Source: Charges and Fields Lab, Dr. Howard
Difficulty: Introductory | Prerequisites: Basic algebra, vector components (x, y notation), familiarity with SI units.
Electric fields are one of the two pillars of electromagnetism (the other being magnetic fields). This topic sits right at the start of your E&M course because nearly everything that follows, from electric potential to capacitors to circuits, builds on how charges create and respond to fields. You should already be comfortable with vectors and basic Coulomb's law. The core idea: charges do not push or pull each other directly. Instead, a charge modifies the space around it (creating a field), and a second charge responds to that field.
A charge creates an electric field in the space around it. Any other charge placed in that space feels a force along the field's direction. The field from a positive charge points outward; from a negative charge, inward. When multiple charges are present, you add their individual field vectors (superposition) to get the net field at any point.
Electric field (E)
A vector field surrounding any electric charge that describes the force a positive test charge would experience per unit charge at each point in space. Units: N/C or V/m.
In simple terms, it is the "influence zone" a charge creates around itself. Drop a tiny positive charge anywhere in that zone and E tells you which way it gets pushed and how hard.
Field lines
Imaginary curves drawn so that the tangent at any point gives the direction of the electric field there. Their density (how closely packed they are) indicates field strength.
Think of them as flow lines on a weather map, except instead of wind they show the direction a positive test charge would drift.
Test charge
A hypothetical small positive charge used to probe an electric field. It must be small enough that it does not disturb the field it is measuring.
Think of it as a tiny sensor: you place it somewhere, see which way it wants to move, and that tells you the field direction.
Superposition principle
The net electric field at any point equals the vector sum of the individual fields produced by each charge. Each charge contributes independently; you simply add the vectors.
In simple terms, fields from different charges do not interfere with each other. They just stack.
Unit vector (r-hat)
A dimensionless vector of magnitude 1 that points from the source charge toward the field point. It gives the direction of the field contribution without affecting its magnitude.
Think of it as a pure direction arrow, one unit long, pointing from the charge to the spot you care about.
Coulomb's constant (k)
The proportionality constant in Coulomb's law and the electric field equation: k = 8.99 x 10^9 N·m²/C². It sets the scale for how strong electrostatic interactions are in SI units.
Point charge
An idealised charge with no physical size, treated as if all its charge sits at a single point. Most introductory problems model charges this way.
A charge affects the space around it, creating an electric field.
A second charge entering that space interacts with the field (not directly with the first charge).
The force on the second charge is along the direction of the field at that location.
This "field as intermediary" picture replaces action-at-a-distance and becomes essential once you reach electromagnetic waves.
Field lines point radially outward from the charge in all directions.
The field is strongest close to the charge (arrows are longer, lines are more densely packed).
A positive test charge placed nearby would be pushed away.
Field lines point radially inward, toward the charge.
Again, the field is strongest near the charge.
A positive test charge placed nearby would be pulled in.
Each charge produces outward-pointing field lines.
At the midpoint between two equal positive charges, the fields are equal and opposite, so they cancel: E = 0.
Away from the midpoint, the fields partially cancel but a net field remains, generally pushing outward and away from the pair.
There is a "null point" on the line connecting them, exactly at the centre for equal charges.
Field lines leave the positive charge and curve toward the negative charge.
The fields between the charges reinforce each other (both point from + toward -).
At the midpoint, the net field is nonzero and points from + to -.
This configuration is called an electric dipole and its field pattern is one of the most commonly tested shapes in introductory E&M.
To find the net field at any point, calculate the field contribution from each charge separately, then add the vectors component by component.
For two charges, that means: compute E₁ (from charge 1), compute E₂ (from charge 2), then E_net = E₁ + E₂.
Symmetry often zeroes out one component. For example, at the midpoint of two equal like charges, both x-components cancel, leaving E = 0 there.
Electric field from a point charge
E = kQ / r² · r̂
Where:
E = electric field vector (N/C)
k = Coulomb's constant = 8.99 × 10⁹ N·m²/C²
Q = source charge (C), with sign included
r = distance from the charge to the field point (m)
r̂ = unit vector pointing from the charge to the field point
Superposition (two charges)
E_net = E₁ + E₂
Compute each field vector separately, then add component by component:
E_net,x = E₁ₓ + E₂ₓ
E_net,y = E₁ᵧ + E₂ᵧ
Unit vector
r̂ = (field point − charge position) / |field point − charge position|
For example, if the charge is at the origin and the field point is at (1, 1), the displacement is (1, 1), its magnitude is √2, so r̂ = (1/√2, 1/√2).
Electric field mapping is the basis of how engineers design capacitors, cathode ray tubes, and electrostatic precipitators (the devices that clean soot from power-plant exhaust). The same superposition principle you practise here with two point charges scales up to model the fields around circuit boards, antennas, and the electrodes in medical devices like defibrillators.
