# Equipotential lines and electric field lines

• ktb
In summary, the conversation discusses finding the electric field lines for an arbitrary dipole in a plane and proving their perpendicularity to the equipotential lines. The equation for the potential of the dipole is given, and the electric field lines are derived using the gradient of the potential. The discussion also addresses the challenge of mapping the function for the electric field into a two-dimensional space. The individual eventually figures out the solution and realizes their previous misunderstanding about electric field lines.
ktb

## Homework Statement

I am given the equation for the potential of an arbitrary dipole. I need to draw the electric field lines for this dipole in a plane, and also show that these lines are perpendicular to the equipotential lines. I have already derived the equation for the electric field using the gradient of the potential and mapped out the equipotential lines.
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## Homework Equations

V d i p ( ⃗r ) = Constant ⃗r ·p⃗ /(r^3)
E⃗ d i p = − ∇⃗ V d i p
= (Constant) 3( ⃗r · p⃗) ⃗r /(r^5) - p⃗/(r^3)
Take p⃗ to equal a unit vector for an orthonormal basis. Such as the unit vector for x in the x, y, z coordinate system.

## The Attempt at a Solution

I know that the gradient of V is always perpendicular to V, so intuitively this makes complete sense. However, I do not know how to show that a scaler quantity (V) is perpendicular to the vector equation I derived for E. I am also unsure how to map such a strange function for E into ℝ2 although obviously I know what it looks like.

Figured it out. Bad misunderstanding on my part about El. field lines.

## 1. What are equipotential lines and electric field lines?

Equipotential lines are imaginary lines that connect points with equal potential in an electric field. Electric field lines, on the other hand, are imaginary lines that indicate the direction of the electric field at a given point.

## 2. How are equipotential lines and electric field lines related?

Equipotential lines and electric field lines are perpendicular to each other at any given point. This means that the electric field lines are always tangent to the equipotential lines.

## 3. What do equipotential lines and electric field lines tell us about an electric field?

Equipotential lines and electric field lines provide valuable information about the strength and direction of an electric field. The spacing between equipotential lines indicates the strength of the electric field, while the direction of the electric field can be determined by the direction of the electric field lines.

## 4. How are equipotential lines and electric field lines used in practical applications?

Equipotential lines and electric field lines are used in various practical applications, such as designing electrical circuits, predicting the behavior of charged particles, and determining the location of electrically neutral points in an electric field.

## 5. Can equipotential lines and electric field lines ever intersect?

No, equipotential lines and electric field lines can never intersect because the electric field at a point can only have one direction and the potential at a point can only have one value. If the lines were to intersect, it would mean that there are two different electric fields or potentials at that point, which is impossible.

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