Lecture

Introduction: the scope of electricity and magnetism

The electromagnetic interaction is one of the four fundamental interactions in nature, and almost all forces we meet in everyday life (elasticity, friction, chemical bonding, intermolecular attraction) are in fact electromagnetic in origin. The force binding atomic nuclei and electrons, as well as lightning, radio waves and light, are all manifestations of this interaction. The course of electricity and magnetism studies electric charges, the electric and magnetic fields they create, electric current, and the interaction of these fields with matter.

The sources of the science are experiment and theory. The ancient Greeks knew that rubbed amber (Greek elektron) attracts light objects. In the 18th century C. Coulomb established the law of interaction of charges, in the 19th century M. Faraday introduced the concept of a field, and J. C. Maxwell united all electromagnetic phenomena in a single system of equations. Modern electrical engineering, electronics, communications and power engineering are built on this foundation. The course begins with electrostatics — charges at rest and their fields.

Electric charge and its properties

Electric charge $q$ is a scalar physical quantity that characterizes the ability of particles and bodies to take part in electromagnetic interaction. Its SI unit is the coulomb (C): the charge passing through a cross-section of a conductor in 1 s at a current of 1 A, $1\ \text{C} = 1\ \text{A}\cdot\text{s}$.

• There are two kinds of charge: positive and negative. Like charges repel, unlike charges attract. The electron carries a negative charge and the proton a positive one.
• Quantization: the charge of any body is a multiple of the elementary charge: $q = \pm Ne$, where $e = 1.602\cdot10^{-19}\ \text{C}$ and $N$ is an integer. R. Millikan proved this in his oil-drop experiments.
• Conservation law: the algebraic sum of the charges of an electrically closed (isolated) system does not change: $\sum q_i = \text{const}$. When bodies are charged, charge is not created from nothing but only redistributed between them.
• Relativistic invariance: the magnitude of a charge does not depend on the reference frame or on the speed of the particle.

Bodies are charged by friction, by contact and by influence (induction). Charges are detected with an electroscope and an electrometer. Substances are divided into conductors (many free charges: metals, electrolytes) and dielectrics, or insulators (almost no free charges: glass, rubber, air).

Coulomb's law

In 1785 C. Coulomb used a torsion balance to measure the force between two charged balls. For point charges (charged bodies whose sizes are very small compared with the distance between them) the result is stated as follows.

Coulomb's law: the force of interaction between two stationary point charges in vacuum is directly proportional to the product of the magnitudes of the charges, inversely proportional to the square of the distance between them, and is directed along the straight line joining the charges.

$$F = k\frac{|q_1|\,|q_2|}{\varepsilon r^2},\qquad k = \frac{1}{4\pi\varepsilon_0} = 8.99\cdot10^{9}\ \frac{\text{N}\cdot\text{m}^2}{\text{C}^2}$$

Here $q_1, q_2$ are the charges (C), $r$ is the distance between them (m), $\varepsilon_0 = 8.85\cdot10^{-12}\ \text{F/m}$ is the electric constant (vacuum permittivity), and $\varepsilon$ is the relative permittivity (dielectric constant) of the medium (vacuum $\varepsilon = 1$, air $\approx 1$, kerosene $\approx 2$, water $\approx 81$). In a dielectric medium the force is $\varepsilon$ times smaller than in vacuum.

In vector form, the force exerted by charge $q_1$ on charge $q_2$ is

$$\vec F_{12} = \frac{1}{4\pi\varepsilon_0\varepsilon}\,\frac{q_1 q_2}{r^3}\,\vec r_{12},$$

where $\vec r_{12}$ is the position vector drawn from $q_1$ to $q_2$. If $q_1q_2 > 0$ the force is repulsive, if $q_1q_2 < 0$ it is attractive. Coulomb forces obey Newton's third law: $\vec F_{21} = -\vec F_{12}$. Electric forces are incomparably stronger than gravitational ones: in a hydrogen atom the Coulomb force between the proton and the electron is about $2\cdot10^{39}$ times greater than their gravitational attraction.

The electric field and its strength

Charges do not act on each other directly but through the electric field (the idea of short-range action, Faraday). The electric field is a special form of matter that exists around every charge and exerts a force on any other charge placed in it. The field of stationary charges is called an electrostatic field.

To probe a field we use a test charge $q_0$ — a small positive point charge that does not noticeably disturb the field. The main force characteristic of the field at a given point is:

The electric field strength is the vector quantity equal to the ratio of the force acting on a positive test charge placed at a given point of the field to the magnitude of that charge: $\vec E = \dfrac{\vec F}{q_0}$.

