Comprehensive Notes on Electric Fields and Charges

Electric Charges and Basic Properties

Electric charge is defined as an intrinsic property of the elementary particles that make up objects, such as electrons ($e$) and protons ($p^+$). It is a scalar quantity, and its SI unit is the Coulomb ($C$). The fundamental unit of charge is represented by the value $1.6 \times 10^{-19}\,C$, where electrons carry a negative charge and protons carry a positive charge. A key behavior of electric charges is that like charges repel each other, while unlike charges attract.

There are several basic properties of electric charge. The first is additivity, which states that if a system contains multiple point charges, such as $q_1, q_2, q_3$, the total charge of the system is the algebraic sum of the individual charges, represented as $q = q_1 + q_2 + q_3$. The second property is the quantization of electric charge. Electric charges are always quantized, meaning the total charge ($q$) is always an integral multiple of the elementary electronic charge ($e$). This is expressed by the formula $q = \text{ne}$, where $n$ is an integer representing the number of electrons.

The third property is the law of conservation of charge, which dictates that charge can neither be created nor destroyed; it can only be transferred from one body to another. Consequently, the total charge of an isolated system remains constant. Finally, charge is invariant, meaning it is independent of the frame of reference. Unlike length, mass, and time, which change as speed approach the speed of light, the charge on a body remains constant regardless of its velocity.

Dynamic States of Charge and Charging Methods

Charges behave differently depending on their state of motion. A charge at rest ($v = 0$) produces only an electric field ($E$). A charge in uniform motion (constant velocity, $v = \text{constant}$) produces both an electric field ($E$) and a magnetic field ($B$). A charge in non-uniform or accelerated motion ($a \neq 0$) produces an electric field, a magnetic field, and radiates energy in the form of electromagnetic (EM) waves. A body can be charged by either adding or removing electrons.

There are three primary methods of charging. The first is by friction or rubbing, where physical contact and movement result in the transfer of electrons. Examples include rubbing a silk cloth against a glass rod, where the glass rod becomes positively charged and the silk becomes negative, or rubbing wool/fur against an ebonite rod, where the rod becomes negative and the fur becomes positive. The second method is conduction, which involves physical touch between a charged conductor and a neutral conductor. In this process, the neutral body acquires the same type of charge as the charged body.

The third method is induction, which occurs without physical touch. When a charged body is brought near a neutral body, it induces an opposite charge on the near side of the neutral body. To make this charge permanent, the neutral body (placed on an insulating stand to prevent earthing) is earthed on the opposite side to neutralize the like charges. Once the earthing wire and then the inducing rod are removed, the formerly neutral body remains charged with the opposite sign to the original inductor.

Coulomb's Law and Electrostatic Force

Coulomb's Law describes the electrostatic or electric force of interaction between two point charges. It states that the force is directly proportional to the product of the magnitudes of the two point charges ($q_1, q_2$) and inversely proportional to the square of the distance ($r$) between them. This inverse square law is expressed by the formula $F = \frac{k q_1 q_2}{r^2}$. The constant $k$ is the Coulomb constant, which has a value of approximately $9 \times 10^9\,Nm^2\,C^{-2}$. The graph of force versus distance is parabolic, showing the inverse relationship $F \times \frac{1}{r^2}$.

Coulomb's Law has several important characteristics and limitations. It follows Newton's Third Law ($F_{12} = -F_{21}$), meaning the forces exerted by two charges on each other are equal and opposite. It is a central force, a long-range force, and a conservative force. However, it is only valid for point charges at rest and for separations greater than nuclear distances ($r > 1\,fm$ or $10^{-15}\,m$). Furthermore, the magnitude of the force depends on the medium in which the charges are placed.

Vector Form of Coulomb's Law

To derive the vector form of Coulomb's Law, consider two point charges $q_1$ and $q_2$ with position vectors $\text{r}_1$ and $\text{r}_2$ from the origin. The displacement vector from $q_1$ to $q_2$ is $\text{r}_{21} = \text{r}_2 - \text{r}_1$. The force exerted on $q_1$ by $q_2$ is given by $\mathbf{F}_{12} = \frac{k q_1 q_2}{|\text{r}_{12}|^2} \text{r}_{12}$. This can be rewritten using the cube of the distance as $\mathbf{F}_{12} = \frac{k q_1 q_2}{|\text{r}_1 - \text{r}_2|^3} (\text{r}_1 - \text{r}_2)$.

Similarly, the force on $q_2$ by $q_1$ is $\mathbf{F}_{21} = \frac{k q_1 q_2}{|\text{r}_2 - \text{r}_1|^3} (\text{r}_2 - \text{r}_1)$. Because $(\text{r}_1 - \text{r}_2) = -(\text{r}_2 - \text{r}_1)$, it is formally proven that $\mathbf{F}_{12} = -\text{F}_{21}$, thereby validating Newton's Third Law in the context of electrostatics.

