Charged Particles in Fields

Overview

Charged Particles in Fields focuses on how charged particles move when placed in an electric field. This combines ideas from Electric Fields, Forces, and Kinematics.

The key principle is:

$$\vec{F}=q\vec{E}$$

An electric field exerts force on a charged particle, causing acceleration.

Definition

For charge $q$ in electric field $\vec{E}$:

$$\vec{F}=q\vec{E}$$

Direction

Positive Charge

• force is in the direction of $\vec{E}$

Negative Charge

• force is opposite to $\vec{E}$

Magnitude

$$F=|q|E$$

Why It Matters

This topic is the bridge between field ideas and motion:

• force from the field produces acceleration
• uniform electric fields lead to constant acceleration
• horizontal-vertical resolution gives standard deflection methods
• energy methods give faster speed calculations in some questions

Key Representations

Acceleration in a Uniform Field

Using Newton’s second law:

$$F=ma$$

Hence:

$$a=\frac{qE}{m}$$

For magnitude:

$$a=\frac{|q|E}{m}$$

Important Trends

• larger charge gives larger acceleration
• stronger field gives larger acceleration
• larger mass gives smaller acceleration

Electrons accelerate very strongly because their mass is very small.

Uniform Electric Field Between Parallel Plates

For plates with potential difference $V$ and separation $d$:

$$E=\frac{V}{d}$$

This field is approximately uniform in the central region.

Hence force is constant:

$$F=qE$$

So acceleration is constant.

Motion Parallel to the Field

If initial velocity is along the field direction:

Positive Charge

• speeds up if moving with the field
• slows down if moving against the field

Negative Charge

Opposite behaviour.

Use SUVAT Equations

Since acceleration is constant:

$$v=u+at$$ $$v^2=u^2+2as$$ $$s=ut+\frac{1}{2}a{t}^2$$

Motion Opposite to the Field

If a particle enters with velocity opposite to its acceleration:

• it decelerates
• it may momentarily stop
• it then reverses direction

This is mathematically the same as one-dimensional motion under constant acceleration.

Motion Perpendicular to the Field

Suppose a particle enters horizontally into a vertical electric field.

Horizontal Direction

No horizontal force:

• velocity remains constant

$$v_x=u_x$$

Vertical Direction

Constant acceleration:

$$a_y=\frac{qE}{m}$$

Thus:

$$v_y=u_y+a_{y}t$$

Usually $u_y=0$.

Combined Motion: Parabolic Path

Because there is:

• uniform motion horizontally
• constant acceleration vertically

the trajectory is a parabola, similar to projectile motion.

Standard Strategy

1. Resolve motion into $x$ and $y$ directions.
2. Solve time using horizontal motion.
3. Use that time in vertical motion.
4. Find displacement, velocity, or angle.

Deflection Between Parallel Plates

If plate length is $L$ and horizontal entry speed is $u_x$:

Time in the Field

$$t=\frac{L}{u_x}$$

Vertical Deflection

$$y=\frac{1}{2}{a}_y{t}^2$$

Vertical Exit Speed

$$v_y=a_{y}t$$

Exit Angle

$$\tan \theta =\frac{v_y}{v_x}$$

where $v_x=u_x$.

Particle Initially at Rest

If released from rest in a uniform field:

$$u=0$$

Then the particle accelerates directly along the force direction.

Use SUVAT or energy methods.

Energy Perspective

Moving through potential difference $\Delta V$:

Change in electric potential energy:

$$\Delta U=q\Delta V$$

If no other forces act:

$$\Delta K=-\Delta U$$

Hence:

$$\frac{1}{2}m{v}^2-\frac{1}{2}m{u}^2=-q\Delta V$$

This is useful for speed calculations.

Electrons in Fields

Electrons are common exam examples.

Charge:

$$q=-e$$

where:

$$e=1.60\times 10^{-19}\text{C}$$

Important Reminder

Electron force is opposite to electric field direction.

Comparison with Gravity

Motion in a uniform electric field is mathematically similar to projectile motion in gravity.

Gravity	Electric Field
acceleration $g$	acceleration $\frac{qE}{m}$
downward	depends on charge sign
same for all masses ideally	depends on $q/m$

See Gravitational vs Electric Fields.

Crossed Fields and Velocity Selector

If a charged particle moves through both an electric field and a magnetic field, the two forces may oppose each other.

In a velocity selector:

• the electric force is $qE$
• the magnetic force is $qvB$
• particles pass undeflected only when these forces balance

$$qE=qvB$$

Hence:

$$v=\frac{E}{B}$$

Only particles with this speed move straight through.

Figure: Velocity selector using crossed electric and magnetic fields.

This idea connects electric-field motion directly to Magnetic Force.

Common Mistakes

1. Using field direction as electron motion direction
2. Forgetting horizontal velocity stays constant
3. Using the wrong sign of acceleration
4. Mixing force and field so that $F=E$
5. Using the energy formula with the wrong sign of $\Delta V$
6. Forgetting to resolve into components
7. Assuming the path is circular instead of parabolic

Quick Exam Method

Parallel Motion

Use:

• $F=qE$
• $a=F/m$
• SUVAT

Deflection Question

1. Find the field:

$$E=\frac{V}{d}$$

2. Find the acceleration:

$$a=\frac{qE}{m}$$

3. Find the time:

$$t=\frac{L}{u_x}$$

4. Solve the vertical motion.

Summary

Core equations:

$$\vec{F}=q\vec{E}$$ $$a=\frac{qE}{m}$$ $$E=\frac{V}{d}$$ $$t=\frac{L}{u_x}$$ $$y=\frac{1}{2}a{t}^2$$ $$\tan \theta =\frac{v_y}{v_x}$$ $$\Delta K=-q\Delta V$$ $$v=\frac{E}{B}$$

Links

• Electric Fields
• Electric Potential and Energy
• Electric Fields Common Exam Traps
• Forces
• Kinematics
• Vectors
• Magnetic Force