The unit of field strength is $\text{N/C}$ (equal to $\text{V/m}$). If $\vec E$ is known, the force on any charge $q$ is $\vec F = q\vec E$: on a positive charge the force points along $\vec E$, on a negative charge it points the opposite way. From Coulomb's law, the field of a point charge is

$$E = \frac{1}{4\pi\varepsilon_0}\,\frac{|q|}{\varepsilon r^2}.$$

The vector $\vec E$ of a positive charge points radially away from the charge, that of a negative charge points towards it.

The superposition principle

The field strength produced by several charges equals the vector sum of the field strengths produced by each charge separately: $\vec E = \vec E_1 + \vec E_2 + \dots + \vec E_n = \sum\limits_i \vec E_i$.

For two vectors the magnitude of the sum is found by the law of cosines: $E = \sqrt{E_1^2 + E_2^2 + 2E_1E_2\cos\alpha}$, where $\alpha$ is the angle between the vectors. For a continuously distributed charge the sum becomes an integral: $\vec E = \dfrac{1}{4\pi\varepsilon_0}\displaystyle\int \frac{dq}{r^2}\,\frac{\vec r}{r}$, using the linear ($\tau = dq/dl$), surface ($\sigma = dq/dS$) and volume ($\rho = dq/dV$) charge densities.

An electric dipole is a system of two charges of equal magnitude and opposite sign separated by a distance $l$. Its main characteristic is the dipole moment $\vec p = q\vec l$ (the vector points from the negative to the positive charge). Many molecules (for example, the water molecule) behave as dipoles.

Graphical representation of the electric field

Faraday proposed depicting a field with field lines (lines of force). A field line is a line whose tangent at every point coincides with the vector $\vec E$ at that point.

• Field lines start on positive charges and end on negative charges (or go off to infinity / come from infinity); electrostatic field lines are not closed.
• Field lines never cross, because at each point $\vec E$ has a unique direction.
• The density of lines (the number of lines crossing a unit area perpendicular to them) is proportional to the field strength: where the lines are crowded, the field is strong.
• A uniform field (the same $\vec E$ at all points) is shown by parallel, equally spaced lines — for example, between the plates of a charged parallel-plate capacitor.

The field of a point charge is shown by radial lines; the field of a dipole by curves leaving the positive charge and entering the negative one; between two equal charges the lines "repel" each other and a point with $E = 0$ appears in the middle.

In the interactive model choose a charge configuration (single charge, dipole, two equal charges, parallel plates), change the charge and watch how the field-line pattern changes. Place the test charge at different points and check that the force on it is tangent to the field line.

Example

In air, positive point charges $q_1 = 2\ \text{nC}$ and $q_2 = 8\ \text{nC}$ are $r = 30\ \text{cm}$ apart. Find: a) the interaction force; b) the field strength at the midpoint between the charges; c) the point where the field strength is zero.

Solution. a) $F = k\dfrac{q_1q_2}{r^2} = 9\cdot10^9\cdot\dfrac{2\cdot10^{-9}\cdot 8\cdot10^{-9}}{0.09} = 1.6\cdot10^{-6}\ \text{N}$ (repulsion).

b) The midpoint is $0.15\ \text{m}$ from each charge: $E_1 = 9\cdot10^9\cdot\dfrac{2\cdot10^{-9}}{0.0225} = 800\ \text{N/C}$ (towards $q_2$), $E_2 = 9\cdot10^9\cdot\dfrac{8\cdot10^{-9}}{0.0225} = 3200\ \text{N/C}$ (towards $q_1$). The vectors are opposite, so the resultant is $E = 3200 - 800 = 2400\ \text{N/C}$, directed towards $q_1$.

c) The point with $E = 0$ lies between the charges, closer to the smaller one. At distance $x$ from $q_1$: $\dfrac{q_1}{x^2} = \dfrac{q_2}{(r-x)^2}\Rightarrow \dfrac{r-x}{x} = \sqrt{\dfrac{q_2}{q_1}} = 2\Rightarrow x = \dfrac{r}{3} = 10\ \text{cm}$.

Applications

Coulomb's law and the field concept are used in electrostatic precipitators, xerography, electrostatic painting, particle accelerators and cathode-ray tubes to control the motion of particles, and in chemistry to explain the interaction of ions and molecules. Lightning protection and the protection of electronic devices from static charge are also based on these laws.