Electric Field and Intensity

An electric field at a point is defined as the electrostatic force per unit test charge acting on a vanishingly small positive test charge ($q_0$) placed at that point. Mathematically, it is expressed as $\text{E} = \frac{\text{F}}{q_0}$. It can also be described as the space around a source charge ($Q$) where its electric influence can be experienced. Using Coulomb's Law, the electric field intensity due to a point charge is $\text{E} = \frac{k Q}{r^2}$. The SI unit for electric field is Newton per Coulomb ($N/C$) or Volt per meter ($V/m$).

In this context, the "source charge" is the charge producing the field, and the "test charge" is a tiny charge used to measure the field without significantly altering it. The electric field is a vector quantity. The relationship $\text{E} \text{ is inversely proportional to } r^2$ indicates that the field intensity decreases as the distance from the source charge increases.

Electric Field Lines and Their Properties

An electric field line (or line of force) is an imaginary curve along which a small positive test charge would tend to move if free to do so. A tangent drawn at any point on these lines gives the direction of the electric field intensity at that point. Field lines are continuous smooth curves without any breaks. They start from positive charges and end at negative charges. If there is only a single charge, the lines start or end at infinity.

Several key properties define these lines. They cannot form closed loops. No two electric field lines can ever cross each other because, if they did, the field would have two different directions at the point of intersection, which is physically impossible. Field lines are always normal (at $90^{\circ}$) to the surface of a conductor in equilibrium. If they were not normal, the parallel component would cause charge flow, contradicting the equilibrium state. Additionally, lines tend to contract lengthwise (explaining the attraction between unlike charges) and expand laterally (explaining repulsion between like charges).

The relative closeness of the lines measures the field strength: close lines indicate a strong field, while lines far apart indicates a weak field. Parallel, equally spaced lines represent a uniform electric field. Crucially, electric field lines do not pass through the interior of a conductor because the electric field inside a conductor in equilibrium is zero ($E_{\text{inside}} = 0$). External charges move to the surface to maintain this equilibrium.

Superposition Principle and Numerical Examples

The Principle of Superposition states that the total electric field at any point due to a group of charges is equal to the vector sum of the electric fields produced by each charge individually at that point, assuming the other charges are absent. This is represented by $\mathbf{E} = \text{E}_1 + \text{E}_2 + \text{E}_3 + ... + \text{E}_n$.

Consider a numerical example where q_1 = 16\,C and q_2 = -9\,C are separated by $8\,cm$. To find the position of the null point (where $E = 0$), we set the magnitudes of the fields equal: $\frac{k(16 \times 10^{-6})}{(8+x)^2} = \frac{k(9 \times 10^{-6})}{x^2}$. Taking the square root of both sides gives $\frac{4}{8+x} = \frac{3}{x}$, leading to $4x = 24 + 3x$, or $x = 24\,cm$. Thus, the null point is $24\,cm$ from $q_2$ and $32\,cm$ from $q_1$. Another example involves two positive charges $q_1 = 1\,C$ and $q_2 = 4\,C$ separated by $10\,cm$. The equilibrium point between them is found using $\frac{k(1)}{x^2} = \frac{k(4)}{(10-x)^2}$, resulting in $10-x = 2x$, or $x = \frac{10}{3}\,cm$.

Electric Dipole and Dipole Moment

An electric dipole consists of a pair of equal and opposite charges ($-q$ and $+q$) separated by a very small distance, typically denoted as $2a$. The electric dipole moment ($p$) is a vector quantity that is the product of the magnitude of one of the charges and the separation distance: $\text{p} = q \times 2a$. The direction of the dipole moment is always from the negative charge toward the positive charge. The SI unit for dipole moment is the Coulomb-meter ($C\,m$).

Dipoles can be found in molecules like $HCl$, $H_2O$, and $C_2H_5OH$. A "non-polar" molecule has no inherent dipole moment because the centers of positive and negative charges coincide. The region of space influenced by the charges of the dipole is referred to as the dipole field.

Field Intensity of a Dipole at Axial and Equatorial Points

At an axial point (a point on the line passing through the two charges at distance $x$ from the center), the net electric field is the difference between the fields of the two charges. The derivation results in $E_{\text{axial}} = \frac{k p 4ax}{(x^2 - a^2)^2}$. For a short dipole where $x \text{ is much greater than } a$, this simplifies to $E_{\text{axial}} = \frac{2 k p}{x^3}$. The net field direction at the axial point is the same as the dipole moment direction.

