How would the electric field vector vary at a large distance

[Naomi13]

[ Moderator note- Edited to re-insert formatting template headers]

Hi guys,
I am stuck at this problem,

Homework Statement

Here it is given that an insulating sphere of radius a, carries a charge density ρ=ρ'( a^2-r^2)cosθ, when r <a. How will the leading order term for the electric field at a distance d, far away from this charge distribution vary?

ρ' is a constant term.

Homework Equations

The Attempt at a Solution

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I was thinking of calculating the total charge first. But here on integrating the given charge density I'm ending up getting zero.
∫ρ dτ= 0. Because dτ=r^2 sinθ dθ dφ dr
I'm unable to understand how to approach the sum, if I'm getting the total charge 0 in the first place?
The answer is that the field must vary by d^-1

 

[BvU]

Hello Naomi,

(Please use and do not erase the homework template -- it's good for you too ! :smile)

Your thinking is just fine. Charge zero does not mean field zero, though:

If the total charge is zero, you might want to look at the subsequent order: after all, a + charge at (0,0,1) and a - charge at (0,0,-1) do cause an electric field, even further away...

 

[Naomi13]

Hello Naomi,

(Please use and do not erase the homework template -- it's good for you too ! :smile)

Your thinking is just fine. Charge zero does not mean field zero, though:

If the total charge is zero, you might want to look at the subsequent order: after all, a + charge at (0,0,1) and a - charge at (0,0,-1) do cause an electric field, even further away...

Hello! Oh I'm so sorry, I'm new so I didn't know.
Well if the charge is zero, how does the field exist? Because the expression of electric field itself has the "Q" term. The answer is that the electric field must vary by d^-1. I'm still not able to solve :((

 

[BvU]

Well if the charge is zero, how does the field exist?

The blue characters are clickable. Did you look there ?

a + charge at (0,0,1) and a - charge at (0,0,-1) do cause an electric field, even further away

Make a drawing and see that the two contributions to the field at some point do not cancel.

The answer is that the electric field must vary by d^-1

Is that so ? How do you know ?

 

[Naomi13]

The blue characters are clickable. Did you look there ?
Make a drawing and see that the two contributions to the field at some point do not cancel.

Is that so ? How do you know ?

Well the answer is provided in my book.

 

[BvU]

Well, if the book says so, who knows . Let's try and find out if it's correct.

What about the remainder of post #4, the other two items ?