Electric Field Lines — Question 2

Question 2

A saddle field produces hyperbolic lines with a singular direction at the null

In a plane, $E_x=ax$ and $E_y=-ay$ with $a>0$. A line in the first quadrant passes through $(L,L)$, where $L>0$.

1. Integrate the field-line equation to find that curve and its arrow orientation.

2. Analyze the field on both coordinate axes and at the origin. Explain why extending the hyperbola through the origin would be incorrect.

Question 2 - Solution

A saddle field produces hyperbolic lines with a singular direction at the null

(a) Method and calculation

$$\frac{dy}{dx}=-\frac yx\Rightarrow\frac{dy}{y}=-\frac{dx}{x}\Rightarrow xy=C.$$$$C=L^2,\quad y=L^2/x;\quad x>0,y>0:\ E_x>0,E_y<0.$$

(b) Interpretation and further calculation

$$y=0:\ \mathbf E=(ax,0),\qquad x=0:\ \mathbf E=(0,-ay).$$$$\mathbf E(0,0)=\mathbf0,\quad xy=L^2\ne0.$$Arrows point away from the null along the x axis and toward it along the y axis. No direction is defined at the null itself.

Physical check

The field-line equation using a ratio is not valid at points where both numerator and denominator vanish.