Electric Field Lines — Question 4

Question 4

Crossing lines contradict a unique nonzero electric-field direction

At a regular point P, a proposed drawing has two field lines crossing: one tangent to $(1,1)$ and the other tangent to $(1,-1)$. The field magnitude at P is stated to be 100 N/C.

1. Show that no nonzero vector can be tangent to both lines.

2. Would replacing the stated field magnitude by zero make the crossing a pair of well-defined field directions? Distinguish a field null from a regular crossing.

Question 4 - Solution

Crossing lines contradict a unique nonzero electric-field direction

(a) Method and calculation

$$\mathbf E=\lambda(1,1)=\mu(1,-1).$$$$\lambda=\mu,\quad\lambda=-\mu\Rightarrow\lambda=\mu=0.$$

(b) Interpretation and further calculation

$$|\mathbf E|=100\,\mathrm{N/C}\ne0:\quad\text{the drawing is inconsistent}.$$$$\mathbf E=0:\quad\mathbf E/|\mathbf E|\ \text{is undefined}.$$A null allows limiting curves to approach it, but does not assign two distinct field directions at that point.

Physical check

This is a deliberately invalid candidate at a nonzero-field point, not a valid field-line map.