vortmax said:I understand that electric field lines do not cross and a simple explanation is that it violates existence-uniqueness. I'm thinking there is a more complete explanation out there though. I think the answer lies in the realm of dynamical systems and non-linear dynamics, but it's been a whlie since I've had that and can't piece together the theorems that would fully explain it.
E-field lines are lines that represent the direction and strength of the electric field surrounding a charged object. They are used to visualize the electric field and its properties.
The fact that e-field lines do not cross is a fundamental property of electric fields. It helps us understand the direction and magnitude of the electric field at any point in space. If e-field lines were allowed to cross, it would imply that the electric field has multiple directions and magnitudes at a single point, which goes against our understanding of electric fields.
We can prove that e-field lines do not cross by conducting experiments and using mathematical equations. For example, we can use a charged object and a test charge to map out the electric field lines. Additionally, the mathematical equation for electric field lines, given by Coulomb's Law, also shows that the lines cannot cross.
In most cases, e-field lines do not cross. However, there are some exceptions, such as in the presence of multiple charges. In this case, the e-field lines from different charges may overlap or interact with each other, but they still do not physically cross.
The fact that e-field lines do not cross allows us to accurately predict and understand the behavior of electric fields. It also helps us in practical applications, such as designing electrical circuits and devices, as we can use the direction and strength of the e-field lines to determine the flow of electricity.