Electric Field Lines

The electric field is an invisible vector quantity spread across all of space. You can write equations for it, but can you see it? Michael Faraday thought you could, at least in a manner of speaking. He developed a brilliant pictorial method that lets you map out the direction and strength of the electric field around any arrangement of charges, all without writing a single formula. These are electric field lines.

Visualising the Field Around a Point Charge

Start with a single positive point charge sitting at the origin. At every point in the surrounding space, the electric field $\mathbf{E}$ points radially outward (away from the charge) and gets weaker with distance. If you were to draw tiny arrows at many points around the charge, each arrow pointing along $\mathbf{E}$ and sized proportional to its strength, you would get a starburst of arrows: long ones close to the charge, shorter ones farther away.

Fig 1.12: Field of a point charge

Now connect the arrows that point along the same radial direction. Each connected chain becomes a smooth curve radiating outward from the charge. These curves are the field lines.

What the Density of Lines Tells You

Once you draw field lines as continuous curves instead of individual arrows, the arrow length (which carried the magnitude information) disappears. Does that mean you have lost track of how strong the field is? Not at all. The field strength is now encoded in how closely packed the lines are:

• Near the charge, where $\mathbf{E}$ is strong, the lines bunch together tightly.
• Far from the charge, where $\mathbf{E}$ is weaker, the same lines fan out and the spacing between them grows.

The key quantity is the number of lines crossing a unit area held perpendicular to the lines. This “line density” is proportional to the magnitude of the electric field at that location.

One important caution: the total number of lines drawn in any diagram is a choice made by the artist. Someone else might draw twice as many. What matters physically is the relative density of lines across different regions, not the absolute count.

From Two Dimensions to Three

Diagrams are drawn on a flat page, but the real world has three dimensions. To properly measure line density, you must count lines crossing a unit area perpendicular to them, not just count lines on a 2D slice.

Fig 1.13: Dependence of electric field strength on distance and its relation to the number of field lines

Consider two small area elements placed perpendicular to the field lines at points R (closer to the charge) and S (farther away). The element at R has more lines passing through each unit of area, confirming that the field at R is stronger than at S.

Solid Angle and the Inverse-Square Dependence

To see precisely why field line density encodes the inverse-square law, you need the idea of a solid angle (the three-dimensional version of an ordinary angle).

Ordinary Angle in Two Dimensions

Place a small straight-line segment of length $\Delta l$ at a distance $r$ from a point O. The planar angle it subtends at O is:

$\Delta\theta = \frac{\Delta l}{r}$

Solid Angle in Three Dimensions

Now replace the line segment with a small flat area $\Delta S$ held perpendicular to the radial direction at a distance $r$ from O. The solid angle subtended by this area at O is:

$\Delta\Omega = \frac{\Delta S}{r^2}$

A solid angle measures the “opening” of a cone. More precisely, if you draw a cone from O through the edges of $\Delta S$ and let it intersect a sphere of radius $R$ centred at O, the solid angle equals the area cut out on that sphere divided by $R^2$.

Connecting Solid Angle to Field Strength

Imagine a cone of solid angle $\Delta\Omega$ emerging from a point charge $q$. At distance $r_1$, this cone cuts across an area $r_1^2 \, \Delta\Omega$. At a larger distance $r_2$, it cuts across a larger area $r_2^2 \, \Delta\Omega$. But the number of radial field lines inside that cone stays the same, call it $n$, because no lines are created or destroyed along the way.

The line density (lines per unit area) at $r_1$ is:

$\frac{n}{r_1^2 \, \Delta\Omega}$

and at $r_2$:

$\frac{n}{r_2^2 \, \Delta\Omega}$

Since $n$ and $\Delta\Omega$ are both constant, the line density scales as $1/r^2$. Because the density of field lines represents the electric field strength, this confirms the $1/r^2$ dependence of the electric field due to a point charge.

Faraday’s Contribution: From “Lines of Force” to Field Lines

Credit for this powerful visual tool goes to Michael Faraday. Without using a single equation, Faraday found a way to map out how the electric field behaves around any arrangement of charges. He originally named these curves lines of force, but the modern term field lines has replaced that phrase. Why the name change? “Lines of force” suggests that something is pulling along the line, which creates confusion when the same idea is applied to magnetic fields. “Field lines” avoids that misleading implication.

Formal Definition of a Field Line

Here is the precise definition: at every point along the curve, the tangent must point in the same direction as the net electric field $\mathbf{E}$ at that location. Since a tangent can point in two opposite directions along any curve, an arrow is drawn on the line to remove the ambiguity and show which way $\mathbf{E}$ actually points.

A field line is a space curve: it exists in full three dimensions, even though diagrams typically show a two-dimensional cross-section.

Field Line Patterns for Common Charge Configurations

Fig 1.14: Field lines due to some simple charge configurations

Single Positive Charge

Field lines point radially outward from the charge in every direction, spreading symmetrically like the spokes of a wheel viewed from the side.

Single Negative Charge

Field lines point radially inward, converging on the charge from all directions. This is the mirror image of the positive charge pattern.

Two Equal Positive Charges Side by Side

Lines emerge from each charge but cannot connect to the other (since field lines never run into a positive charge). Instead, they curve away from the space between the two charges. A neutral point exists at the midpoint, where the two fields cancel and the net field is zero. The pattern beautifully illustrates the mutual repulsion between like charges.

Electric Dipole (Equal and Opposite Charges)

Lines emerge from the positive charge $+q$ and curve around through space to end on the negative charge $-q$. The resulting pattern illustrates the mutual attraction between opposite charges. Close to each charge the lines look radial; farther out, they sweep in graceful arcs connecting the two.

Four Properties That Field Lines Must Obey

Regardless of how complicated a charge arrangement might be, electric field lines always follow these rules:

(i) Every field line begins on a positive charge and terminates on a negative charge. When only one type of charge is present, lines have no local endpoint: those leaving a positive charge head outward toward infinity, while those approaching a negative charge arrive from infinity.

(ii) In any region free of charges, field lines are continuous, unbroken curves. Lines can only begin or end where a charge sits. In empty space, they pass through smoothly.

(iii) Two field lines can never cross each other. At any point in space the electric field has one definite direction. If two lines crossed, the tangent at the intersection would point in two directions simultaneously, which is physically impossible.

(iv) Electrostatic field lines never form closed loops. This is a direct consequence of the conservative nature of the electrostatic field. In a conservative field, the work done in moving a charge around any closed path is zero. A closed field line would allow a test charge to gain energy endlessly by cycling around the loop, violating energy conservation.