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Homework Help: Can electric field lines intersect in free space?

  1. Jun 17, 2012 #1
    Can electric field lines intersect in free space?
    I found the answer on the internet, but I will like to understand why not?

    Graphics will be appreciate it a lot!

  2. jcsd
  3. Jun 17, 2012 #2
    Nope, it has to do with mathematics...the Existence and Uniqueness Theorem. Field lines can't intersect because, if they did, the equations describing electric and magnetic fields would violate this mathematical theorem. Solutions to the differential equations describing electricity and magentism must be unique! :D
  4. Jun 17, 2012 #3


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    Field lines are artificial constructs we use to show points of equal potential in space. Electric field lines show the potential of the field at given points. Though we only draw a few, there are actually an infinite number than can be drawn between those few.

    An easy-to-grasp analogy is that of contour lines showing altitude on a map. You have a map with a land area that has a mountain on it. You draw a line connecting all points on the land surface that are 100ft above sea level. You draw another line, inside that one that connects all points on the surface that are 200ft above sea level.

    These two lines will always be closed loops, and will never intersect.

    You can see why they could never intersect if you think about what they're representing. If the 100ft contour ever crossed the 200ft contour, it would mean that, at that point on the Earth, the altitude of the land is both 100ft and 200ft simultaneously.

    Now, it's not perfect analogy. There are some arrangements on a map where they sort of could have two contour lins intersect. A vertical cliff or a cave could cause this, but that's symptomatic of a land map. You can't do that with electric fields. You can't have a single point in space that has two values for the electric field.
  5. Jun 17, 2012 #4
    Hmm, you are describing equipotential lines.

    I usually think of electric field lines as pointing along the gradient of the potential, i.e. going from positive potential to negative in the same way that magnetic field lines go from north to south.

    In any case, this type of field line cannot cross either. Think of the field line indicating the direction of the slope on a hill. That can only point in one direction. When you come to a saddle point (the pass between two hills), then the top is flat, i.e. no gradient, no field line. Move off the top a little bit and the slope starts to become steeper, but it will point in only one direction.

    This situation will occur with a quadrupole arrangement. Take 2 positive and 2 negative charges (+Q and -Q).

    Put +Q at (A ,0) and (-A, 0).
    Put -Q at (0, A) and (0, -A).

    Then at (0,0) there is no potential gradient, hence no electric field.

    By symmetry you expect field lines to run along the x and y axes, and they seem to cross at (0,0). But if you look closely the do not reach (0,0) because there is no field, hence no field line.

    As DaveC pointed out, field lines are artificial constructs to help visualize invisible electric and magnetic fields. Just drawing lines gives no impression of the field strength, so this representation is incomplete. Don't try to overstretch this means of visualizatin by constructing pathological cases.

    Equipotential lines are a bit better for this as the density of lines gives an idea of the gradient (slope).
  6. Jun 17, 2012 #5

    A charge source can send an electric field to a point. At the point, the electric field is represented by a vector. A different charge source in a different position can also do the same thing. These vectors will intersect and make a new vector. Does that answer your question?

  7. Jun 17, 2012 #6
    Thank you all for your help!!
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