At an equatorial point (a point on the perpendicular bisector at distance $x$ from the center), the vertical components of the fields cancel out, and the horizontal components add up. The field is given by $E_{\text{equitorial}} = \frac{2 E \text{cos}(\theta) = k q 2a}{(x^2 + a^2)^{3/2}}$. For a short dipole ($x \text{ much greater than } a$), this simplifies to $E_{\text{equitorial}} = \frac{k p}{x^3}$. Notably, the magnitude of the field at an axial point is twice that at an equatorial point for the same distance ($E_{\text{axial}} = 2 E_{\text{equitorial}}$). The direction of the field at the equatorial point is opposite to the direction of the dipole moment.

Torque and Electric Permittivity

When an electric dipole is placed in a uniform electric field, it experiences a torque but no net translational force because the forces on $+q$ and $-q$ are equal and opposite ($F_{\text{net}} = 0$). The torque ($\tau$) is defined as the product of one of the forces and the perpendicular distance between them: $\tau = q E \times 2l \text{sin}(\theta) = p E \text{sin}(\theta)$. In vector form, $\mathbf{\tau} = \text{p} \times \text{E}$. This torque causes the dipole to rotate to align with the field. If the field is non-uniform, the dipole experiences both torque and a net translational force.

Electric permittivity () is a physical property of a material that measures its ability to resist the formation of an electric field within it; it determines the force between two charges in that medium. Permittivity of free space (vacuum) is denoted as _0 = 8.85 \times 10^{-12}\,C^2\,N^{-1}\,m^{-2}. Relative permittivity (_r), also known as the dielectric constant ($K$), is the ratio of the permittivity of a medium to that of free space (K = \frac{}{_0}). For vacuum/air, $K = 1$; for conductors, K = ; and for insulators, $K > 1$.

Charge Distribution and Electric Flux

Continuous charge distributions are classified into three types. Linear charge distribution (
) applies to lines or wires and is measured as charge per unit length (
= \frac{q}{L}, unit $C/m$). Surface charge distribution () applies to planar surfaces and is charge per unit area ( = \frac{q}{A}, unit $C/m^2$). Volumetric charge distribution (
0) applies to spheres or volumes and is charge per unit volume (
0 = \frac{q}{V}, unit $C/m^3$).

An area vector is used in flux calculations, where the vector's magnitude equals the surface area and its direction is always perpendicular (normal) to the surface. For 3D closed surfaces, the normal is always taken as outwards. Electric flux () is a measure of the total number of electric field lines passing through a given area. It is a scalar quantity with the SI unit $N\,m^2/C$. It is calculated as  = \text{E} \times \text{A} = |\text{E}| |\text{A}| \text{cos}(\theta). Flux is maximum when the field is perpendicular to the area ($90^{\circ}$ to surface, $0^{\circ}$ to area vector) and zero when parallel to the area.

Gauss' Law and Its Proof

Gauss' Law states that the total electric flux through any closed surface is equal to \frac{1}{_0} times the net charge enclosed by that surface: _{\text{tot}} = \frac{q_{\text{net}}}{_0}. The closed surface used for these calculations is a hypothetical surface called a Gaussian surface. To prove Gauss' Law for an isolated positive point charge $Q$, assume a spherical Gaussian surface of radius $r$ centered on the charge. Since the electric field $E$ is radial and parallel to the area element $dA$ at all points ($\theta = 0^{\circ}$), the flux integral becomes  E \, dA = E  dA = (\frac{1}{4 _0} \frac{Q}{r^2}) (4 r^2) = \frac{Q}{_0}.

Important properties of Gauss' Law: flux is independent of the shape or size of the surface; it depends only on the charge inside the surface; it is independent of the positions of the charges inside. While flux depends only on internal charges, the electric field intensity ($E$) at any point is the result of all charges, both inside and outside. Incoming flux is considered negative while outgoing flux is positive. If the net flux is zero, it does not necessarily mean the field is zero, but if the field is zero everywhere on the surface, the flux must be zero.

Applications of Gauss' Law

Gauss' Law is used to derive the electric field for various symmetric charge distributions. For an infinitely long straight charged wire with linear density
, a cylindrical Gaussian surface is used. The resulting field is E = \frac{
}{2 _0 r}, showing that the field decreases linearly with distance. For an infinite thin plane sheet of charge with surface density , a cylindrical Gaussian surface passing through the sheet yields E = \frac{}{2 _0}. This field is independent of the distance from the sheet. If two oppositely charged plates are parallel, the field between them is E = \frac{}{_0}, and the field outside is zero.

For a uniformly charged thin spherical shell (hollow sphere) of radius $R$:

1. Inside the shell ($r < R$), the enclosed charge is zero, so the electric field is $E = 0$.
2. On the surface ($r = R$), the field is maximum, E = \frac{k Q}{R^2} = \frac{}{_0}.
3. Outside the shell ($r > R$), the field behaves as if all the charge were concentrated at the center: $E = \frac{k Q}{r^2}$. The graph of $E$ versus $r$ remains at zero until $r = R$, then follows an inverse square decay ($E \times \frac{1}{r^2}$).