d=1) \\ &=& 2\pi r \ \ \ \ \ \ (d=2) \\ &=& 4\pi r^2 \ \ \ \ (d=3) \end{eqnarray} As a result, $$ E = \frac{Q}{A_d \epsilon_0} $$ and the $r$ dependence comes in through the factor $A_d$ in the denominator. It is constant for $d=1$, falls off as $1/r$ for $d=2$, and falls off as $1/r^2$ for $d=3$.

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  • $\begingroup$ Thanks for the answer—I learnt something I had no idea about, but is there any symmetry operation that we can use to deduce the independence of the field on the x-coordinate? I'm asking this because that's what my textbook asks. $\endgroup$
    – archie
    Commented Jan 9 at 17:29

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