Key Principles and Applications of Electric Charges and Fields
Electric charges are fundamental properties of matter that lead to electric forces and fields. These properties govern the interactions between particles and influence various physical processes at macroscopic and microscopic levels. The study of electric charges and fields forms the foundation of electrostatics and is essential for understanding advanced concepts in electromagnetism.
Electric Charge: Definition and Properties
Electric charge is an intrinsic property of particles that results in electrical and magnetic interactions. It exists in two types: positive and negative. Like charges repel, and unlike charges attract each other. Charge is a scalar quantity and follows the principle of conservation, meaning the total charge in an isolated system remains constant.
The SI unit of electric charge is the coulomb (C), and the smallest possible charge carried by an electron is $-1.6 \times 10^{-19}$ C. Charge is also quantized, existing as integral multiples of this elementary value. For more on continuous distributions, see Electric Field Lines Explained.
Units and Dimensional Formula of Charge
In the SI system, the unit of charge is coulomb (C). In the CGS system, it is statcoulomb (stat C). The dimensional formula for electric charge is $[A T]$, where $A$ is current and $T$ is time.
| Quantity | Value/Unit |
|---|---|
| SI Unit | Coulomb (C) |
| CGS Unit | Statcoulomb |
| Dimensional Formula | [A T] |
Coulomb’s Law
Coulomb’s Law defines the force between two stationary point charges. The force is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. It acts along the straight line joining the charges.
The mathematical expression is $F = k \dfrac{q_1 q_2}{r^2}$, where $k = \dfrac{1}{4\pi \varepsilon_0} = 9 \times 10^{9}\ \mathrm{N\,m^2\,C^{-2}}$ in vacuum. Here, $\varepsilon_0$ is the permittivity of free space.
Principle of Superposition
When multiple charges are present, the force on any single charge is the vector sum of the forces exerted by each of the other charges, acting independently. This is called the principle of superposition and is fundamental to computing net electric force in a system of charges.
Relative Permittivity and Dielectric Constant
The relative permittivity (dielectric constant) of a medium is the ratio of the permittivity of the medium ($\varepsilon$) to the permittivity of free space ($\varepsilon_0$). It shows how the force between charged objects is affected by the surrounding medium.
$K = \dfrac{\varepsilon}{\varepsilon_0}$, and $K \geq 1$ for any medium. These concepts are important in materials with different electric properties, as further explained in Magnetic Materials Explained.
Electric Field: Concept and Formula
The electric field is a vector quantity that describes the force per unit positive charge at any point in space due to other charges. It is given by $E = F/q$, where $F$ is the force experienced by a test charge $q$.
For a point charge $Q$ at a distance $r$, $E = k \dfrac{Q}{r^2}$, directed radially away from $Q$ if $Q$ is positive or towards $Q$ if negative. The SI unit of electric field is $\mathrm{N\,C^{-1}}$ or $\mathrm{V\,m^{-1}}$.
Properties and Representation of Electric Field Lines
Electric field lines are imaginary lines used to represent the direction and strength of the electric field around charges. They start from positive charges and end at negative charges. The tangent at any point gives the direction of the electric field.
- No two electric field lines cross each other
- Field lines do not form closed loops in electrostatics
Continuous Charge Distribution
When charges are distributed over a line, surface, or volume, the system is modeled as a continuous charge distribution. It is described by linear ($\lambda$), surface ($\sigma$), and volume ($\rho$) charge densities.
| Type | Formula |
|---|---|
| Linear Charge Density ($\lambda$) | $\lambda = \dfrac{Q}{l}$ |
| Surface Charge Density ($\sigma$) | $\sigma = \dfrac{Q}{S}$ |
| Volume Charge Density ($\rho$) | $\rho = \dfrac{Q}{V}$ |
Electric Dipole and Dipole Moment
An electric dipole consists of two equal and opposite charges separated by a fixed distance. The dipole moment ($\vec{p}$) measures the strength and direction of a dipole and is given by $\vec{p} = q \times 2\vec{a}$, where $2a$ is the separation.
The SI unit of dipole moment is coulomb-meter (C m). The direction of the dipole moment is from negative to positive charge. Detailed analysis for field calculations can be found in Understanding Electric Potential.
Electric Field Due to a Dipole
The electric field of a dipole at a point on its axial line at distance $r$ is $E_{axial} = \dfrac{1}{4\pi\varepsilon_0} \dfrac{2p}{r^3}$. On the equatorial line, $E_{equatorial} = \dfrac{1}{4\pi\varepsilon_0} \dfrac{p}{r^3}$.
Torque and Work on a Dipole in an Electric Field
A dipole placed in a uniform electric field experiences a torque $\tau = p E \sin \theta$, where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$. The work done in rotating a dipole from angle $\theta_1$ to $\theta_2$ is $W = pE (\cos\theta_1 - \cos\theta_2)$.
Electric Flux
Electric flux through a surface is the total number of electric field lines passing through that surface. It is a scalar quantity and quantifies how much the electric field penetrates a given area. The formula for a uniform electric field is $\phi = E \cdot A \cos\theta$.
Gauss’s Law
Gauss’s Law establishes the relationship between the electric flux through a closed surface and the net charge enclosed within that surface. In mathematical form: $\phi_E = \oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{enclosed}}{\varepsilon_0}$.
Applications of Gauss’s Law
Gauss’s Law is used to compute the electric field due to symmetric charge distributions such as infinite lines, planes, and spherical shells. It simplifies calculations when high symmetry is present in the charge arrangement.
