# Why field lines must begin or end on electric charges?

• pc2-brazil
In summary, using Gauss' law, it can be concluded that field lines must begin or end on electric charges because if they did not, changing the size of a Gaussian surface could result in a violation of Gauss' law, where the flux does not remain constant. This reasoning applies to both positive and negative charges, and can also be extended to cases with multiple charges or continuous charge distributions.
pc2-brazil

## Homework Statement

Use Gauss' law to explain why field lines must begin or end on electric charges.

## Homework Equations

Gauss' law:
$$\epsilon_0 \phi_E=q$$

## The Attempt at a Solution

I'm not sure whether this solution is incomplete or inconsistent.

First, I take a gaussian surface enclosing one positive charge +q. There, фE > 0. So, field lines are outwards from the gaussian surface. If I change the size of the surface, the flux must remain constant. If a field line finished abruptly, it would be possible to change the size of the gaussian surface for the line not to cut it; this would change the flux, violating Gauss' law.
If the field lines started from a point different from the positive charge, changing the size of the gaussian surface would also make it possible not to enclose some line, also changing the total flux. This reasoning also applies to the case where the charge inside the surface is negative.

Now, suppose that the surface encloses a negative charge -Q along with the positive charge +q:

If q = Q, as the charges will cancel each other out, the net enclosed charge will be zero, therefore Gauss' law states that the flux will be zero too. So, the number of field lines leaving a surface must be equal to the number of field lines entering the surface. And the field lines leaving the positive charge must be entering the negative charge. Otherwise, if a line finished in a point inside the surface which is not on any charge, it would be possible to modify the gaussian surface in order to change the flux even though it is still enclosing the two charges.

I'm not sure how to proceed if q is different from Q:
I think that, if q > Q, following Gauss' law, the flux will be positive but smaller than it was in the situation where there is only the positive charge +q. It means that some lines which come from the positive charge are not leaving the surface. These lines must be entering the negative charge, or else they would have to break in the middle of the way and changing the surface would change the flux. I think that this same reasoning can be applied if q < Q; the difference is that the flux will be negative (therefore, there will be more lines entering the surface than lines leaving it).

I think your solution is convincing, but not absolutely rigorous. What if there's 3 charges, or a continuous charge distribution? I would approach the problem by identifying the starting or ending point of a field line, and drawing a Gaussian sphere around the start/end point. If I shrink the sphere, the charge enclosed approaches 0 no matter what the charge distribution looks like, but the flux doesn't. This violates Gauss' law.

## 1. Why do electric field lines begin or end on charges?

Electric field lines begin or end on charges because electric charges create electric fields. These fields represent the force that a charge will experience when placed in that location. The direction of the field lines indicates the direction of the force that a positive charge would experience if placed in that field. Therefore, since charges create electric fields, the field lines must begin or end on charges.

## 2. How do electric field lines show the strength of an electric field?

Electric field lines show the strength of an electric field by their density. The closer the lines are to each other, the stronger the electric field is in that region. This is because the electric field lines represent the direction and magnitude of the electric field at any given point. The more lines there are in a certain area, the stronger the electric field is in that area.

## 3. What happens to the electric field lines around a charge when another charge is introduced?

When another charge is introduced, the electric field lines around the original charge will be affected. If the introduced charge has the same sign as the original charge, the electric field lines will repel each other and spread out. If the introduced charge has the opposite sign, the electric field lines will attract each other and move towards the opposite charge.

## 4. Can electric field lines cross?

No, electric field lines cannot cross. This is because the direction of the electric field at any given point can only have one direction. If the field lines were to cross, it would imply that the electric field at that point has two different directions, which is not possible.

## 5. Why do electric field lines not start or end in empty space?

Electric field lines do not start or end in empty space because electric fields can only exist in the presence of electric charges. Empty space does not have any charges, so there can be no electric field lines in that region. The field lines can only begin or end on charges because that is where the electric field is created.

• Introductory Physics Homework Help
Replies
28
Views
339
• Introductory Physics Homework Help
Replies
12
Views
1K
• Introductory Physics Homework Help
Replies
11
Views
773
• Introductory Physics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
23
Views
945
• Introductory Physics Homework Help
Replies
1
Views
233
• Introductory Physics Homework Help
Replies
10
Views
1K
• Introductory Physics Homework Help
Replies
18
Views
2K
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
3K