A beam of non-relativistic charged particles moves without deviation through a region $A$ of length

Question

A beam of non-relativistic charged particles moves without deviation through a region $A$ of length $a = 10 	ext{ cm}$ where there are transverse mutually perpendicular electric and magnetic fields with strength $E = 10 	ext{ kV/m}$ and induction $B = 10 	ext{ mT}$. When the magnetic field is switched off, the trace of the beam on a screen $S$, located at a distance $b = 20 	ext{ cm}$ from the end of region $A$, shifts by $\Delta x = 5 	ext{ mm}$. Find the specific charge $q/m$ of the particles in units of $10^7 	ext{ C/kg}$.

Professor PC Thomas Classes · Answer

1. Velocity Selection:
$qE = qv_xB \implies v_x = \frac{E}{B}$
2. Total Deflection ($\Delta x = 	ext{Inside Field } y_1 + 	ext{Outside Field } y_2$):
$\Delta x = \frac{1}{2} a_y t^2 + b 	an	heta$
3. Substitute Kinematic Variables ($a_y = \frac{qE}{m}$, $t = \frac{a}{v_x}$, $	an	heta = \frac{v_y}{v_x}$):
$\Delta x = \frac{1}{2} \left(\frac{qE}{m}ight) \left(\frac{a}{v_x}ight)^2 + b \left(\frac{\frac{qE}{m} \frac{a}{v_x}}{v_x}ight)$
4. Factor and substitute $v_x = \frac{E}{B}$:
$\Delta x = \frac{qE a}{m v_x^2} \left(\frac{a}{2} + bight) = \frac{q B^2 a}{m E} \left(\frac{a}{2} + bight)$
5. Solve for Specific Charge:
$\frac{q}{m} = \frac{2 E \Delta x}{B^2 a (a + 2b)}$