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An electric field is a region of space in which a charged particle experiences a force. This concept is central to understanding how charges interact at a distance, and it underpins everything from simple circuits to particle accelerators. Electric fields are invisible, but their effects are measurable and predictable using well-defined mathematical relationships.
Electric field strength (E) is defined as the force per unit positive charge placed at that point:
E = F / Q
where F is the force in newtons (N) and Q is the charge in coulombs (C). The unit of electric field strength is N C⁻¹ (newtons per coulomb), which is equivalent to V m⁻¹ (volts per metre).
Electric field strength is a vector quantity — it has both magnitude and direction. By convention, the direction of the electric field is the direction in which a positive test charge would move if placed in the field. This means field lines point away from positive charges and towards negative charges.
The definition uses a "unit positive charge" as a theoretical probe. In practice, the test charge must be small enough not to disturb the field it is measuring. If you placed a large charge near another charge, the large charge would redistribute the source charge and distort the field. The test charge is an idealisation that lets us map the field without changing it.
Electric field lines are a visual tool for representing the direction and relative strength of an electric field.
Rules for field lines:
A point charge (or a charged sphere, when viewed from outside) produces a radial field. The field lines spread outward in all directions from a positive charge (or converge inward towards a negative charge). The field strength decreases with distance from the charge — the lines spread further apart.
The key feature of a radial field is that it follows an inverse-square law: the field strength is proportional to 1/r². At twice the distance, the field is one quarter as strong. At three times the distance, it is one ninth as strong.
Between two parallel plates connected to a potential difference, the electric field is uniform. The field lines are parallel, equally spaced, and perpendicular to the plates. This means the field strength is the same at every point between the plates (ignoring edge effects).
At the edges of the plates, the field lines curve outward — this is called the fringe effect. In exam questions, you are usually told to ignore edge effects and treat the field as perfectly uniform.
For a uniform field between parallel plates separated by a distance d with a potential difference V across them:
E = V / d
This is an extremely useful relationship. It tells you that doubling the voltage doubles the field strength, and doubling the plate separation halves it.
Two parallel plates are separated by 5.0 cm and have a potential difference of 2000 V across them. Calculate the electric field strength between the plates.
E = V / d = 2000 / 0.050 = 40 000 V m⁻¹ = 4.0 × 10⁴ V m⁻¹
Note the conversion of 5.0 cm to 0.050 m — always work in SI units.
A pair of parallel plates separated by 1.2 cm are used in an ink-jet printer to deflect charged droplets. The maximum field strength the air can sustain without breakdown (sparking) is 3.0 × 10⁶ V m⁻¹. What is the maximum voltage that can be applied across the plates?
V = Ed = 3.0 × 10⁶ × 0.012 = 36 000 V = 36 kV
In reality, ink-jet printers use much lower voltages (a few kV) with very small plate separations (a few mm) to achieve the required deflection.
The direction of the electric field is defined as the direction a positive charge would accelerate. This means:
When solving problems involving negative charges (such as electrons), remember that the force on the charge is in the opposite direction to the field. An electron placed in a uniform field between parallel plates accelerates towards the positive plate — against the field direction.
Rearranging E = F / Q gives:
F = EQ
For a uniform field (E = V / d), this becomes:
F = VQ / d
This force is constant throughout the uniform field, meaning a charged particle between parallel plates experiences a constant acceleration — exactly like a mass in a uniform gravitational field near the Earth’s surface.
An electron (charge = 1.6 × 10⁻¹⁹ C, mass = 9.11 × 10⁻³¹ kg) enters a uniform electric field of strength 5.0 × 10³ V m⁻¹. Calculate the force on the electron and its acceleration.
Force: F = EQ = 5.0 × 10³ × 1.6 × 10⁻¹⁹ = 8.0 × 10⁻¹⁶ N
Acceleration: a = F / m = 8.0 × 10⁻¹⁶ / 9.11 × 10⁻³¹ = 8.8 × 10¹⁴ m s⁻²
This is an enormous acceleration — roughly 10¹⁴ times greater than g. This illustrates why electric forces dominate over gravitational forces at the atomic and subatomic scale.
A proton (charge = 1.6 × 10⁻¹⁹ C, mass = 1.67 × 10⁻²⁷ kg) enters horizontally between two parallel plates with a horizontal velocity of 1.0 × 10⁶ m s⁻¹. The plates are 4.0 cm long and separated by 2.0 cm with a potential difference of 200 V. Calculate the vertical deflection as the proton exits the plates.
E = V / d = 200 / 0.020 = 10 000 V m⁻¹
F = EQ = 10 000 × 1.6 × 10⁻¹⁹ = 1.6 × 10⁻¹⁵ N
a = F / m = 1.6 × 10⁻¹⁵ / 1.67 × 10⁻²⁷ = 9.58 × 10¹¹ m s⁻²
Time to traverse plates: t = L / v = 0.040 / 1.0 × 10⁶ = 4.0 × 10⁻⁸ s
Vertical deflection: s = ½at² = ½ × 9.58 × 10¹¹ × (4.0 × 10⁻⁸)² = ½ × 9.58 × 10¹¹ × 1.6 × 10⁻¹⁵ = 7.7 × 10⁻⁴ m = 0.77 mm
Since this is much less than half the plate separation (10 mm), the proton exits without hitting a plate.
There is a close analogy between electric and gravitational fields:
| Property | Electric Field | Gravitational Field |
|---|---|---|
| Field strength definition | Force per unit charge (E = F/Q) | Force per unit mass (g = F/m) |
| Uniform field | Between parallel plates | Near Earth’s surface |
| Radial field | Around a point charge | Around a point mass |
| Field lines | Start on +, end on − | Always point towards mass |
| Force can be | Attractive or repulsive | Attractive only |
| Source property | Charge (positive or negative) | Mass (always positive) |
This analogy is helpful throughout this topic. Many of the mathematical relationships have the same structure, differing only in whether they involve charge or mass. A charged particle moving horizontally between parallel plates follows a parabolic path — exactly like a ball thrown horizontally in a gravitational field.
Cathode ray oscilloscopes (CROs): Electrons are accelerated and then deflected by uniform electric fields between parallel plates. Two sets of plates (horizontal and vertical) allow the electron beam to be steered to any point on the screen. The deflection is proportional to the applied voltage, which is how the CRO displays waveforms.
Electrostatic precipitators: Used in coal-fired power stations to remove particulate pollution from flue gases. Particles pass through a strong electric field between plates, become charged, and are attracted to collecting plates where they accumulate and can be removed. This technology removes over 99% of particulate matter.
Capacitive touchscreens: Your finger changes the local electric field pattern on the screen surface. Sensors detect this change and determine the touch location. The field principles are exactly those covered in this lesson.