# Equipotential lines and electric field lines

• ktb
In summary, the conversation discusses deriving the equation for the electric field of an arbitrary dipole, mapping out the equipotential lines, and showing that the electric field lines are perpendicular to the equipotential lines. The equations for the potential and electric field are provided, along with the clarification that the unit vector for an orthonormal basis is taken as the unit vector for x in the x, y, z coordinate system. The individual is unsure about how to show the perpendicularity of the scalar quantity V and the vector equation for E, as well as how to map the function for E into ℝ2. However, they later figure out the misunderstanding and apologize for the unnecessary post.
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.

## 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 by me about El. field lines. Sorry for the unnecessary post.

## 1. What are equipotential lines?

Equipotential lines are imaginary lines drawn on a map or diagram that connect points of equal potential. In the context of electric fields, they represent points in space where the potential energy of a charged particle would be the same, regardless of its position along the line.

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

Equipotential lines and electric field lines are closely related, as they both represent different aspects of the same phenomenon: the electric field. Electric field lines show the direction and strength of the electric field, while equipotential lines show the points in space where the electric potential energy is the same.

## 3. How are equipotential lines and electric field lines drawn?

Equipotential lines and electric field lines are typically drawn using mathematical equations and numerical calculations. The equations involve the distance between charges, the strength of the charges, and other factors. They can also be drawn using physical models and experimental data.

## 4. What is the significance of equipotential lines and electric field lines?

Equipotential lines and electric field lines are important tools for understanding and visualizing the behavior of electric fields. They can help us understand how charged particles move, how much energy they have, and how they interact with each other and their surroundings.

## 5. How do equipotential lines and electric field lines relate to work and energy?

Equipotential lines and electric field lines are closely related to work and energy. The work done on a charged particle moving along an equipotential line is zero, as the potential energy at all points on the line is the same. Electric field lines show the direction in which a charged particle would move to decrease its potential energy, and therefore the direction in which work is done on the particle.

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