describe the effect of a uniform electric field on the motion of charged particles

Uniform Electric Fields – A‑Level Physics (Cambridge 9702)

1. Syllabus Context

This note addresses the Cambridge 9702 syllabus blocks 18.1 – 18.5 and the associated assessment objectives (AO).

Syllabus Block	Content Covered	AO
18.1 Electric fields – definition, field lines, F = qE	Definition, direction of field lines (positive → negative), force on a charge, superposition, line‑density conventions	AO1, AO2
18.2 Uniform electric fields – E = ΔV/d	Derivation from the potential gradient, vector form \(\mathbf{E}= -abla V\), condition that \(\mathbf{E}\) is parallel to the plate normal, numerical example, equipotentials	AO1, AO2
18.3 Electric force between point charges – Coulomb’s law	Explicit statement of Coulomb’s law, comparison with the uniform‑field case	AO1, AO2
18.4 Electric field of a point charge – E = kQ/r²	Inverse‑square law, field‑line density, superposition to obtain an approximately uniform region	AO1, AO2
18.5 Electric potential – V = –∫E·dl, equipotentials	Relation between uniform field and equipotential planes, work‑energy link	AO1, AO2
Practical skills (AO3)	Mini‑experiment to measure E between parallel plates, data analysis, error discussion	AO3

2. Key Concepts

• Uniform electric field: \(\mathbf{E}\) has the same magnitude and direction at every point in the region. In practice this is realised between large, parallel conducting plates.
• Field‑line conventions:
  • Lines start on positive charges (or the positive plate) and end on negative charges (or the negative plate).
  • Line density ∝ field strength.
  • Lines never cross; the tangent to a line gives the direction of \(\mathbf{E}\) at that point.
• Superposition: The net electric field at any point is the vector sum of the fields produced by all sources.
• Electric field from potential: \[ \mathbf{E}= -abla V\qquad\text{(general form)} \] For a uniform field parallel to the displacement between two equipotential surfaces, \[ E = \frac{\Delta V}{d}, \] where \(d\) is the separation of the surfaces.
• Force on a charge: \(\displaystyle \mathbf{F}=q\mathbf{E}\) (same direction as \(\mathbf{E}\) for \(q>0\), opposite for \(q<0\)).
• Resulting acceleration: \[ \mathbf{a}= \frac{\mathbf{F}}{m}= \frac{q\mathbf{E}}{m}. \]

3. Deriving the Uniform‑Field Expression

From the definition of electric potential difference,

\[ \Delta V = -\int_{A}^{B}\mathbf{E}\cdot d\mathbf{l}. \]

If \(\mathbf{E}\) is uniform and parallel to the path (as between parallel plates), the integral reduces to

\[ \Delta V = -E\,d \;\;\Longrightarrow\;\; E = \frac{\Delta V}{d}. \]

Numerical example:

• Potential of the positive plate: \(+200\;\text{V}\)
• Potential of the negative plate: \(0\;\text{V}\)
• Plate separation: \(d = 5.0\times10^{-3}\;\text{m}\)
• Uniform field magnitude: \[ E = \frac{200\;\text{V}}{5.0\times10^{-3}\;\text{m}} = 4.0\times10^{4}\;\text{V m}^{-1}, \] directed from the positive to the negative plate.

4. Motion of Charged Particles in a Uniform Field

4.1 General Procedure

1. Resolve the initial velocity \(\mathbf{v}_0\) into components parallel (\(v_{0\parallel}\)) and perpendicular (\(v_{0\perp}\)) to \(\mathbf{E}\).
2. Apply one‑dimensional kinematics to each component:
   • Parallel component (constant acceleration \(a = qE/m\)): \[ v_{\parallel}=v_{0\parallel}+at,\qquad x_{\parallel}=v_{0\parallel}t+\tfrac12 a t^{2}. \]
   • Perpendicular component (uniform motion): \[ v_{\perp}=v_{0\perp},\qquad x_{\perp}=v_{0\perp}t. \]
3. The overall trajectory is the vector sum of the two motions; it lies in the plane defined by \(\mathbf{v}_0\) and \(\mathbf{E}\).

4.2 Specific Cases

Case 1 – \(\mathbf{v}_0\) parallel (or anti‑parallel) to \(\mathbf{E}\)

• Motion is one‑dimensional along the field lines.
• Equations: \[ v = v_0 + \frac{qE}{m}t,\qquad x = v_0 t + \frac12\frac{qE}{m}t^{2}. \]

Case 2 – \(\mathbf{v}_0\) perpendicular to \(\mathbf{E}\)

• Uniform sideways motion combined with constant acceleration across the field.
• Trajectory is a parabola (direct analogue of projectile motion): \[ x = v_{0x}t,\qquad y = \frac12\frac{qE}{m}t^{2}. \]

Case 3 – Arbitrary angle \(\theta\) to the field

• Combine the results of Cases 1 and 2.
• Using \(v_{0x}=v_0\cos\theta\) and \(v_{0y}=v_0\sin\theta\), the path can be written as \[ y = \frac{qE}{2m v_{0x}^{2}}\,x^{2} + \frac{v_{0y}}{v_{0x}}\,x, \] a segment of a parabola.

5. Energy Considerations

• Work done by the field when a charge moves a distance \(s\) along the field: \[ W = qE s = q\Delta V. \]
• Work‑energy theorem: \[ W = \Delta K = \frac12 m v_f^{2} - \frac12 m v_i^{2}. \]
• Corresponding change in electric potential energy: \[ \Delta U = q\Delta V = -W. \]

6. Relation to Electric Potential & Equipotentials

• In a uniform field the equipotential surfaces are parallel planes, each separated by a constant potential difference \(\Delta V\).
• \(\mathbf{E}\) is always perpendicular to equipotentials. Consequently:
  • Moving **along** an equipotential: \(\mathbf{F}=0\) → no change in kinetic energy.
  • Moving **across** equipotentials (i.e., in the direction of \(\mathbf{E}\)): kinetic energy changes according to the equations in Section 5.

7. Comparison with Point‑Charge Fields

• Coulomb’s law (syllabus 18.3): \[ \mathbf{F}=k\frac{q_1 q_2}{r^{2}}\hat{\mathbf{r}},\qquad k = \frac{1}{4\pi\varepsilon_0}. \]
• Electric field of a point charge (syllabus 18.4): \[ \mathbf{E}=k\frac{Q}{r^{2}}\hat{\mathbf{r}}. \]
• Key differences:
  • Magnitude varies with distance (\(\propto 1/r^{2}\)) → non‑uniform.
  • Field lines radiate outward (or inward) from the charge, unlike the parallel lines of a uniform field.
  • Superposition of many point‑charge fields can produce an approximately uniform region (e.g., between large parallel plates where edge effects are negligible).

8. Practical Investigation (AO3)

8.1 Objective

Measure the magnitude of a uniform electric field between parallel plates using a calibrated electrostatic probe.

8.2 Apparatus

• Two large, parallel conducting plates with adjustable separation.
• Variable DC power supply (0–500 V).
• Digital voltmeter (to read plate potentials).
• Electrostatic field probe (small charged sphere attached to a spring balance) calibrated in N C⁻¹.
• Ruler or calipers for plate separation.
• Connecting wires, earth‑earth safety.

8.3 Method (outline)

1. Set the plate separation to a known value \(d\) (measure with calipers).
2. Connect the plates to the power supply; set a potential difference \(\Delta V\) and record it with the voltmeter.
3. Place the field probe midway between the plates, ensuring it does not touch either plate.
4. Read the force \(F\) on the probe from the spring balance; calculate the field as \[ E_{\text{meas}} = \frac{F}{q_{\text{probe}}}. \]
5. Repeat for at least three different \(\Delta V\) values while keeping \(d\) constant.
6. Repeat the whole set with a second plate separation (e.g., double the original distance) to verify the linear relation \(E = \Delta V/d\).

8.4 Sample Data Table

Run	Plate separation, \(d\) (mm)	\(\Delta V\) (V)	Probe charge, \(q\) (µC)	Measured force, \(F\) (mN)	\(E_{\text{meas}} = F/q\) (V m⁻¹)	Theoretical \(E = \Delta V/d\) (V m⁻¹)
1	5.0	100	2.0	0.80	4.0 × 10⁴	4.0 × 10⁴
2	5.0	250	2.0	2.00	1.0 × 10⁵	5.0 × 10⁴
3	10.0	250	2.0	1.00	5.0 × 10⁴	2.5 × 10⁴

8.5 Analysis & Error Checklist

• Plot \(E_{\text{meas}}\) against \(\Delta V/d\); the gradient should be close to 1.
• Possible sources of error (and their likely effect):
  • Inaccurate plate separation (parallax, plate warping) – systematic under‑ or over‑estimate of \(E\).
  • Edge effects – field near the plate edges is weaker, causing measured \(E\) to be slightly low.
  • Uncertainty in probe charge (leakage, charging method) – random error in \(E_{\text{meas}}\).
  • Instrumental limits (voltmeter resolution, spring‑balance calibration) – systematic bias if not zero‑checked.
  • Air currents or temperature gradients – random fluctuations in the force reading.
• Discuss each error, classify as systematic or random, and suggest mitigation strategies (e.g., use the central region of the plates, repeat measurements, calibrate the probe before each set).

9. Common Misconceptions

• “Electric fields always increase the speed of a charge.” – Only the component of \(\mathbf{E}\) parallel to the motion changes speed; a perpendicular component changes direction only.
• “A neutral particle feels a force in an electric field.” – Net force is zero because the forces on the positive and negative charges within the particle cancel.
• “All charged particles follow the same path in a uniform field.” – Trajectories depend on charge sign, magnitude, mass, and the initial velocity vector.
• “Equipotential surfaces are always curved.” – In a uniform field they are flat, parallel planes.

10. Summary Table

Initial Velocity Direction	Resulting Motion	Key Equations
Parallel (or anti‑parallel) to \(\mathbf{E}\)	Linear acceleration (or deceleration) along the field lines	\(v = v_0 + \dfrac{qE}{m}t,\; x = v_0 t + \dfrac12\dfrac{qE}{m}t^{2}\)
Perpendicular to \(\mathbf{E}\)	Parabolic trajectory – uniform sideways motion, accelerated motion across the field	\(x = v_{0x}t,\; y = \dfrac12\dfrac{qE}{m}t^{2}\)
Arbitrary angle \(\theta\)	Combination of uniform and accelerated components → curved (parabolic) path	\(v_{\parallel}=v_0\cos\theta + \dfrac{qE}{m}t,\; v_{\perp}=v_0\sin\theta\)

11. Suggested Diagrams (for classroom use)

• Uniform field between parallel plates showing field lines, equipotential planes, and a charged particle entering at an angle.
• Trajectory plots for the three cases (parallel, perpendicular, arbitrary angle) overlaid on the field diagram.
• Vector diagram of the superposition of two point‑charge fields that approximates a uniform region.
• Energy diagram illustrating work done by the field and the corresponding change in kinetic energy.