How to calculate the direction of field in equipotential surfaces?

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How to calculate the direction of field in equipotential surfaces?

Let us assume there is an eletrically charged object somewhere in space. It causes an electric field, defined as the attracting or repellent force some other particle with unit charge (1 Coulomb) would experience from it.

Eletric potential is the potential energy which that other unit-charge particle would build up when approaching from infinite distance. So you need to compute an integral: The field is inversely proportional to the square of distance, whereas potential is inversely proportional to distance.

The electric field is a vector (it has direction), whereas potential is a scalar. (If you are more deeply into analytics: You would integrate the inner product of the field vector with the infinitesimal steps of movement through space, which are also vectors. The inner product of two vectors is a scalar, that's why potential is a scalar. You can choose any path of approach through space to integrate over, the potential at the end point will always be the same.)

A dipole means there are two electrically charged objects slightly displaced from each other, with equally sized charges of opposite sign. The dipole moment is charge multiplied by displacement. (That is, higher charges closer to each other will do the same as lower charges more strongly displaced. From a distance, the difference blurs anyway.)

For a dipole antenna, assume electric charges sit at the opposite ends of the dipole rod.

Equipotential surfaces are surfaces of same electric potential. Or equipotential lines, if you use an only two-dimensional model (which is totally adequate). The equipotential surfaces or lines are always perpendicular to the field lines.

Theoretically you can draw as many equipotential lines (or surfaces) as there are conceivable values for the electric potential, that is an infinite number. The straight line (or flat surface) that goes through the middle of the dipole is only one of the many, the one for potential zero.

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