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a particle with charge q is to be brought from far away to a point near an electric dipole. no work is done if the final position of the particle is on:

the line through the charges of the dipole
a line that is perpendicular to the dipole moment
a line that makes an angle of 45° with the dipole moment
a line that makes an angle of 30° with the dipole moment
none of the above

Explanation:

To determine where no work is done when moving a charged particle near an electric dipole, we use the concept of equipotential surfaces. The electric potential due to a dipole depends on the angle \(\theta\) (between the position vector and the dipole moment) and the distance \(r\). For a line perpendicular to the dipole moment (equatorial line of the dipole), the electric potential is zero everywhere (since the potential due to the two charges cancels out). When moving a charge along an equipotential surface, the work done \(W = q\Delta V\) (where \(\Delta V\) is the potential difference) is zero because \(\Delta V = 0\).

• For the line through the charges (axial line), the potential is non - zero and changes with position, so work would be done.
• For lines making \(30^{\circ}\) or \(45^{\circ}\) with the dipole moment, the potential is non - zero and varies along these lines, so work would be done.

So, no work is done when the particle moves along a line perpendicular to the dipole moment.

Answer:

a line that is perpendicular to the dipole moment