Coulomb Law and Electric Field
Coulomb's Law and Electrostatic Force
Coulomb's Law quantifies the mutual electrostatic force between two stationary point charges as proportional to the product of charges and inversely proportional to the square of their separation.
$$F = k \frac{|q_1 q_2|}{r^2}$$
Determines the magnitude of force between two static charges.
$F$=Electrostatic force(N)
$k$=Coulomb's constant = $\frac{1}{4\pi\epsilon_0}$ ≈ 8.99 × 10⁹(N·m²/C²)
$q_1, q_2$=Charge magnitudes(C)
$r$=Distance between centers of the point charges(m)
$r \to 0$
→Force approaches infinity (point charge idealization breaks down)
$q_1, q_2$ same sign
→Repulsive force (positive product)
$q_1, q_2$ opposite sign
→Attractive force (negative product)
Inverse Square Scaling: Doubling $r$ reduces force to $1/4$; tripling $r$ reduces it to $1/9$. The pattern is $F' = F/n^2$ when distance is multiplied by $n$.
Permittivity of Free Space: $\epsilon_0 = 8.85 \times 10^{-12}$ C²/(N·m²) gives $k = 1/(4\pi\epsilon_0) = 9 \times 10^9$ N·m²/C².
Newton's Third Law: Coulomb's force is mutual — $\vec{F}_{12} = -\vec{F}_{21}$. Both charges experience equal and opposite forces regardless of their magnitudes.
Dimensional Check: $[k][q^2]/[r^2]$ = (N·m²/C²)(C²)/(m²) = N. Always verify your answer has units of force.
The vector form of Coulomb's Law gives both the magnitude and direction of the force using unit vectors along the line joining the charges.
$$\vec{F}_{21} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r}_{21}$$
Gives the force on charge 2 due to charge 1, including direction via the unit vector.
$\vec{F}_{21}$=Force on $q_2$ due to $q_1$(N)
$\hat{r}_{21}$=Unit vector from $q_1$ toward $q_2$(dimensionless)
Both charges positive
→Force is along $\hat{r}_{21}$ (repulsion, away from source)
Opposite charges
→Product $q_1 q_2 < 0$ reverses direction (attraction, toward source)
Unit Vector Direction: $\hat{r}_{21}$ points from the source charge ($q_1$) to where the force is felt ($q_2$). Swapping subscripts reverses direction: $\hat{r}_{12} = -\hat{r}_{21}$.
Sign Convention: In the vector form, keep the signs of $q_1$ and $q_2$. The formula automatically gives the correct direction — positive product means repulsion, negative means attraction.
When a dielectric material (insulator) fills the space between charges, the electrostatic force is reduced by the material's relative permittivity $\epsilon_r$.
$$F = \frac{1}{4\pi\epsilon_0 \epsilon_r} \frac{q_1 q_2}{r^2}$$
Coulomb's force in a medium is weaker than in vacuum by a factor of $\epsilon_r$.
$\epsilon_r$=Relative permittivity (dielectric constant) of the medium(dimensionless)
Vacuum ($\epsilon_r = 1$)
→Reduces to the standard Coulomb's law
Water ($\epsilon_r \approx 78.5$)
→Force is reduced to about $1/78$ of vacuum value
Air ($\epsilon_r \approx 1.0006$)
→Essentially the same as vacuum — safely use $\epsilon_r = 1$
Force Ratio Shortcut: $F_{\text{medium}} = F_{\text{vacuum}} / \epsilon_r$. No need to recalculate from scratch — just divide.
Why It Decreases: The dielectric's molecules partially align with the external field, creating an opposing internal field that effectively shields the charges from each other.
Key Values: Vacuum = 1, Air ≈ 1, Glass ≈ 5–10, Water ≈ 78.5. Water's high $\epsilon_r$ is why ionic compounds dissolve easily in it.
The Concept of the Electric Field
The electric field intensity (E) is defined as the force per unit positive test charge placed at a point, representing the property of space established by a source charge.
$$\vec{E} = \frac{\vec{F}}{q_0}$$
Defines field strength as a vector quantity independent of the test charge magnitude.
$\vec{E}$=Electric field vector(N/C (equivalently V/m))
$\vec{F}$=Force on test charge(N)
$q_0$=Magnitude of the positive test charge(C)
Positive charge in field
→Force is parallel to $\vec{E}$ ($\vec{F} = q\vec{E}$)
Negative charge in field
→Force is antiparallel to $\vec{E}$ ($\vec{F} = q\vec{E}$, but $q < 0$)
Field Exists Independently: The source charge creates a field whether or not another charge is there to feel it. The test charge merely probes what already exists.
