A Point Charge in an Electric Field — Question 1

Question 1

A deflector followed by free drift measures charge-to-mass ratio

A nonrelativistic particle enters a uniform field $E\hat{\mathbf y}$ with velocity $v\hat{\mathbf x}$. The field occupies horizontal length L and is followed by a field-free drift distance D. Ignore gravity, edge fields and source disturbance. A screen measures signed displacement Y from the undeflected path.

1. Derive Y including both the in-field deflection and drift contribution.

2. Infer $q/m$ for $E=200$ N/C, $v=2.00\times10^5$ m/s, $L=.100$ m, $D=.200$ m and $Y=-.00500$ m.

Question 1 - Solution

A deflector followed by free drift measures charge-to-mass ratio

(a) Method and calculation

$$t_L=L/v,\quad a_y=qE/m,\quad y_L=\tfrac12a_y(L/v)^2,\quad v_{y,L}=a_yL/v.$$$$Y=y_L+v_{y,L}D/v=\frac{qE}{mv^2}L(D+L/2).$$

(b) Interpretation and further calculation

$$\frac qm=\frac{Yv^2}{EL(D+L/2)}=-4.00\times10^7\,\mathrm{C/kg}.$$$$Y<0,E>0\Rightarrow q<0.$$Ignoring drift would overestimate the magnitude inferred for $q/m$, because the particle keeps its acquired transverse velocity after leaving the field.

Physical check

Horizontal speed remains constant because the prescribed field has no x component.