"Field lines from opposite charges cancel." They do not. Field lines from a positive charge and a negative charge point in the same general direction between them (both from + toward -), so the fields reinforce there, not cancel. It is two like charges whose fields cancel at the midpoint.
"The electric field exists only where there is a charge." The field fills all of space around a charge. A charge is the source of the field, but the field itself extends outward to every point.
"Field lines can cross." They cannot. At any point in space there is exactly one net field direction. If lines crossed, that point would have two directions, which is physically impossible.
"A stronger charge means longer field lines." The lines themselves are infinitely long (they start or end at charges or at infinity). A stronger charge means more field lines, not longer ones. Density of lines, not length, encodes strength.
⚠️ You will be asked to sketch field lines for single charges and charge pairs. Know the patterns cold: outward for +, inward for -, dipole loops for unlike charges, mutual repulsion pattern for like charges.
⚠️ Expect a calculation where you find the net E-field at a point due to two or more charges. The method is always the same: find each field vector using E = kQ/r² · r̂, then add components.
⚠️ Watch the sign of Q. For a negative charge, the field vector points toward the charge (the negative sign flips r̂). If you forget the sign, your field direction will be backwards.
⚠️ The zero-field (null) point between two like charges is a classic exam question. For equal charges, it is exactly at the midpoint. For unequal charges, it shifts toward the weaker charge.
True or false: Electric field lines point away from a negative charge. (False, they point toward it.)
Fill in the blank: At the exact midpoint between two equal positive charges, the net electric field is ____. (Zero.)
True or false: The superposition principle says you multiply the fields from each charge to get the total. (False, you add them as vectors.)
Fill in the blank: The SI unit of electric field is ____ or equivalently ____. (N/C or V/m.)
True or false: Field lines from two unlike charges cancel between them. (False, they reinforce between them.)
Q: A single +1 nC charge sits at the origin. What is the electric field at the point (0 m, 0.5 m)?
A: r = 0.5 m, r̂ = ⟨0, 1⟩. E = (8.99 × 10⁹)(1 × 10⁻⁹) / (0.5)² · ⟨0, 1⟩ = 35.96 · ⟨0, 1⟩ = ⟨0, 35.96⟩ N/C. The field points straight up.
Q: That same +1 nC charge is at the origin. What is the field at (1 m, 1 m)?
A: r = √2 m, r̂ = ⟨1/√2, 1/√2⟩. E = (8.99 × 10⁹)(1 × 10⁻⁹) / 2 · ⟨1/√2, 1/√2⟩ = 4.495 · ⟨0.707, 0.707⟩ = ⟨3.178, 3.178⟩ N/C.
Q: Two +1 nC charges are placed at (-0.5 m, 0) and (0.5 m, 0). What is the net field at the origin (0, 0)?
A: By symmetry, the field from the left charge points right (+x) and the field from the right charge points left (-x). Both have the same magnitude, so E_net = ⟨0, 0⟩ N/C. The fields cancel completely.
Q: A +1 nC charge is at (-0.5 m, 0) and a -1 nC charge is at (0.5 m, 0). What is the net field at the origin?
A: The +1 nC charge produces a field pointing to the right (+x) at the origin: E₁ = ⟨35.96, 0⟩ N/C. The -1 nC charge also produces a field pointing to the right (toward itself, which is +x direction): E₂ = ⟨+35.96, 0⟩ N/C. The fields reinforce: E_net = ⟨71.92, 0⟩ N/C.
Q: Explain why the net electric field at the midpoint of two unlike charges is nonzero, while it is zero for two like charges of equal magnitude.
A: For like charges, the fields at the midpoint point in opposite directions (both away from their respective sources) and have equal magnitudes, so they cancel. For unlike charges, the field from the positive charge points away from it (toward the midpoint and beyond) while the field from the negative charge points toward it (also in the same direction, from + toward -). Both contributions point the same way, so they add rather than cancel.
This material connects directly to electric potential (voltage). The electric field is the negative gradient of the potential: E = -dV/dr. Once you are comfortable with fields, the jump to potential is a short one.
Superposition of fields also sets the stage for Gauss's law, where you will use the symmetry of field patterns (like the radial field of a point charge) to calculate fields for entire charge distributions without summing individual contributions.
The dipole field pattern you studied here reappears in molecular physics (polar molecules like water are electric dipoles) and in antenna theory.
electric field, E-field, field lines, field vectors, Coulomb's law, point charge, superposition, vector addition, dipole, like charges, unlike charges, opposite charges, test charge, field direction, field magnitude, N/C, V/m, Coulomb's constant, k constant, charge configuration, radial field, PHY-222, Classical Physics II, PhET charges and fields