Electrostatics

Charge, conductors, charge conservation, and insulators

Charge can be either positive or negative, with a neutral state representing zero charge. Like charges repel each other, while opposite charges attract. Charge is quantized and measured in Coulombs. In conductors, such as metals, charges move freely, facilitating the transfer of electrical energy.

Conversely, insulators restrict the movement of charges, which is why nonmetals are typically poor conductors. A fundamental principle is charge conservation: charge cannot be created or destroyed, only transferred between objects.

Coulomb’s law

Coulomb’s Law describes the electrostatic force between two charged particles. It states that this force is directly proportional to the product of the two charges ($q_1$ and $q_2$) and inversely proportional to the square of the distance ($r$) between them, as expressed by the equation:

$$F=\frac{k{q}_1{q}_2}{r^2}$$

where $k$ is Coulomb’s constant (approximately $9\times 10^{9}N\cdot m/C^2$).

If both charges have the same sign, the force is repulsive, whereas if they have opposite signs, the force is attractive.

Electric field E

The electric field is a vector quantity that represents the force per unit charge at any point, measured in Newtons per Coulomb ($N/C$).

One common way to visualize the electric field is through field lines, which indicate both the direction and relative strength of the field:

• Closely spaced lines signify a stronger field
• More widely spaced lines indicate a weaker field.
• For a single positive charge, these lines radiate outward, and for a negative charge, they converge inward.

Adding electrical charges

The electric fields ${\mathbf{E}}_1$ and ${\mathbf{E}}_2$ at the origin $O$ add to ${\mathbf{E}}_{\text{tot}}$.

The electric field strength at the origin due to $q_1$ is labeled ${\vec{E}}_1$ and is calculated:

$$E_1=k\frac{q_1}{r_1^2}=(8.99\times 10^9\text{N}\cdot {\text{m}}^2/{\text{C}}^2)\cdot \frac{5.00\times 10^{-9}\text{C}}{(2.00\times 10^{-2}\text{m}{)}^2}$$

$$E_1=1.124\times 10^5\text{N/C}$$

Similarly, ${\vec{E}}_2$ is:

$$E_2=k\frac{q_2}{r_2^2}=(8.99\times 10^9\text{N}\cdot {\text{m}}^2/{\text{C}}^2)\cdot \frac{10.0\times 10^{-9}\text{C}}{(4.00\times 10^{-2}\text{m}{)}^2}$$

$$E_2=0.5619\times 10^5\text{N/C}$$

The magnitude of the total field $E_{\text{tot}}$ is:

$$E_{\text{tot}}=\sqrt{E_1^2+E_2^2}=\sqrt{(1.124\times 10^5{)}^2+(0.5619\times 10^5{)}^2}=1.26\times 10^5\text{N/C}$$

The direction is:

$$\theta ={\tan }^{-1}\left(\frac{E_1}{E_2}\right)={\tan }^{-1}\left(\frac{1.124\times 10^5}{0.5619\times 10^5}\right)=63.4^{\circ }$$

or $63.4^{\circ }$ above the x-axis.

In a dipole system, field lines emerge from the positive end and enter the negative end. For two like charges, the lines repel one another, with the overall pattern reversing for negative charges compared to positive ones.

When multiple charges are present, the net electric field at any location is obtained by taking the vector sum of the individual fields. In specific configurations, such as within a capacitor, the electric field between the plates is typically uniform except near the edges, where fringe effects occur. Similarly, the electric field around a long, charged wire or a cylinder extends radially outward, and in the case of a conducting cylinder, the field inside is zero.

Absolute potential and potential difference

The absolute potential at a point is the energy per unit charge that a test charge would have due to a source charge. It is expressed by the equation $V=U/q_0$ = $kq/r$, where U is the electric potential energy of the test charge $q_0$, $q$ is the charge generating the potential, and $r$ is the distance from this charge. When multiple charges are present, the total potential is the algebraic sum of the individual potentials (with positive charges creating positive potentials and negative charges creating negative potentials). The standard unit of potential is the Volt ($V$), which is equivalent to Joules per Coulomb (J/C).

The potential difference ($\Delta$V) is the difference between the potentials at two points, calculated as $\Delta$$V=V_B-V_A$. This measure is important in various applications, such as the voltage across the plates of a capacitor or between the positive and negative terminals of a battery.

Equipotential lines and electrical dipoles

Equipotential lines are contours along which the potential remains constant. Because there is no change in potential when moving along these lines, no work is done, and they are always perpendicular to electric field lines.

An electric dipole consists of a positive and a negative charge separated by a distance. In an electric field, a dipole naturally aligns itself so that its positive end points in the direction of the field. To determine the potential created by a dipole at a specific point, one calculates the individual potentials due to each charge and then sums them.

Electrostatic energy, electric potential at a point in space

Electrostatic induction is a process where a charged object causes a redistribution of charges in a nearby neutral object without any direct conduction of charge between them. In this phenomenon, the electric field of the charged object polarizes the neutral object, shifting its internal charge distribution. A common example is when a sweater rubbed against hair induces the hair to stand up; here, the charged sweater creates an electric field that rearranges the charges in the hair, causing a noticeable effect even though no electrons are physically transferred. This process is static, meaning that the induced polarization occurs simply due to the presence of the charged species.

A point charge is an idealized model in physics in which an electric charge is considered to be concentrated at one specific point in space, even though it actually occupies a small but finite volume. This assumption greatly simplifies the computation of electric fields and potentials in many electrostatic problems.

Gauss’s law states that the net electric flux ($\Phi$E) through any closed surface is equal to the total charge ($q$) enclosed divided by the permittivity of free space (${ϵ}_0$), or mathematically, ${ϵ}_0$$E=q/$${ϵ}_0$. The electric flux is calculated as the product of the electric field ($E$) and the area ($A$) it penetrates, adjusted by the cosine of the angle ($\theta$) between the field direction and the normal to the surface ($\Phi$$E=EAcos$$\theta$). This law implies that if no charge is contained within a closed surface, the net electric flux through that surface is zero. The **Faraday cage, **now familiar to many from post-apocalyptic books and movies as the key to preventing electronics from destruction in an EMP event, is a practical application of Gauss’s law; it is a closed conducting shell where the internal electric field is nullified by the rearrangement of surface charges that cancel any external fields.