Electric Field Lines

What a field line is

An electric field line is a curve drawn so that the tangent to it at every point gives the direction of the net electric field at that point. An arrow on the curve is necessary, since a tangent by itself allows two opposite directions. A field line is a space curve — the figures are drawn in a plane, but the lines exist in three dimensions. Faraday invented the picture and called them lines of force; the modern term is field lines.

How magnitude is represented

• Start by drawing arrows at many points, each along $\vec{E}$ there and of length proportional to $E$. Joining arrows that point along one another gives a field line, and the length information is then carried instead by the density of lines.
• Lines crowd together where the field is strong and are far apart where it is weak. Only the relative density matters, since any number of lines may be drawn in a region.
• Density means lines per unit cross-sectional area taken perpendicular to the lines, since space is three-dimensional.
• The picture is self-consistent with the inverse-square law: within a given solid angle $\Delta\Omega$ the same number $n$ of radial lines passes through area $r_1^2\Delta\Omega$ at $P_1$ and $r_2^2\Delta\Omega$ at $P_2$, so the number of lines per unit area is $\dfrac{n}{r^2\Delta\Omega}$ — a $1/r^2$ dependence, exactly matching $E$ for a point charge.
• In a region of uniform field the lines are equally spaced parallel straight lines.

The four properties CBSE asks you to list

• Field lines start at positive charges and end at negative charges. For a single isolated charge they start or end at infinity.
• In a charge-free region, field lines are continuous curves without breaks.
• Two field lines can never cross each other.
• Electrostatic field lines do not form closed loops.

The two reasons that carry the marks

• Why lines never cross: if two lines crossed, the field at the point of intersection would have two different directions at the same instant. The electric field at a point has a unique direction, so this is impossible.
• Why lines never form closed loops: because the electrostatic field is conservative. Around a closed loop the line integral of $\vec{E}$ is zero, so the work done in carrying a charge round the loop is zero. Along a closed field line the force would always have a component along the motion and the work would be non-zero — a contradiction.

Standard patterns described in words

• Single positive charge: straight lines radiating outward uniformly in all directions.
• Single negative charge: the same radial lines, but directed inward towards the charge.
• Two equal positive charges: lines leave each charge and bend away from the other; between them is a neutral point on the line joining them, where no line passes, and the region between the charges is comparatively empty of lines — the picture of mutual repulsion.
• Dipole ($+q$ and $-q$): every line leaving $+q$ curves round and terminates on $-q$. The lines are densest along the axis in the region between the charges — the picture of mutual attraction. On the equatorial plane the lines run parallel to the dipole axis, from the $+q$ side towards the $-q$ side, which is why the equatorial field is antiparallel to $\vec{p}$.

Point charge near an uncharged conducting plate

This is a repeatedly set 1-mark sketch. The two things being marked are the induced charge and the right angle at the surface.

• Positive charge $+q$ near the plate: lines radiate from $+q$ in all directions; those heading towards the plate curve so as to meet its near face perpendicularly. The near face acquires an induced negative charge, and the lines terminate on it. The far face carries an equal induced positive charge.
• Negative charge $-Q$ near the plate: lines converge radially into $-Q$; the near face of the plate now carries induced positive charge and the far face induced negative charge, and the lines leave the plate's near surface at right angles and run into $-Q$.
• The rule behind both: field lines always meet the surface of a conductor normally, because any tangential component would drive the free charges along the surface until it vanished.
• Note that the plate as a whole stays neutral — the charge is only redistributed by induction.

Field lines and flux

The number of field lines crossing a unit area held normal to the field is a measure of the field strength there, and the number crossing a tilted area of magnitude $\Delta S$ is proportional to $E\,\Delta S\cos\theta$. That statement is the bridge to the next section: it is exactly the definition of electric flux.