Test Charge Must Be Small: If $q_0$ is too large, its own field rearranges the source charges, changing the very field you're trying to measure.
Dimensional Check: $[F]/[q]$ = N/C. The field can also be expressed as V/m (equivalent unit).
Electric Field of a Point Charge
A single source charge $Q$ creates a field that spreads radially with a magnitude determined by the distance $r$ from the charge center.
$$E = k \frac{|Q|}{r^2}$$
Calculates the field strength generated by a single point charge at distance $r$.
$E$=Electric field strength(N/C)
$Q$=Source charge(C)
$r$=Distance from source charge(m)
$Q > 0$
→Field points radially outward from the charge
$Q < 0$
→Field points radially inward toward the charge
$r \to \infty$
→Field approaches zero — charge influence fades with distance
Radial Symmetry: At a fixed distance $r$, the magnitude of $E$ is the same in every direction. Only the direction (radially outward or inward) changes with the sign of $Q$.
Proportionality Shortcut: If you know $E_1$ at distance $r_1$, then at distance $r_2$: $E_2 = E_1 (r_1/r_2)^2$. No need to recalculate from scratch.
Combining Force and Field: Once you know $\vec{E}$ at a point, the force on any charge $q$ placed there is simply $\vec{F} = q\vec{E}$.
Superposition Principle
The total electric field at any point is the vector sum of individual fields produced by each charge independently — this is the superposition principle.
$$\vec{E}_{net} = \sum_i \vec{E}_i$$
Total field is the cumulative vector sum of all contributing source fields.
$\vec{E}_{net}$=Net electric field vector at the point(N/C)
$\vec{E}_i$=Field contribution from the $i$-th charge(N/C)
Two equal like charges, midpoint
→Fields cancel → $\vec{E}_{net} = 0$ at midpoint
Two equal unlike charges, midpoint
→Fields add (both point toward the negative charge)
Vector Addition Required: You cannot just add magnitudes. Break each $\vec{E}_i$ into components, sum $x$-components and $y$-components separately, then recombine.
Zero-Field Point (Two Like Charges): Between two unequal positive charges $q_1 > q_2$, the null point is closer to the smaller charge. Set $E_1 = E_2$ and solve for position.
Zero-Field Point (Two Unlike Charges): The null point lies outside the pair, on the side of the smaller magnitude charge.
Electric Field Lines
Electric field lines are a visual map of the field: they show the direction of $\vec{E}$ at every point and their spacing indicates the field strength.
Origin and Termination: Field lines start on positive charges and end on negative charges. An isolated positive charge has lines going to infinity.
Tangent Rule: The tangent to a field line at any point gives the direction of $\vec{E}$ at that point.
Density ∝ Strength: Where lines are closely packed, the field is strong; where they spread apart, the field is weak.
Non-Crossing Rule: Field lines never intersect. If they did, $\vec{E}$ would have two directions at the crossing point, which is physically impossible.
Uniform Field: Parallel, equally spaced lines (e.g., between parallel plates of a capacitor) represent a uniform electric field — same $\vec{E}$ everywhere.
Common field line patterns include the radial field of a single charge, the dipole pattern of two opposite charges, and the repulsion pattern of two like charges.
Single Positive Charge: Lines radiate outward uniformly in all directions.
Single Negative Charge: Lines point radially inward from all directions.
Dipole (±q): Lines curve from the positive charge to the negative charge. The field is strongest along the axis connecting them.
Two Like Charges: Lines repel in the middle region. A neutral zone (zero-field point) exists between them.
Parallel Plates: Uniform, parallel lines in the central region; fringe effects at the edges.
Applications of Electrostatics
A photocopier (xerography) uses electrostatic principles: a selenium-coated drum retains charge in dark areas (image pattern), attracts oppositely charged toner, and transfers it to paper.
Photoconductor: Selenium is an insulator in the dark but becomes a conductor when exposed to light. Light areas lose their charge; dark areas retain it.
Toner Transfer: Negatively charged toner sticks to positively charged (dark) areas on the drum, then transfers to paper and is fused by heated rollers.
Inkjet printers use a charging electrode and deflection plates (a parallel-plate capacitor) to steer ink droplets: uncharged drops hit the paper, while charged drops are deflected into a gutter.
Deflection Mechanism: Charged droplets experience a force $\vec{F} = q\vec{E}$ between the deflection plates, curving them away from the paper into a collection gutter.
Print Control: The computer turns the charging electrode on/off — uncharged droplets fly straight to paper, charged ones are diverted.