Connection field lines/potential/vector field

In summary: So, in summary, the derivative of the field line is parallel to the vector field and the equipotential lines are perpendicular to the field lines in three-dimensional space.
  • #1
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Hey guys,
I'm following a course on vector calculus and I'm having some trouble connecting things. Suppose we have a three-dimensionale vectorfield F(x,y,z)=(F1,F2,F3) and suppose we have a potential phi for this. So: F=grad(phi).
The field lines of a vector field are defined as d(r)/dt = lamba(t)F(r(t)). In words: the derivative of the field line (which is parametrized bij r) is parallel to F(r(t)).

Now for my question. Suppose we have equipotential lines, so phi=c with c a constant. What's the connection between these equipotential lines, the field lines and the vector field in terms of being parallel or right-angled?

I think the answer should be that the field lines are right-angled with the equipotential lines, but I don't know how to deduce this.

Thanks in advance!
 
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  • #2
Let f(x,y,z) be the value of the potential field at each point (x,y,z). Then the vector [itex]grad f= \nabla f[/itex] points in the direction of the "field lines", the lines of fastest increase of the function f. Further, the rate of increase of f in the direction of unit vector [itex]\vec{v}[/itex] is given by [itex]\nabla f\cdot \vec{v}[/itex]. That is, the direction in which the derivative is 0, the "equipotential lines" (strictly speaking, in three dimensions, they would be equipotential surfaces) is exactly the direction in which that dot product is 0, the direction in which the vector [itex]\vec{v}[/itex] is perpendicular to [itex]\nabla f[/itex] and so perpendicular to the field lines.
 

1. What are connection field lines?

Connection field lines are imaginary lines that represent the direction and strength of an electric, magnetic, or gravitational field. They connect points in space with the same field strength, and the density of the lines indicates the intensity of the field at that point.

2. How are connection field lines related to potential?

Connection field lines and potential are closely related in that the potential at a point is directly proportional to the density of the field lines at that point. This means that where the field lines are closer together, the potential is higher, and where they are further apart, the potential is lower.

3. What is a vector field?

A vector field is a mathematical concept used to describe the magnitude and direction of a physical quantity at every point in space. It is represented by a vector at each point, with the length of the vector indicating the strength of the field and the direction pointing towards the direction of the field.

4. How are vector fields used in connection field lines?

Vector fields are used to visualize and analyze connection field lines. By plotting vectors at different points, we can get a better understanding of the direction and strength of the field at each point. This helps us to identify patterns and make predictions about the behavior of the field.

5. What are some real-world applications of connection field lines and vector fields?

Connection field lines and vector fields have a wide range of applications in different fields of science and engineering. Some examples include understanding the behavior of electric and magnetic fields in electronic devices, predicting weather patterns using atmospheric pressure fields, and analyzing fluid flow in engineering design.

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