An electric charge affects the space around it even before anything else is brought near. This influence on space is the electric field. Place a second charge in that field and it feels a force. The related quantity — electric potential — tells you how much energy a charge gains or loses by moving through the field. Together, electric field and potential are the two fundamental ways to describe the electromagnetic influence of charges on their surroundings, and they underlie everything from lightning rods to capacitors to the electrical signals in your nervous system.
Electric field E: the force per unit positive test charge at a point in space. E = F/q. Unit: N/C or V/m. A vector quantity — it points in the direction a positive charge would be pushed.
Electric potential V: the work done per unit positive charge to move it from infinity to that point against the electric field. V = W/q. Unit: volts (V = J/C). A scalar quantity.
Electric Field: E = F/q
The electric field at a point is defined as the force that would be exerted on a small positive test charge q placed there:
The unit is N/C (newtons per coulomb) or equivalently V/m (volts per metre). The electric field is a vector — it has both magnitude and direction. The direction at any point is the direction of force on a positive charge. Negative charges experience force in the opposite direction to E.
For a point charge Q at distance r:
where k = 8.99 × 10⁹ N·m²/C². This follows directly from Coulomb's Law. For positive Q, E points radially outward; for negative Q, E points radially inward.
Electric Field Lines
Electric field lines are a visual representation of the field:
• They start on positive charges and end on negative charges (or extend to infinity).
• They never cross (the field has a unique direction at every point).
• Their density (how close together they are) indicates field strength — closer lines mean stronger field.
• They are always perpendicular to conducting surfaces and to equipotential surfaces.
For a uniform field between two parallel plates (separation d, potential difference V):
The field lines are parallel and equally spaced — field strength is the same everywhere between the plates.
Electric Potential: V = kQ/r
Electric potential V at a point is the work done per unit charge to bring a positive test charge from infinity to that point:
Potential is a scalar — it has magnitude but no direction. This makes it easier to work with than the vector field in many situations. The potential at a point due to multiple charges is the algebraic (scalar) sum of individual potentials:
The unit of potential is the volt (V = J/C). A potential difference of 1 V between two points means 1 joule of work is done moving 1 coulomb of charge between them.
The Relationship Between E and V
Electric field and potential are not independent — they are related by:
The electric field is the negative gradient of potential. This means the field points from high potential to low potential (downhill in the energy landscape). The negative sign ensures that positive charges are pushed from high to low potential (their natural tendency).
For a uniform field: E = V/d (magnitude), where V is the potential difference across distance d.
Equipotential Surfaces
Equipotential surfaces are surfaces on which the electric potential is constant everywhere. Moving a charge along an equipotential surface requires no work (W = qΔV = 0 when ΔV = 0). Key properties:
• Always perpendicular to field lines.
• For a point charge: spherical shells centred on the charge.
• For a uniform field: flat planes perpendicular to the field direction.
• Conductors in electrostatic equilibrium are equipotential surfaces — if they weren't, charges would rearrange until they were.
Worked Examples
Example 1: Field from a point charge
Find E at 0.30 m from a +5 μC charge.
Example 2: Potential from a point charge
Find V at 0.20 m from a +8 μC charge.
Example 3: Energy of a test charge
A +2 μC test charge is moved from V = 100 V to V = 400 V. How much work is done?
Example 4: Field between parallel plates
Two plates separated by 5 mm with a potential difference of 500 V between them.
Capacitors: Storing Energy in Electric Fields
A capacitor stores energy in the electric field between two conducting plates. Capacitance C (farads, F) relates charge Q and potential difference V:
Energy stored in a capacitor:
For a parallel-plate capacitor with plate area A, separation d, and dielectric constant ε_r:
where ε₀ = 8.85 × 10⁻¹² F/m is the permittivity of free space. Capacitors are fundamental components in every electronic circuit — they store energy for flash photography, filter signals in audio equipment, stabilise power supplies, and store energy for rapid discharge in defibrillators.
In atomic and nuclear physics, the joule is inconveniently large. Instead, the electron-volt (eV) is used: the energy gained by one electron moving through a potential difference of 1 volt. 1 eV = 1.6 × 10⁻¹⁹ J. Typical atomic bond energies: a few eV. Typical nuclear reactions: millions of eV (MeV). LHC proton energy: 6.5 × 10¹² eV = 6.5 TeV.
Real-World Applications
CRT displays: electrons are accelerated through a potential difference, acquiring kinetic energy KE = eV, then deflected by electric fields to scan the screen.
Electrocardiograms (ECG): measure potential differences on the skin surface caused by the electrical activity of the heart — tiny voltages (~1 mV) that reveal cardiac function.
Van de Graaff generators: build up very high potential (millions of volts) by continuously separating charge, creating large electric fields that can accelerate particles in linear accelerators.
Lightning conductors: provide a low-resistance path for charge to flow, preventing the build-up of very high potential differences that would lead to damaging lightning strikes through buildings.
Frequently Asked Questions
What is an electric field?
The electric field E at a point is the force per unit positive charge: E = F/q. It is a vector (pointing in the direction a positive charge would be pushed). For a point charge Q: E = kQ/r². Unit: N/C or V/m. The field describes how space is modified by the presence of charge.
What is electric potential?
Electric potential V is the work done per unit positive charge to bring it from infinity to that point: V = W/q. It is a scalar (no direction). For a point charge Q: V = kQ/r. Unit: volts (V = J/C). The potential difference between two points equals the work done per coulomb in moving charge between them.
What is the relationship between electric field and potential?
E = −dV/dr. The electric field is the negative gradient of potential — it points from high to low potential. For a uniform field: E = V/d. This means if you know how potential varies in space, you can find the field, and vice versa. Field lines are always perpendicular to equipotential surfaces.
What are equipotential surfaces?
Equipotential surfaces are surfaces of constant electric potential. No work is done moving a charge along an equipotential. They are always perpendicular to field lines. Around a point charge they are concentric spheres. Between parallel plates they are flat planes parallel to the plates.
What is the unit of electric field?
N/C (newtons per coulomb) or equivalently V/m (volts per metre). The two units are identical: 1 N/C = 1 V/m. This follows from E = F/q = (N)/(C) and E = V/d = (V)/(m). The V/m form is often more intuitive when dealing with potential differences across known distances.
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Written by
Dr. Sarah KimThermodynamics researcher with a PhD from MIT, specializing in statistical mechanics and energy transfer. Passionate about connecting molecular physics to everyday phenomena.
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