Electric Field Lines — Question 8

Question 8

A family of line shapes leaves the field scale undetermined

Field lines in a region have shapes $y=C\exp(x/L)$ with $L>0$, and arrows point toward increasing x. One compatible electrostatic field is assumed to have constant positive x component $E_0$.

1. Derive its y component from the line slope.

2. Compare the line shapes and force magnitudes for $E_0=100$ N/C and $E_0=300$ N/C at $y=L$. Explain what information is missing from an uncalibrated line drawing.

Question 8 - Solution

A family of line shapes leaves the field scale undetermined

(a) Method and calculation

$$\frac{dy}{dx}=\frac yL=\frac{E_y}{E_x},\quad E_x=E_0\Rightarrow E_y=E_0y/L.$$$$\mathbf E=E_0(1,y/L),\quad y=L:\ |\mathbf E|=\sqrt2E_0.$$

(b) Interpretation and further calculation

$$|\mathbf E|=100\sqrt2\text{ or }300\sqrt2\,\mathrm{N/C},\quad F_2/F_1=3\ \text{for the same probe}.$$$$E_y/E_x=y/L\quad\text{in both cases}.$$Identical oriented curves can represent different overall field strengths unless their density convention is calibrated.

Physical check

The tangent ratio sets direction; multiplying a field by a positive constant preserves every oriented field line.