Physics · Electricity and magnetism · Grade 8

Electric field and potential

What does the “invisible mountain” around a charge look like: field lines and equipotential surfaces

The situation

On a dry day the little “snap” of a spark when you touch a door handle, or the hair standing on end when a balloon is rubbed against it — all of this is the work of the invisible electric field around charges. A field cannot be seen, but it can be measured through the force it exerts on a test charge. Physicists describe the field with two kinds of “map”: field lines, and equipotential lines joining points of equal potential — just like the hills and contour lines on a geographical map. How is the field distributed among the charges, and how much work does it take to move a charge from one point to another?

Topic

Electric field and potential. Field lines and equipotential surfaces around charged spheres

Key ideas

Electric field strength
Potential and work
Gauss's theorem

Formulas

\vec{E} = \frac{\vec{F}}{q_{0}}, E = k \frac{| q |}{r^{2}}, \vec{E} = \sum_{i} {\vec{E}}_{i}

Field strength is the force acting on a test charge; the field of a point charge; the principle of superposition.

φ = k \frac{q}{r}, A_{12} = q_{0} (φ_{1} - φ_{2}) = - Δ W_{p}

The potential of a point charge; the work of moving a charge does not depend on the path and equals the decrease in potential energy ( $W_{p} = q_{0} φ$ ).

\vec{E} = - \nabla φ, E = \frac{U}{d}

The field points in the direction in which the potential falls fastest; in a uniform field $U = E d$ .

Φ_{E} = \oint \vec{E} \cdot d \vec{S} = \frac{Q_{inside}}{ε_{0}}

Gauss's theorem (Ostrogradsky–Gauss): the flux through a closed surface depends only on the charge inside it.

E_{plane} = \frac{σ}{2 ε_{0}}, E_{line} = \frac{λ}{2 π ε_{0} r}, M = p E \sin α

From Gauss's theorem: the fields of an infinite plane and an infinite line of charge; the torque on a dipole in an external field ( $p = q l$ ).

Theory

Electric field, field strength and superposition

Every charge has an electric field around it — it exerts a force on other charges. To measure the field we use a small positive test charge $q_{0}$ (it must not change the field appreciably): $\vec{E} = \vec{F} / q_{0}$ , with unit N/C = V/m. The field of a point charge is $E = k q / r^{2}$ . The field of several charges is the vector sum (the principle of superposition) — the “Superposition” button shows the contribution of each charge and the sum at the test charge. Charges may be distributed along a line ( $λ$ , C/m), over a surface ( $σ$ , C/m²) or through a volume ( $ρ$ , C/m³); their field is the sum (integral) of the fields of the elementary charges.

Field lines and the graphical representation of the field

Field lines start on a positive charge and end on a negative one; the tangent at every point is along $\vec{E}$ ; the density of the lines is proportional to the field strength; they never cross (at every point $\vec{E}$ is unique). A uniform field (between plates) has parallel, equally spaced lines. The colour map shows the potential (red — positive, blue — negative), and the thin lines are equipotentials with equal steps of $Δ φ$ : they are perpendicular to the field lines, and where they are close together the field is strong. In “Relief” mode the potential is shown as hills and valleys: a positive test charge moves “downhill”. The potential of a dipole falls off far away as $φ \sim 1 / r^{2}$ , and that of a quadrupole as $\sim 1 / r^{3}$ — compare the slopes on the “lg φ — lg r” graph (multipole moments).

Work, potential and potential energy

The electrostatic force is a conservative force: the work of moving a charge does not depend on the shape of the path, only on the starting and end points: $A_{12} = q_{0} (φ_{1} - φ_{2})$ . Around a closed path the work is zero — the circulation of the field $\oint \vec{E} \cdot d \vec{l} = 0$ . Drag the test charge: the “Work along the path” $\int \vec{F} \cdot d \vec{l}$ is always equal to $q_{0} Δ φ$ . The potential is $φ = W_{p} / q_{0}$ ; the potential difference (voltage) is $U = φ_{1} - φ_{2}$ . Field strength and potential are related by $\vec{E} = - \nabla φ$ (the potential gradient): in a uniform field $E = U / d$ . The energy of a system of charges, $W = \sum_{i < j} k q_{i} q_{j} / r_{i j}$ , is shown in the readouts; the field energy and its density are in the “Capacitor and capacitance” experiment.

