Question

Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge-to-mass ratio?

Hint:

Force qE pushes positive charges along E and negative charges against E, so same-side tracks share a sign. Deflection y is proportional to q/m, so the most-bent particle has the largest charge-to-mass ratio.

Answer 1

Figure 1.33 shows three charged particles entering the same uniform electric field and bending along curved tracks. The field points from the upper plate to the lower plate (downward in the figure).

Step 1: Concept. A charge in a uniform field feels a force \(\vec F = q\vec E\). A positive charge is pushed along \(\vec E\); a negative charge is pushed opposite to \(\vec E\). So particles that bend one way are positive and those bending the opposite way are negative.

Step 2: Signs of the charges. Particles 1 and 2 deflect towards the negative (lower) plate, i.e. towards the same side, so they carry charge of the same sign that is attracted to the negative plate. Particle 3 deflects towards the positive (upper) plate, the opposite side. Comparing the directions, particles 1 and 2 are negative and particle 3 is positive.

Step 3: Charge-to-mass ratio. For the same field and roughly the same path length, the sideways deflection is

\[y = \frac{1}{2}\,a\,t^{2}, \qquad a = \frac{qE}{m}\]\[y \propto \frac{q}{m}\]

The particle that bends the most has the largest deflection \(y\), hence the largest \(q/m\).

Step 4: Particle 3 shows the greatest curvature (largest deflection) in the figure, so it has the highest charge-to-mass ratio.

\[\boxed{1,2 \text{ negative}; \ 3 \text{ positive}; \ \text{highest } q/m: \text{particle } 3}\]

Answer 2

Curvature (radius-of-bending) argument:

Step 1: Treat each track like projectile motion across the field region. The transverse force \(qE\) gives a transverse acceleration \(a = qE/m\), so the path curves like a parabola.

Step 2: Sign from bending direction. The constant force always points the same way for a given sign of charge. Two particles that veer towards the same plate must carry the same sign; the third, veering towards the opposite plate, has the opposite sign. In Fig. 1.33 particles 1 and 2 turn one way and particle 3 turns the other way, so particles 1 and 2 are negative and particle 3 is positive.

Step 3: Read q/m from how sharply it bends. A sharper bend (smaller radius of curvature, larger sideways shift) means a larger acceleration. Since \(a = (q/m)E\) and \(E\) is common to all three,

\[\frac{q}{m} \propto (\text{amount of deflection}).\]

Step 4: Particle 3 is deflected most sharply, so it has the highest \(q/m\).

\[\boxed{1,2 \to (-),\ 3 \to (+),\ \text{max } q/m = \text{particle } 3}\]