- Electric field near an infinite line of charge
- Field due to an infinite plane of charge
- Field outside and inside a uniformly charged sphere
For more practice, review Electrostatics Mock Test.
Solved Example: Calculation of Charge Transfer Using Quantization
If $1 \times 10^{10}$ alpha particles are emitted per second, and each alpha particle carries a charge $q = 2 \times 1.6 \times 10^{-19}$ C, the total charge accumulating per second is $Q = 1 \times 10^{10} \times 3.2 \times 10^{-19} = 3.2 \times 10^{-9}$ C/s. For a total charge of $8\, \mu$C, time taken $t = \dfrac{8 \times 10^{-6}}{3.2 \times 10^{-9}} = 2.5 \times 10^3$ s.
Summary of Key Formulae
This section summarizes the essential formulae for electric charges and fields needed for JEE Main.
| Concept | Formula |
|---|---|
| Coulomb’s Law | $F = k\dfrac{q_1 q_2}{r^2}$ |
| Electric Field (Point Charge) | $E = k \dfrac{Q}{r^2}$ |
| Flux through Surface | $\phi = E A \cos\theta$ |
| Gauss’s Law | $\oint \vec{E} \cdot d\vec{A} = \dfrac{Q_{enclosed}}{\varepsilon_0}$ |
| Dipole Moment | $\vec{p} = q \times 2\vec{a}$ |
Mastering electric charges and fields is essential for further studies in electrostatics and for success in JEE Main. Practice concepts with revision notes and problems related to this chapter, such as those found in Communication System Revision Notes.
FAQs on Understanding Electric Charges and Fields
1. What is electric charge?
Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric and magnetic field.
Electric charges can be positive or negative and are measured in coulombs (C). Key points include:
- Protons carry positive charge, electrons carry negative charge.
- Like charges repel; unlike charges attract.
- Charge is quantized and conserved.
2. State and explain Coulomb’s law of electrostatics.
Coulomb’s law quantifies the electrostatic force between two point charges.
The force between charges is:
- Directly proportional to the product of their magnitudes
- Inversely proportional to the square of the distance between them
- Depends on the medium
The formula is: F = (1/(4πε₀)) × (q₁q₂/r²), where F is force, q₁ and q₂ are charges, r is distance, and ε₀ is permittivity of free space.
3. What is quantisation of charge?
Quantisation of charge means charge exists only in discrete packets called elementary charge.
Main points:
- The smallest unit of charge is the electron charge (e = 1.6 × 10-19 C).
- Any charge (Q) is given by: Q = n × e (where n is an integer).
- Charge cannot take any arbitrary value; it’s always a whole-number multiple of 'e'.
4. What do you mean by electric field and how is it represented?
An electric field is a region where a charge experiences a force due to another charge.
Main features:
- Electric field (E) at a point is defined as the force experienced by a unit positive charge placed at that point (E = F/q).
- Represented by field lines pointing from positive to negative charges.
- The direction of field lines shows the path a positive charge would take.
5. Define electric field intensity and give its SI unit.
Electric field intensity refers to the force per unit positive charge at a point in the field.
- Formula: E = F/q
- SI unit is newton per coulomb (N/C) or volt per meter (V/m)
- Shows both magnitude and direction (vector quantity)
6. What is the principle of superposition of electric charges?
The superposition principle states that the total electric force on a charge due to a group of other charges is the vector sum of the individual forces exerted by each charge.
Key points:
- Total force = sum of all individual forces (vector addition)
- This applies because electric forces are linear
7. Explain the concept of electric field lines and their properties.
Electric field lines provide a visual representation of electric fields and their behavior.
Main properties:
- Begin on positive charges and end on negative charges
- Never intersect each other
- Denser field lines mean a stronger field
- Tangent at any point gives field direction
8. What is meant by the term ‘conductors’ and ‘insulators’ with respect to electric charge?
Conductors and insulators are materials classified by their ability to allow electric charges to flow.
- Conductors: Allow free movement of charges (e.g., metals like copper, aluminium)
- Insulators: Do not allow charges to move freely (e.g., rubber, glass)
- This property affects how materials respond to electric fields
9. What is the difference between point charge and continuous charge distribution?
Point charge refers to a single, small-sized charge, whereas continuous charge distribution involves charges spread over a region.
- Point charge: Treated as if all charge is concentrated at a point (idealized concept)
- Continuous distribution: Charge spread over a line, surface, or volume
- Different mathematical methods are used for calculations involving each
10. State Gauss’s law and mention its significance.
Gauss’s law relates the total electric flux through a closed surface to the charge enclosed by that surface.
- Mathematically: ∮E·dA = Qenclosed/ε₀
- Useful for calculating electric fields for symmetric charge distributions (like sphere, cylinder)
- Simplifies complex problems
11. What are the applications of Gauss’s law?
Gauss’s law is primarily used to calculate electric fields of symmetric charge distributions.
Applications include:
- Field due to an infinite line of charge
- Field inside and outside a charged sphere
- Field due to infinite plane sheet of charge
- Understanding shielding and electrostatic equilibrium
12. What is the value of the permittivity of free space, and why is it important?
The permittivity of free space (ε₀) is a fundamental constant crucial in electrostatics.
- Value: ε₀ = 8.854 × 10-12 C²/N·m²
- Appears in Coulomb’s law and Gauss's law
- Determines the strength of the electric force in vacuum
