Q.Figure 1.30 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?

Concept: Charge-to-Mass Ratio

A charged particle entering a uniform electric field experiences a constant force $F = qE$ perpendicular to its initial velocity. This produces a parabolic trajectory, much like projectile motion under gravity. The vertical deflection depends on the acceleration $a = \frac{qE}{m}$, so for a given field and entry speed, larger deflection means higher $\frac{q}{m}$.

The field points downward (from positive to negative plate). Particles 1 and 2 curve upward, so they experience an upward force — they must be negatively charged. Particle 3 curves downward, experiencing a downward force — it is positively charged. …

Why particles curve in an electric field

A charged particle entering a uniform electric field experiences a constant force perpendicular to its initial velocity. This is exactly analogous to projectile motion under gravity: the horizontal component of velocity remains unchanged, while the perpendicular component grows linearly with time. The result is a parabolic trajectory.

The key insight is that the amount of bending depends on the acceleration, which in turn depends on the charge-to-mass ratio $\frac{q}{m}$. A particle with large $q$ feels a strong force; a particle with small $m$ responds with large acceleration. The combination $\frac{q}{m}$ determines how sharply the path curves.

Step-by-step analysis

1. Identify the field direction.

   The positive plate is at the top, the negative plate at the bottom. The electric field $\vec{E}$ points from positive to negative, so downward in the figure.

2. Determine the force on each particle.

   A positive charge experiences force $\vec{F} = q\vec{E}$ in the direction of the field (downward). A negative charge experiences force opposite to the field (upward).

3. Read the deflections.

   • Tracks 1 and 2 curve upward (toward the positive plate) $\implies$ the force is upward $\implies$ the particles are negative.
   • Track 3 curves downward (toward the negative plate) $\implies$ the force is downward $\implies$ the particle is positive.

4. Compare the magnitudes of deflection.

   All three particles enter horizontally with (presumably) similar speeds and traverse roughly the same horizontal distance through the field. The vertical displacement $y$ in a uniform field is given by

   $y = \frac{1}{2} a t^2 = \frac{1}{2} \frac{qE}{m} \left(\frac{x}{v_0}\right)^2,$ …

Method: Electrostatic Trajectory Analysis

This method uses the curvature and direction of deflection in a uniform electric field to determine charge sign and relative charge-to-mass ratio ($q/m$).

Step 1: Determine the direction of the electric field

• The field lines are drawn from positive to negative plates.
• In the figure, the top plate is positive ($+$) and the bottom plate is negative ($-$).
• Therefore, the electric field $\vec{E}$ points vertically downward.

Step 2: Find the sign of each charge from deflection

• A positive charge experiences a force in the same direction as $\vec{E}$ (downward).
• A negative charge experiences a force opposite to $\vec{E}$ (upward).

From the tracks:

• Particle A curves upward → force opposite to $\vec{E}$ → negative charge.
• Particle B curves downward → force same as $\vec{E}$ → positive charge.
• Particle C curves downward → force same as $\vec{E}$ → positive charge.

Step 3: Compare charge-to-mass ratios using curvature

• In a uniform electric field, the vertical acceleration of a charged particle is:

• The curvature (how sharply the path bends) is proportional to $|q|/m$ — larger $q/m$ means more deflection for the same initial speed.

From the figure: …

Here are the common mistakes students make on this specific question, along with the conceptual fixes to avoid them.

Mistake 1: Confusing the direction of the electric field with the direction of force

• The Error: Students see the field arrows pointing downward and assume all particles must move downward. They then mislabel the charges based on the initial direction of the track rather than the curvature.
• Why it happens: They forget that the force on a charge depends on the sign: $\vec{F} = q\vec{E}$. A positive charge feels a force in the same direction as $\vec{E}$ (downward). A negative charge feels a force opposite to $\vec{E}$ (upward).
• How to avoid: Always draw the force vector first.
  • If the particle bends toward the direction of $\vec{E}$, it is positive.
  • If the particle bends away from $\vec{E}$ (opposite direction), it is negative.
  • In this figure, $\vec{E}$ is downward. Particle (1) bends downward → positive. Particles (2) and (3) bend upward → negative.

Mistake 2: Forgetting that the deflection depends on both charge and mass

• The Error: Students assume the particle that bends the most has the largest charge, or that the particle that bends the least has the largest mass. They pick the wrong particle for the highest $q/m$ ratio.
• Why it happens: They treat the deflection as a function of charge alone, ignoring Newton’s Second Law. The acceleration (and thus curvature) is given by:

The deflection depends on the **ratio** $q/m$, not just $q$.

• How to avoid: Remember the key formula for motion in a uniform field (similar to projectile motion under gravity):
  • The vertical deflection $y$ is proportional to acceleration: $y \propto a \propto \frac{q}{m}$.
  • More deflection $\implies$ higher $|q|/m$ (for the same initial velocity and field strength).
  • In the figure, particle (3) has the greatest curvature (bends the most sharply upward). Therefore, it has the highest charge-to-mass ratio.

Mistake 3: Ignoring the initial velocity direction

• The Error: Students think the particles are starting from rest or moving straight into the field. They then misinterpret the parabolic path.
• Why it happens: They don’t realize the particles enter the field with an initial horizontal velocity (from left to right). The field only provides a vertical force, creating a parabolic trajectory (like a horizontal projectile). …