Physics · Electricity and magnetism · Grade 7

Conductor, dielectric, lightning

In a thunderstorm, why is it safe inside a car but dangerous under a lone tree?

The situation

On a stifling summer day a jet-black cloud appeared over the city: lightning flashed and thunder rumbled. A driver on the road stayed inside the car and waited — a correct decision. A thin metal rod on top of a tower — a lightning rod — takes the lightning itself and carries the current to the ground. The same physics works in the classroom: a charged ball redistributes the charges in a metal sphere without touching it, and a rubbed comb lifts bits of paper. How do a conductor and a dielectric behave in an electric field — and why are you safe from lightning inside a car?

Topic

Conductors, dielectrics and lightning. A lightning rod in a thunderstorm and a person inside a Faraday cage

Key ideas

Electrostatic induction
Inside a conductor E = 0
Lightning conductor

Formulas

{\vec{E}}_{in} = 0, φ = const

In electrostatic equilibrium there is no field inside a conductor, all its points are at the same potential, and excess charge sits only on the surface.

E = \frac{σ}{ε_{0}}

The field next to the surface of a conductor is perpendicular to the surface and proportional to the surface charge density $σ$ .

Q = - q \frac{R}{d}

The charge induced on an earthed sphere of radius $R$ by a charge $q$ at distance $d$ (method of images).

E_{in} = \frac{3}{ε + 2} E_{0}

Inside a dielectric sphere in a uniform field $E_{0}$ the field is reduced, but does not fall to zero.

E_{tip} = β E_{0}, R_{str} \approx 10 I^{0.65}

At a tall, sharp tip the field is magnified $β$ times. If the lightning current is $I$ (kA), it “jumps” to the nearest object at a distance $R_{str}$ (m).

Theory

Conductor and dielectric

In metals each atom makes one or two of its electrons “common” — these free electrons can move throughout the whole body. That is why metal, salt water, damp soil and the human body are conductors. In glass, rubber, plastic, dry paper and air the charges are bound to their own molecule: they only shift or turn slightly in place — these are dielectrics (insulators). In “Induction” mode, switch between “Metal sphere” and “Dielectric sphere” to see the difference.

Electrostatic induction and the field inside a conductor

When a charged ball approaches, the free electrons in the metal sphere shift: charge of the opposite sign gathers on the near side and charge of the same sign on the far side (electrostatic induction), while the total charge remains zero. The electrons move until the field inside becomes zero — this happens in less than a billionth of a second (slowed down in the scene). Place the probe inside the sphere: $E = 0$ ; on the $E (x)$ graph there is also no field inside the sphere, and $φ (x)$ is constant — the whole sphere is equipotential. The electroscope shows the potential of the sphere. Charging by induction: with the ball nearby, connect the sphere to earth (a charge $Q = - q R / d$ remains), disconnect it from earth, then take the ball away. In a hollow sphere the excess charge is only on the outer surface: the proof plane takes no charge from the inner surface, and the field in the cavity is zero however close the external charge is — this is electrostatic shielding (Faraday cage).

How charge is distributed over the surface

On the surface of a conductor the charge is distributed unevenly: where the curvature is large (the radius is small), the density $σ$ and the field $E = σ / ε_{0}$ are large. Two spheres connected by a wire have equal potentials: $k q_{1} / R_{1} = k q_{2} / R_{2}$ , so $σ_{1} / σ_{2} = R_{2} / R_{1}$ — the field is stronger on the surface of the smaller sphere. At a sharp tip the field begins to ionise the air — a corona discharge (sailors called it “St Elmo's fire”). At university level: in a system of conductors the potentials depend linearly on the charges, $φ_{i} = \sum_{j} α_{i j} q_{j}$ (this gives rise to the capacitance coefficients), and the field energy is $W = \frac{1}{2} \sum_{i} q_{i} φ_{i}$ . Capacitance and capacitors are covered in the “Capacitor” experiment.

A dielectric in an electric field

In a field a dielectric becomes polarised: in non-polar molecules the electron cloud shifts, and polar molecules (for example, water) turn along the field. As a result bound charges appear on the surface and the field inside is reduced depending on $ε$ (in a sphere to $3 E_{0} / (ε + 2)$ ), but does not fall to zero as in a conductor. A polarised piece is pulled towards the stronger side in a non-uniform field: $F = \frac{α}{2} \partial (E^{2}) / \partial y$ . This is why a comb lifts a piece of paper. Dry paper is a pure dielectric: it jumps only from a few millimetres away. Ordinary paper absorbs moisture from the air and conducts slightly, so it jumps from farther away; metal foil, a conductor, is polarised even more strongly. The foil touches the comb, instantly picks up charge and is repelled, while the paper sticks and takes up charge slowly. Some crystals become polarised when pressed (piezoelectrics — a gas lighter), and some are polarised spontaneously (ferroelectrics) — in detail in the “Capacitor” experiment.

Thunderstorm, lightning and the lightning rod

Inside a cumulonimbus cloud, hailstones and ice crystals collide and charges are separated: light positive crystals gather at the top (≈ 10 km) and heavy negative particles in the middle (≈ 6 km). A positive charge is induced on the ground beneath the cloud, and the field reaches several kV/m (in fair weather only ≈ 0.1 kV/m). A leader descends from the cloud in steps; when its tip comes within a distance $R_{str}$ of some object, an upward leader rises from the object, the channel is completed and a return stroke of tens of thousands of amperes passes — the air is heated to ≈ 30 000 K and expands violently: thunder. The job of a lightning rod is to take the lightning itself and carry the current safely to the ground along a wire; objects inside the “rolling sphere” zone are not struck. A car body is a Faraday cage: the current flows along the metal surface and there is no field inside. A lone tree, however, attracts lightning — standing under it is dangerous.

Model and simplifications

The field of a sphere is calculated with the exact solution (metal — image charges, dielectric — Legendre series); the charged ball is treated as a point charge, and the electroscope scale is approximate. The comb charge is treated as a line charge at the tips of its teeth (where the field does not exceed the air breakdown limit); the pieces are thin plates standing along the field: dry paper — a dielectric ( $ε \approx 3$ ), ordinary paper — a weak conductor because of moisture, foil — a conductor. The cloud is two point charges (+Q at 10 km, −Q at 6 km), the ground is a conductor; the tower is also taken as a conductor (damp brick conducts in a static field), and the field is calculated with ring charges (boundary element method). The corona discharge threshold comes from Peek's formula; its space charge is not taken into account. In the scene distances above 60 m are shortened, and the lightning is slowed down several tens of times.

Did you know?

The lightning rod was invented in the middle of the 18th century by the American scientist and statesman Benjamin Franklin.
A car is protected from lightning not by its rubber tyres but by its metal body: the current flows along the surface of the body into the ground. That is why a car with a soft top (a convertible) or a plastic body gives no such protection.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Electrification, Coulomb's law — Grade 7
Electric circuit, Ohm's law — Grade 7
Power sources, EMF, internal resistance — Grade 7
Work and power of an electric current — Grade 8
Electric motor — Grade 8
Aurora and the Lorentz force — Grade 8
Electric field and potential — Grade 8
Capacitor and capacitance — Grade 8
Circuits and Kirchhoff's rules — Grade 8
Electric current in metals — Grade 8