Electric Field Lines — Question 3

Question 3

A planar radial sketch does not encode three-dimensional line density by arc spacing

A positive point source is represented by a fixed number of radial field lines per solid angle. Compare concentric spheres of radii $r$ and $3r$. On each sphere the same set of lines crosses a patch subtending fixed solid angle $\Delta\Omega$. Use patch area $A=r^2\Delta\Omega$ and field strength proportional to line count per perpendicular area.

1. Find the patch-area and field-strength ratios.

2. Explain why measuring adjacent-ray spacing on a flat printed diagram would incorrectly suggest an inverse-first-power field.

Question 3 - Solution

A planar radial sketch does not encode three-dimensional line density by arc spacing

(a) Method and calculation

$$A_1=r^2\Delta\Omega,\quad A_2=(3r)^2\Delta\Omega=9A_1.$$$$E_2/E_1=(N/A_2)/(N/A_1)=1/9.$$

(b) Interpretation and further calculation

$$\text{planar arc spacing }s=r\Delta\theta\Rightarrow s_2/s_1=3.$$$$E\propto1/(\text{perpendicular area per line}),\quad E\not\propto1/s\ \text{for a 3D point source}.$$A flat cross-section suppresses spreading in the second transverse direction.

Physical check

Line count is a drawing convention; density comparisons require one consistent normalization and the correct dimensional geometry.