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## 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.