Angle of escaping electric field lines

[Matt Chu]

Homework Statement

Two charges 2q and -q are located at x = 0 and x = a respectively. There are field lines extending from the positive charge and lines going inwards to the negative charge. Some of these lines go from the positive charge to the negative, but some go off to infinity from the positive charge. The question asks to find the angle at which the lines go towards infinity. (It also mentions to use a "cleverly chosen Gaussian surface.)

Homework Equations

Gauss's law
∫ E⋅dA = q_en/ε₀

The Attempt at a Solution

I tried to maybe using Gauss's law on the two charges individually, then on both charges together. I'm not sure exactly what a "cleverly chosen Gaussian surface" is, so I don't know exactly how to proceed.

 

[haruspex]

Some of these lines go from the positive charge to the negative

How many, as a fraction of total?

 

[Matt Chu]

How many, as a fraction of total?

That's a part of the question, I guess. I assume it is related to the distance between the charges, but I'm not sure what the relationship is exactly.

 

[BvU]

Hello Matt,

Does this picture (borrowed from here) give you a hint ?

 

[haruspex]

That's a part of the question, I guess. I assume it is related to the distance between the charges, but I'm not sure what the relationship is exactly.

If you use a consistent relationship between field line density and field strength, how does the number of lines emerging from 2q compare with the number arriving at -q?

 

[BvU]

Let me give another hint: if the problem statement mentions infinity, then perhaps a cleverly chosen Gauss volume is allowed to extend to ininity as well

 

[Matt Chu]

If you use a consistent relationship between field line density and field strength, how does the number of lines emerging from 2q compare with the number arriving at -q?

Just from the definition of flux it's obvious that the electric field is twice as strong from the charge twice as strong. Therefore there are twice as many lines emerging from the 2q charge as there are going into the -q charge. But that doesn't necessarily tell me how many lines are going from the positive charge to the negative charge.

 

[haruspex]

But that doesn't necessarily tell me how many lines are going from the positive charge to the negative charge.

Why not?

 

[Matt Chu]

Let me give another hint: if the problem statement mentions infinity, then perhaps a cleverly chosen Gauss volume is allowed to extend to ininity as well

That's what I thought, to use a large sphere such that the two charges appear like one point charge and comparing that to the electric field found by using Gauss's law on the individual charges, but that doesn't really give me any relevant information.

 

[BvU]

How many E field lines cross the surface of a gaussian volume that contains zero charge ?

 

[haruspex]

That's what I thought, to use a large sphere such that the two charges appear like one point charge and comparing that to the electric field found by using Gauss's law on the individual charges, but that doesn't really give me any relevant information.

It gives me an answer.
So what fraction of the field lines from 2q go to -q?

 

[Matt Chu]

It gives me an answer.
So what fraction of the field lines from 2q go to -q?

It would seem that half of the field lines go from 2q to -q. But the angle at which they leave, can that be assumed to be constant regardless of their distance?

 

[haruspex]

It would seem that half of the field lines go from 2q to -q. But the angle at which they leave, can that be assumed to be constant regardless of their distance?

How will the field lines be arranged extremely close to the 2q charge?

 

[Matt Chu]

How will the field lines be arranged extremely close to the 2q charge?

They all extend outwards evenly in every direction.

 

[haruspex]

They all extend outwards evenly in every direction.

Right, and half go to the -q. So... ?

 

[Matt Chu]

Right, and half go to the -q. So... ?

The angle is 90 degrees from the horizontal?

 

[haruspex]

The angle is 90 degrees from the horizontal?

Yes.

 

[Matt Chu]

Yes.

Huh, that was easy. Lol

 

[haruspex]

Huh, that was easy. Lol

It would have been more interesting with a different ratio.

 

[BvU]

How many E field lines cross the surface of a gaussian volume that contains zero charge ?

Just for the record: was this a useful hint or just misleading ?

I wonder what the intended 'cleverly chosen gaussian volume' is...

(because the gaussian volume that actually has zero E field lines crossing it has a rather specific shape -- but a plane perpendicular to the line connecting the charges and cutting the 2q in half has a net $\int \vec E\cdot\vec {dS} = 0\$)