Explain why the divergence of each of the functions must be zero

[brad sue]

Homework Statement

the question is:
Without doing any calculations, explain why the divergence of each of the functions must be zero(Hint: consider what electric field these functions physically correspond to)
P.S: I make the unit vectors in bold.

A₁=x

A₂=ρ(1/ρ) with ρ>0

A₃=r(1/r²) with r>0

Homework Equations

I think we need to argue about the charge

The Attempt at a Solution

for A1 , the electric field of A! is a plane, which constant, so divergence is zero.

for A2, it seems to be a line of charge but I don know how to argue for the divergence to be zero.

Same for A3

Can someone see what is going on?
B

 

[Mindscrape]

Divergence shows the flux through some kind of area, so it indicates how much "stuff" flows in compared to out. This is basically going to be your argument for every single one, is that somehow as much stuff flows in as out.

Are the vectors rho and r supposed to represent spherical and cylindrical coordinates?

 

[m2003]

ecause these functions represent electric fields, which are conservative vector fields, their divergence must be zero. This is a consequence of the fundamental theorem of calculus, which states that for a conservative vector field, the line integral of the field over a closed loop is equal to zero. In other words, the net flow of the field into or out of a closed surface is zero. This is true for all three functions given, as they represent electric fields that are constant or decrease with distance. Additionally, since the functions are defined in terms of a single variable (x, ρ, or r), there is no variation in the field in the other dimensions, further supporting the idea that the divergence must be zero.