# Electrostatics and Magnetostatics

# Charged ring equivalent to a dipole

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

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Let a thin ring (A), of center O and radius , be divided in two equally evenly charged parts and .

## Question

By using the notion of electrostatic dipole, determine the potential and the electrostatic field at a point M, very far away from O ( ).

### Hint

Try to cut the circle in small elementary dipoles.

### Solution

Let's cut the circle in small elementary dipoles delimited by the angle .

The elementary dipole moment is :

The symmetry shows that the resulting dipole vector is oriented as .

It is worth :

Let's consider a point M located in the plane (Oyz), and let's write the angle between axis and (OM) line.

The potential created by the ring becomes :

The electric field is expressed by: