Uniform Electric Fields (Parallel Plates)

1. Definitions (Must Know)

A. Uniform electric field

A uniform electric field is a field where electric field strength E has constant magnitude and direction (approximately true between large, parallel plates).

B. Electric field strength, E

Electric field strength, E, is force per unit positive charge:

E = F/q

Unit: N C⁻¹ (equivalently V m⁻¹).

C. Potential difference, Δ V

Potential difference, Δ V, is work done per unit charge by an external force:

Δ V = W/q

Unit: V, where 1 V = 1 J C⁻¹.

D. Potential gradient (1D)

Along a chosen axis x, the electric field is the negative potential gradient:

E = -dV/dx

2. Key Ideas (What Earns Marks)

• Between parallel plates separated by distance d with potential difference Δ V:
  • E = Δ V/d (magnitude)
  • direction: from the positive plate to the negative plate
• Field and potential are linked by:
  • E = -dV/dx (1D)
  • for a uniform field, E ≈ -Δ V/Δ x
• Force on a charge in a uniform field:
  • F = qE
• Motion is kinematics with constant acceleration:
  • a = qE/m
  • negative charge accelerates opposite to the field direction.

3. Detailed Explanations

A. Why |E| = |Δ V|/d between parallel plates

Choose the x-axis along the field. Electric field is the negative potential gradient:

Eₓ = -dV/dx

The gradient is constant in a uniform field. Across plate separation d,

Δ V = -Eₓd

Therefore the magnitude relationship used for plate calculations is

|E| = |Δ V|/d

The minus sign carries the direction: the field points from higher potential to lower potential.

B. Motion of a charged particle in a uniform field

Because F = qE is constant, the acceleration is constant:

a = F/m = qE/m

So you can use kinematics:

• If the initial velocity is parallel to the field: 1D constant acceleration.
• If the initial velocity is perpendicular to the field: uniform velocity in one direction + uniform acceleration in the other (a “projectile-like” path).

Data table

Example trajectory (proton, values from worked example)
Horizontal distance, x (cm)	Vertical deflection, y (mm)
0	0
1	0
2	0
3	0.1
4	0.2
5	0.3

4. Common Mistakes

• Using E = V/d without converting d to metres.
• Forgetting q can be negative (direction of acceleration flips).
• Mixing up units: 1 V/m = 1 N/C.
• Using potential difference when the question needs displacement along the field direction.

5. Exam Tips

• Always define axes and the sign convention (which direction is +).
• Use magnitudes first, then state the direction separately if signs are messy.
• Sanity check: a positive charge accelerates towards lower potential (towards the negative plate).

6. Worked Examples

Example 1: Find E and the force on an electronCore

Parallel plates have Δ V = 500 V across separation d = 0.020 m.

1. Find E.
2. Find the force on an electron (q = -1.60 × 10⁻¹⁹ C).

Show Answer

E = V/d = 500/0.020 = 2.5 × 10⁴ V/m

Force magnitude:

|F| = |q|E = (1.60 × 10⁻¹⁹)(2.5 × 10⁴) = 4.0 × 10⁻¹⁵ N

Direction: electron accelerates towards the positive plate (opposite to field direction).

Example 2: Acceleration of a proton in a uniform fieldCore

A proton (q = + 1.60 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg) is in a uniform field E = 1.0 × 10⁴ V/m. Find its acceleration.

Show Answer

a = qE/m; = (1.60 × 10⁻¹⁹)(1.0 × 10⁴)/1.67 × 10⁻²⁷; ≈ 9.6 × 10¹¹ m s⁻²

Example 3: Deflection while passing between platesCore

A proton enters a region of uniform electric field with horizontal speed u = 2.0 × 10⁶ m s⁻¹. The field region has length L = 0.050 m and field strength E = 1.0 × 10⁴ V/m. Find the vertical deflection when it exits the field (initial vertical velocity is zero).

Show Answer

Time in field: t = L/u = 0.050/2.0 × 10⁶ = 2.5 × 10⁻⁸ s

Acceleration: a = qE/m ≈ 9.6 × 10¹¹ m s⁻²

Deflection: y = 1/2at²; = 1/2(9.6 × 10¹¹)(2.5 × 10⁻⁸)²; ≈ 3.0 × 10⁻⁴ m

Example 4: Find potential difference from E and dCore

Parallel plates are separated by d = 12 mm. The field strength between them is E = 3.0 × 10⁴ V m⁻¹.

Find the potential difference between the plates (magnitude).

Show Answer

For a uniform field, E =; Delta V/d so: ; Delta V = Ed = (3.0; times10⁴)(12; times10⁻³) = 3.6; times10² V

Example 5: Find separation from a required fieldCore

You need a uniform field of E = 2.0 × 10⁵ V m⁻¹ using a potential difference of ; Delta V = 1.0 kV. What plate separation d is required?

Show Answer

d =; frac; Delta VE =; frac1.0; times10³2.0; times10⁵ = 5.0; times10⁻³ m = 5.0 mm

7. Mind Stretchers

Mind stretcher 1: Show that 1 V/m = 1 N/CExtension

Show Answer

1 V/m = 1 J/C/1 m = 1 N m/C m = 1 N/C

Mind stretcher 2: Why is the field approximately uniform between plates?Extension

Explain why the electric field is approximately uniform in the central region between two large, parallel plates.

Show Answer

In the central region, the field lines are roughly straight, parallel, and equally spaced, so the field has nearly constant direction and magnitude.

Non-uniformity occurs mainly near the edges due to fringing, but this is small if the plate size is much larger than the separation.

Mind stretcher 3: Simulation Bridge: Electric Field ExplorerExtension

Concept Explorer: Electric Field Explorer

Switch between parallel-plate and point-charge models, move the probe, and test E-V-r relationships with guided prompts.

BetaA LevelFieldsBest for: A Level electric fields revision

• E = V/d
• V = kQ/r
• E = kQ/r²
• Field Direction

Open the full interactive simulation on its own page

Use the standalone simulation page for the live controls, SVG scene, run modes, and scoring flow.

Open Interactive Simulation

The lesson stays lightweight and links out to the dedicated simulation page.

Explore uniform fields in the Electric Field Explorer.

8. Practice, Quiz and Next Step

Close your notes and use Uniform Electric Fields (Parallel Plates) in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: An unfamiliar data set or physical system requires you to apply Uniform Electric Fields (Parallel Plates) while stating the model, regime and assumptions.

1. Retrieve: define uniform electric fields (parallel plates) in your own words, including units, sign or conditions where relevant.
2. Represent: Choose and label an appropriate diagram, graph, table or symbolic model; derive or justify the relationship used.
3. Apply: Reach a conclusion, then evaluate it using units, uncertainty, a limiting case and one practical or modelling limitation.

Check the response before looking back

• The model, regime, coordinates and assumptions are explicit.
• The derivation or multi-step reasoning is visible rather than implied.
• The conclusion is tested against units, data quality and a limiting case.
• A practical control, uncertainty or model limitation is evaluated where applicable.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theA-Level Physics course hub orpractice browser for an independent re-test.