# Non Uniform Electric Field Lines Problem

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1. Aug 26, 2015

### Abid Mir

Ok we know that the electric field(uniform or non uniform) is a conservative field.

Imagine three horizontal electric field lines in '+X' direction separated by unequal distances lets say line 1 and 2 is separated by distance 'a' and line 2 and 3 is separted by some distance 'b' such that line 1 and 3 are separated by distance 'a+b' where ' a<b ' ( non uniform electric field lines). Now imagine a rectangular loop which encloses half distance of ' a ' and again half distance of 'b'. Now if we calculate the total work done in moving a unit charge in the rectangular loop we see that the work done comes non zero as a<b and electric field in the region of a is greater than the region of b. That shows that these electric field lines cannot exist. BUT while drawing non uniform electric field we draw them the same way as these lines are drawn. So is our way of drawing non uniform electric field lines incorrect?

2. Aug 26, 2015

### BvU

I should think so. Remember that equipotential lines are perpendicular to electric field lines. How would you draw the equipotential lines in the situation you describe ?

3. Aug 26, 2015

### Abid Mir

Well u are talking about a different thing. Pls go through my question again

4. Aug 26, 2015

### BvU

No I am not talking about a different thing. You can't just imagine field lines. They have to satisfy certain conditions, precisely to constitute a conservative field.
Go back in your Griffiths and re-read the section Divergence and curl of electrostatic fields (3rd ed it's 2.2).

 In fact, in the next section (2.3 in mine) he explicitly adresses your conumdrum:
Boy, this Griffiths book is gold !

Last edited: Aug 26, 2015
5. Aug 26, 2015

### jerromyjon

I can't wait until I get to read one.

I just imagine it as a topological map, there has to be continuity in the elevations (potentials).

6. Aug 26, 2015

### Staff: Mentor

Yes, such lines cannot exist and if some illustration draws them that way then the illustration is wrong.

7. Aug 26, 2015

### Abid Mir

Omg i need to get my hands on the griffiths

8. Aug 26, 2015

### Korak Biswas

Will you please elaborate how you did the calculation? Probably then it will be easier to answer your question.

9. Aug 26, 2015

### Abid Mir

Integrate E.dl over the whole loop and work comes out to be positive.

10. Aug 26, 2015

### Korak Biswas

Ok. But what exactly do you mean by field lines? Are they lines of force? Do you define electrostatic field as the areal number density of field lines?

11. Aug 26, 2015

### BvU

Ah! My mistake. I encountered the reference to Griffiths in the other thread on the "magnetic force does no work" issue and didn't realize it was someone else who quoted from that book. So my reference to "your Griffiths" is misplaced.

12. Aug 26, 2015

### BvU

Are you familiar with the Gauss theorem ?

13. Aug 26, 2015

### Abid Mir

Yeah im

14. Aug 26, 2015

### Abid Mir

The
they represent electric field lines of force but non uniform

15. Aug 26, 2015

### Chandra Prayaga

Perhaps a diagram by you would make the situation clearer, but it looks like you are talking of an electrostatic field that is not possible. One of the conditions for an electrostatic field is that the curl of the field must be zero, which indeed translates into the statement that the work done around a closed loop must be zero. so if you are imagining field lines which give non-zero work over a closed loop, then that is an impossible electrostatic field

16. Aug 27, 2015

### Korak Biswas

I think pictorial representation of field lines is qualitative. Field lines of an electric field are nothing but collection of vectors at every point of space having the same direction of movement of a positive charge placed in that field. So actually field lines exist everywhere .There should not be any gap between them unless you make a part of the space shielded somehow (for instance with a hollow conducting sphere). But there is a way to define electrostatic field in terms of number of field lines. Electrostatic field at a point is defined as the number of field lines per unit area at that point. This is acceptable probably because areal density of field lines varies as $\frac{1}{r^2}$.To understand it keep a positive charge at the origin .Then large number of ($N \rightarrow\infty$) field lines will be 'emitted' from that point radially outwards. But at distance r the areal number density will be $\frac{N}{4\pi r^2}$. And as N (although very large) is constant (because there is no other source of field lines), you can say that areal number density of field lines goes as $\frac{1}{r^2}$. So you may identify it as electric field. So uniform electric field means uniform areal number density and non uniform field means non uniform areal number density of field lines. That's why we represent non uniform electric field with lines having non uniform distances in between them. This is not quantitative at all. If you want a quantitative representation, you have to take care of number density of field lines along with the direction at every point.