Final answer:
Equipotential lines must be perpendicular to electric field lines because work done along these lines is zero, which implies that force (aligned with the electric field) is perpendicular to displacement (along equipotential lines). Moreover, illustrations of these lines in electric fields emphasize their perpendicular nature.
Step-by-step explanation:
Why Equipotential Lines Must Be Perpendicular to Electric Field Lines
An equipotential line is a line or a surface over which the electric potential is constant. This means that no work is required to move a charge along an equipotential line or surface since the potential energy of the charge remains constant. The concept of work being zero is critical to understanding why equipotential lines must be perpendicular to electric field lines. Since work is defined as the component of a force along the direction of displacement times the magnitude of that displacement, and because the work done in an equipotential area is zero, it implies that the force (which is aligned with the electric field) must be perpendicular to the displacement (which is along the equipotential lines). As such, equipotential lines and surfaces are always perpendicular to electric field lines.
When considering the relationship between equipotential lines and electric field lines, option c) is correct: 'Work done along equipotential lines is zero, and work is perpendicular to force.' Moreover, another way to deduce the perpendicular nature of equipotential lines to electric field lines is through the use of visual aids, as shown in figures where electric field and equipotential lines are mapped out. These illustrations often demonstrate that by drawing one set of lines (either electric field or equipotential), the other set can be determined by ensuring that they are perpendicular to each other.
Regarding different equipotential lines, they cannot cross each other. If they were to cross, it would imply two different potential values at the same point in space, which is not possible—each point in space can only have one value of electric potential.