What the Electric Field Is
Consider a point charge $Q$ fixed at the origin. Place a second charge $q$ at a point $P$ a distance $r$ away, and $Q$ exerts a force on $q$ as given by Coulomb's law. NCERT poses a sharper question: if $q$ is removed, what is left at $P$? The answer the early scientists gave is that $Q$ produces an electric field everywhere in the surrounding space. When another charge is brought to $P$, the field already present there acts on it and produces the force.
Operationally, the field is the force per unit positive test charge. If a small positive test charge $q_0$ at a point experiences a force $\mathbf{F}$, then the electric field there is
$$\mathbf{E} = \frac{\mathbf{F}}{q_0}, \qquad \mathbf{F} = q\,\mathbf{E}$$
Equation $\mathbf{F} = q\mathbf{E}$ also fixes the SI unit: with force in newtons and charge in coulombs, $E$ is measured in newtons per coulomb (N/C). NCERT notes the alternate unit volt per metre (V/m), introduced in the next chapter — the two are numerically identical. Because force is a vector, the electric field is a vector field: it has a magnitude and a direction at every point in three-dimensional space.
The field does not depend on the test charge
Although $E$ is defined as $F/q$, the field is independent of $q$. Since $F$ is proportional to $q$, the ratio $F/q$ cancels the dependence. The field is a property of the source and the point in space — not of whatever charge you happen to drop there.
To avoid the test charge disturbing the source, NCERT writes the exact definition as a limit: $\;\mathbf{E} = \displaystyle\lim_{q \to 0}\frac{\mathbf{F}}{q}.$
Field of a Point Charge
Setting $q_0$ to unity in $F = q_0 E$ shows that the electric field due to a charge $Q$ is numerically equal to the force a unit positive charge would feel. Substituting Coulomb's law gives the field of a point charge at distance $r$:
$$\mathbf{E} = \frac{1}{4\pi\varepsilon_0}\,\frac{Q}{r^{2}}\,\hat{\mathbf{r}} = k\,\frac{Q}{r^{2}}\,\hat{\mathbf{r}}, \qquad k = \frac{1}{4\pi\varepsilon_0} \approx 9 \times 10^{9}\ \text{N m}^2\,\text{C}^{-2}$$
Here $\hat{\mathbf{r}}$ is the unit vector pointing from the source charge to the field point. NCERT lists the consequences plainly: for a positive source charge the field points radially outward; for a negative source charge it points radially inward. Because the magnitude depends only on $r$, the field has the same value everywhere on a sphere centred on the charge — it possesses spherical symmetry.
$E = kQ/r^2$ is the field; $F = qE$ is the force
$E = kQ/r^2$ involves only the source charge $Q$ — it is the field the source sets up, with or without anything placed in it. The force on a charge $q$ that you then introduce is $F = qE$. Writing $F = kQ/r^2$ (omitting the second charge) or treating $E$ as if it depended on $q$ are both classic errors.
Field = cause set up by the source. Force = effect felt by a charge placed in it.
Superposition of Fields
For a system of charges $q_1, q_2, \ldots, q_n$, the field at a point $P$ is again defined as the force on a unit positive test charge placed there without disturbing the sources. Combining Coulomb's law with the superposition principle, the total field is the vector sum of the fields each charge would produce alone:
$$\mathbf{E}(\mathbf{r}) = \mathbf{E}_1 + \mathbf{E}_2 + \cdots + \mathbf{E}_n = \frac{1}{4\pi\varepsilon_0}\sum_{i=1}^{n}\frac{q_i}{r_{iP}^{2}}\,\hat{\mathbf{r}}_{iP}$$
Each $\hat{\mathbf{r}}_{iP}$ points from charge $q_i$ to $P$, and $r_{iP}$ is the distance between them. Because $\mathbf{E}$ is a vector, the contributions must be added head-to-tail (or by components), not as bare numbers — a point the worked example below makes concrete.
Two point charges $q_1 = +10^{-8}\,\text{C}$ and $q_2 = -10^{-8}\,\text{C}$ are placed $0.1\,\text{m}$ apart. The field is required at three points around them.
At a point A located $0.05\,\text{m}$ from each charge along the line joining them, both fields point the same way: $E_{1A} = E_{2A} = 3.6\times10^{4}\ \text{N C}^{-1}$, so they add to $E_A = 7.2\times10^{4}\ \text{N C}^{-1}$ directed toward the right. At a point B beyond the positive charge the two fields oppose, giving $E_B = 3.2\times10^{4}\ \text{N C}^{-1}$ to the left. The arithmetic differs only because the directions differ — the same superposition rule governs all three points.
The vector-sum rule used here is developed in full in Forces Between Multiple Charges — work through it before tackling field problems.