Gauss's theorem (Ostrogradsky–Gauss)

The flux of the field $Φ = \oint \vec{E} \cdot d \vec{S}$ is the number of field lines crossing the surface. The theorem: the flux through a closed surface equals $Q_{inside} / ε_{0}$ — the shape of the surface and any outside charges do not matter (the lines of an outside charge enter and leave again). When there is symmetry, the theorem gives the field immediately: inside a charged sphere $E = 0$ , outside it is $k Q / r^{2}$ ; inside a uniformly charged ball $E = k Q r / R^{3}$ (potentials: constant $k Q / R$ inside the sphere, $k Q / r$ outside — on the $φ (r)$ graph); the field of an infinite line is $λ / (2 π ε_{0} r)$ (cylindrical surface); of an infinite plane, $σ / 2 ε_{0}$ (box surface); between two oppositely charged planes it is $σ / ε_{0}$ , and outside it is zero. In differential form $\nabla \cdot \vec{E} = ρ / ε_{0}$ , that is, $\nabla^{2} φ = - ρ / ε_{0}$ (Poisson's equation); in a region without charge $\nabla^{2} φ = 0$ (Laplace) — this very equation gives the distribution in an electrolytic tank.

A uniform field, a dipole and the motion of a charge

If the plates are infinite, $E = σ / ε_{0}$ between them; with real (finite) plates the field “spreads out” at the edges and is slightly weaker at the centre — measure it. A dipole ( $p = q l$ ) in a uniform field experiences a torque $M = p E \sin α$ and turns to line up with the field; in a non-uniform field it is also pulled towards the stronger side. Between the plates an electron acquires an acceleration $a = e E / m$ and traces a parabola like a projectile (this is how the cathode-ray tube and the oscilloscope worked). Conductors and dielectrics in a field are covered in the experiments “Conductor, dielectric, lightning” and “Capacitor and capacitance” (the electric displacement $\vec{D}$ , the boundary conditions and the refraction of field lines are also there). The method of images: the field of a charge near an earthed plane equals the field of “the charge plus its opposite image in the mirror” (Charges → “Charge and conducting plane”). Another method is separation of variables: for example, the potential of a dielectric ball in a field is found as a series of Legendre polynomials (in the same “Conductor…” experiment).

Lab work: the electrolytic tank; the model

Electrodes are placed in a tank filled with water and a voltage is applied; with a probe and a voltmeter points of equal potential are found — they form an equipotential line. The field lines are drawn perpendicular to them. The small current in the water does not distort the shape of the field: the potential obeys the same Laplace equation (here it is solved on a grid with a 5 mm step, and the walls of the tank do not conduct current). The model: in “Charges” mode the field is calculated in the plane of the charges (a cross-section of the three-dimensional Coulomb field); the plates are taken to be very tall (a 2D problem); the electron's motion is slowed down by a factor of $10^{8}$ , and relativistic corrections are ignored. If charges move, the potentials obey the d'Alembert (wave) equations in the Lorenz gauge and changes propagate at the speed of light — the retarded (Liénard–Wiechert) potentials and waves are in the experiment “Maxwell's equations” (antenna).

Did you know?

The concept of field lines was introduced in the 19th century by the English physicist Michael Faraday; he regarded the field as a real thing existing in the space between charges.
The Earth's surface is negatively charged: in the open air near the ground there is a field of about 100–150 V/m, that is, the air at a height of 2 m is at a potential of roughly 200–300 V relative to the ground. We do not notice it: our body is a conductor, it alters the field and is itself at one potential.

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

Conductor, dielectric, lightning — 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
Capacitor and capacitance — Grade 8
Circuits and Kirchhoff's rules — Grade 8
Electric current in metals — Grade 8
Electrolysis, Faraday's laws — Grade 8
Electric current in vacuum and gases — Grade 8