Learn first: Electric charge and Coulomb's law
Charges do not act on each other directly but through the electric field around them. A field is a special form of matter: it always exists around a charge, it has energy, and it exerts a force on other charges. A field is described by two quantities: field strength (the force characteristic) and potential (the energy characteristic). This topic is the key to capacitors, current, cathode-ray devices and even the structure of the atom.
To “feel” a field we place a small positive test charge q in it and measure the force on it. The ratio of force to charge does not depend on the test charge; it is a property of the field at that point:
E = F/q, unit N/C = V/m
Field strength is a vector pointing the same way as the force on a positive test charge. On a negative charge the force points against the field: F = qE.
The field of a point charge follows from Coulomb’s law:
E = k·|q|/(εr²), k = 9·10⁹ N·m²/C²
Field strength is inversely proportional to the square of distance: doubling the distance makes E four times smaller. Outside a charged sphere the field is as if all the charge were at the centre; inside, the field is zero.
A charged body is acted on by an electric force and its weight. The resultant is 32 mN and horizontal; the weight is 24 mN. Find the electric force.
A field is pictured with field lines: at each point the tangent to a line shows the direction of E, and the density of lines shows its size.
The field of several charges is found by the superposition principle: each charge’s field is independent of the others, and the resultant field is their vector sum:
E = E₁ + E₂ + ... (as vectors)
A uniform field has the same strength at every point; its lines are parallel and equally spaced. Such a field exists between the plates of a parallel-plate capacitor.
Charges of +4 nC and −4 nC are 20 cm apart in air. Find the field strength at the midpoint.
The work done by electrostatic forces does not depend on the path, only on the start and end points (as with gravity). So a charge in a field can be given a potential energy W.
Potential: φ = W/q, unit volt (V); for a point charge φ = kq/(εr)
Voltage (potential difference): U = φ₁ − φ₂ = A/q
Work of the field: A = qU
In a uniform field: U = E·d, i.e. E = U/d
The potential energy of a charge is W = qφ. A positive charge moves by itself towards lower potential, a negative one (an electron, say) towards higher potential; either way its potential energy falls and its kinetic energy grows, just as a body falls downwards under gravity. The zero of potential is chosen freely (usually at infinity or at the Earth), but the physically meaningful quantity, the potential difference, does not depend on that choice.
Atomic physics uses the electronvolt: 1 eV = 1.6·10⁻¹⁹ J, the energy an electron gains crossing a potential difference of 1 V.
An electron at rest is accelerated through 100 V. What energy and speed does it gain (m = 9.1·10⁻³¹ kg)?
A conductor contains free electrons. In an external field they shift and create surface charges (electrostatic induction), whose field exactly cancels the external field inside:
A dielectric has no free charges, but its molecules are polarised in a field: positive and negative charges shift slightly, or polar molecules turn along the field. The polarisation charges weaken the field ε times:
E = E₀/ε, ε is the relative permittivity (air ≈ 1, glass ≈ 7, water ≈ 81)
In a uniform field a particle feels a constant force, so its acceleration is constant:
a = qE/m
The deflecting plates of a cathode-ray tube work on this principle.
An electron enters between capacitor plates parallel to them at v₀ = 10⁷ m/s. The field is E = 10⁴ V/m and the plates are 5 cm long. How far is the electron deflected?