Physical Significance of the Field
For electrostatics, NCERT is candid: the electric field is convenient but not strictly necessary, since the measurable quantity — the force on a charge — can be found directly from Coulomb's law and superposition. The field is, for static charges, simply an elegant way of characterising the electrical environment, telling us at each point what force a unit positive charge would feel.
Its deeper meaning emerges only beyond electrostatics, with charges in accelerated motion. No signal can travel faster than the speed of light $c$, so the effect of a sudden movement of one charge cannot reach a distant charge instantaneously — there is a time delay between cause and effect. The field accounts for this elegantly: an accelerating charge launches electromagnetic waves that propagate at speed $c$ and only later exert a force on the distant charge.
Because of this, fields are regarded as physical entities with their own dynamics, able to carry energy, and not merely as mathematical bookkeeping. The concept, introduced by Faraday, is now central to physics.
| Aspect | What NCERT establishes |
|---|---|
| Definition | Force per unit positive test charge, $E = F/q_0$, taken in the limit $q_0 \to 0$. |
| Independence | $E$ is independent of the test charge; it depends only on the source and the point in space. |
| Source vs. test charge | The charge producing the field is the source charge; the charge probing it is the test charge. |
| Spatial nature | A vector defined at every point of 3-D space; varies from point to point. |
| Finite propagation | Effects of a changing source travel outward at speed $c$, not instantaneously. |
Electric Field Lines
Since $\mathbf{E}$ is a vector at every point, we can draw arrows whose length is proportional to the field strength at each location. For a point charge these arrows point radially outward and shorten with distance because $E \propto 1/r^2$. Connecting arrows that follow one another produces a continuous curve — a field line. Faraday named these lines of force; NCERT prefers the term field lines as more accurate.
Replacing arrows with lines seems to discard the information once carried by arrow length. It does not. The field magnitude is now encoded in the density of lines: where the field is strong, near a charge, the lines crowd together; where it is weak, far away, the lines spread apart. The number of lines crossing a unit area held perpendicular to them is proportional to the field magnitude there.
A neat self-consistency follows. As distance from a point charge increases, the field falls as $1/r^2$ while the area enclosing the charge grows as $r^2$, so the number of lines crossing any enclosing surface stays constant regardless of distance. The absolute number of lines one chooses to draw is arbitrary — only their relative density carries physical meaning.
Properties of Field Lines
A field line is, in general, a space curve drawn so that the tangent to it at any point gives the direction of the net field there; an arrowhead fixes which of the two tangent directions is meant. NCERT lists four governing properties, set out below.
| Property | Statement | Reason |
|---|---|---|
| Endpoints | Lines start on positive charges and end on negative charges; for a single charge they start or end at infinity. | Field points away from $+$ and toward $-$. |
| Continuity | In a charge-free region the lines are continuous curves with no breaks. | The field is defined and finite at every such point. |
| No crossing | Two field lines can never cross each other. | A crossing would give two field directions at one point — absurd. |
| No closed loops | Electrostatic field lines do not form closed loops. | Follows from the conservative nature of the electrostatic field. |
| Direction | The tangent at a point gives the direction of $\mathbf{E}$ there. | Lines are built by joining field arrows. |
| Density | Closeness of lines is proportional to field strength. | Lines per unit perpendicular area $\propto E$. |
Field lines never cross — and never loop
Two examiner favourites. First, no two field lines intersect: at a crossing the tangent would define two directions for $\mathbf{E}$, which is impossible. Second, electrostatic lines do not form closed loops — that is reserved for magnetic field lines. A diagram showing crossing lines or a closed electrostatic loop is automatically wrong.
Closer lines ⇒ stronger field. Lines begin on $+$, terminate on $-$ (or at infinity).
Field-Line Patterns
NCERT illustrates three standard configurations. A single charge gives the radial pattern of Figure 1 above. The remaining two — the dipole and a pair of like charges — reveal attraction and repulsion at a glance.
Electric Field & Field Lines in one screen
- Definition: $E = F/q_0$ — force per unit positive test charge, taken as $q_0 \to 0$. Unit: N/C ($=$ V/m). It is a vector field, independent of the test charge.
- Point charge: $E = kQ/r^2$, radially out for $+Q$, in for $-Q$; spherically symmetric. Don't confuse the field $kQ/r^2$ with the force $qE$.
- Superposition: total field is the vector sum $\mathbf{E} = \sum k q_i/r_{iP}^2\,\hat{\mathbf{r}}_{iP}$.
- Significance: a real physical entity that mediates the force and propagates changes at speed $c$.
- Field lines: tangent gives field direction; density ∝ strength; start on $+$, end on $-$; never cross; never form closed loops in electrostatics.
- Patterns: radial (single charge), curving $+$ to $-$ (dipole), bending apart with a null point (two like charges).