Electric Field Lines — Question 7

Question 7

A particle with transverse initial velocity does not trace a field line

A uniform field points along $+x$. A positive particle starts at the origin with velocity $v_0\hat{\mathbf y}$. Its constant electric acceleration is $a\hat{\mathbf x}$ with $a>0$, and other forces are absent.

1. Find its trajectory and compare its tangent with the straight horizontal field lines.

2. Find the trajectory slope at $t=v_0/a$ and explain what the field-line direction predicts about the motion.

Question 7 - Solution

A particle with transverse initial velocity does not trace a field line

(a) Method and calculation

$$x=\tfrac12at^2,\quad y=v_0t\Rightarrow x=\frac{a}{2v_0^2}y^2.$$$$\frac{dy}{dx}=\frac{v_y}{v_x}=\frac{v_0}{at}\ne0\quad(t>0).$$

(b) Interpretation and further calculation

$$t=v_0/a:\quad dy/dx=1,\quad\theta_v=45^\circ.$$$$\mathbf a=(a,0)\parallel\mathbf E,\qquad\mathbf v=(at,v_0)\not\parallel\mathbf E.$$A field line fixes the instantaneous acceleration direction for a positive charge, not its velocity direction.

Physical check

Only special initial conditions and geometries make an actual trajectory coincide with a field line.