Equipotential Lines: Can They Cross?

[mich_v87]

Is it possible for two different equipotential lines or two lines of force to cross ??

 

[vanesch]

Is it possible for two different equipotential lines or two lines of force to cross ??

Ha, subtle. A priori, no, you'd think and in general this is correct. However, there can be saddle points and the potential line with the value equal to the potential of the saddle point will show a "crossing".
However, clearly, potential lines with DIFFERENT potential values cannot cross. Idem for field lines (except of course at singularities, like point charges, where they all meet).

Interesting thought:

If V(x,y) is the (electrostatic) potential, then you can construct an analytic complex function V_c(x+i y) such that Re( V_c(x+i y)) = V(x,y) and Im(V_c(x+i y)) = constant gives you the field lines.

cheers,
Patrick.

 

[quicknote]

It is not possible for two different equipotential lines to cross. This is because equipotential lines represent points in a field where the potential is equal. If two equipotential lines were to cross, it would mean that the potential at that point is equal to two different values, which is not physically possible.

However, it is possible for two lines of force to cross. Lines of force represent the direction of the electric field at a point, and they can cross at points where the electric field is changing in both magnitude and direction. This often occurs near charges or in regions of varying electric field strength.

In summary, while equipotential lines cannot cross due to the definition of potential, lines of force can cross if the electric field is changing in both magnitude and direction. This is an important distinction to make when analyzing electric fields and their associated lines.