Question

Can two equipotential lines cross? Why or why not?

Short answer

Provide a justification for your answer. Answer: No, two equipotential lines cannot cross each other. This is because if they were to cross, there would be a contradiction in their potentials at the intersection point. At the point of intersection, the potential would have both V1 and V2 simultaneously, which is not possible as V1 is not equal to V2. Thus, equipotential lines cannot cross each other.

Step-by-step solution

Equipotential lines definition

Equipotential lines are lines where the potential is the same at every point along the line. In other words, no work is required to move a charge along an equipotential line because the electric potential is constant.

Imagine two equipotential lines cross

Suppose two equipotential lines with different potentials were to cross. Let's denote the potentials of the two lines as V1 and V2, where V1 is not equal to V2.

Determine the potential at the crossing point

At the point of intersection, along the line with potential V1, the potential would be V1. However, since the point is also on the line with potential V2, the potential would also be V2.

Arriving at a contradiction

Since V1 is not equal to V2, we have a contradiction. The potential at the point of intersection cannot be both V1 and V2 simultaneously. This can only imply that two equipotential lines cannot cross.

Conclusion

In conclusion, two equipotential lines cannot cross each other because it would lead to a contradiction in their potentials at the intersection point, with different values for the potential at the same point, which is not possible.