Gauss’S Law In Electrostatics

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Gauss’s law in electrostatics

Gauss’s law is one of the fundamental laws in electrostatics.
It relates the electric field with the charge distribution.
It is valid for any closed surface.
The mathematical form of Gauss’s law is given by:
- ∮ E · dA = ε₀ * Σ Q / ε₀ = Q_in / ε₀
Here, E is the electric field, dA is the surface element, ε₀ is the permittivity of free space, Q is the total charge enclosed by the surface, and Q_in is the charge inside the surface. slide 2:

Gauss’s law in electrostatics - Application

Gauss’s law can be used to determine the electric field due to various charge distributions.
It simplifies the calculation of electric fields in situations with symmetry.
The principle of superposition can be applied to find the total electric field.
Gauss’s law is valid in both uniform and non-uniform electric fields. slide 3:

Gauss’s law in electrostatics - Examples

Example 1: Electric field due to a point charge
- Consider a point charge Q at the origin.
- The electric field at a distance r from the charge is given by E = k * Q / r^2, where k is the Coulomb’s constant.
Example 2: Electric field due to a uniformly charged sphere
- Consider a uniformly charged sphere of radius R and total charge Q.
- The electric field inside the sphere is zero, and outside it is given by E = k * Q / r^2, where r is the distance from the center of the sphere. slide 4:

Gauss’s law in electrostatics - Derivation

Gauss’s law can be derived using the concept of electric flux.
Electric flux through a closed surface is defined as the dot product of the electric field vector and the surface area vector.
Mathematically, electric flux Φ is given by Φ = ∮ E · dA, where the integral is taken over the closed surface. slide 5:

Gauss’s law in electrostatics - Derivation (contd.)

Applying the divergence theorem to the electric flux, we get ∮ E · dA = ∮ (∇ · E) dV, where the integral is taken over the volume enclosed by the closed surface.
Using the definition of divergence (∇ · E), the equation becomes ∮ (∇ · E) dV = ∮ ρ / ε₀ dV, where ρ is the charge density.
Simplifying further, we have ∇ · E = ρ / ε₀. slide 6:

Gauss’s law in electrostatics - Derivation (contd.)

The equation ∇ · E = ρ / ε₀ is known as Gauss’s law in differential form.
By applying Gauss’s law to different charge distributions, we can determine the electric field in different situations.
Gauss’s law is a powerful tool for solving electrostatic problems. slide 7:

Gauss’s law in electrostatics - Problem

Problem: Find the electric field due to a uniformly charged infinite plane sheet.
Solution:
- Consider a plane sheet of charge density σ.
- The electric field above the sheet is E = σ / (2ε₀), and below the sheet is E = -σ / (2ε₀).
- The direction of the field is perpendicular to the sheet. slide 8:

Limitations of Gauss’s law

Gauss’s law has a few limitations:
- It is applicable only in electrostatic situations.
- It does not account for time-varying fields.
- It does not provide any information about the detailed distribution of the electric field.
- It assumes the medium is linear and isotropic. slide 9:

Summary

Gauss’s law relates the electric field with the charge distribution.
It simplifies the calculation of electric fields in situations with symmetry.
Gauss’s law is applicable in both uniform and non-uniform electric fields.
It can be derived using the concept of electric flux and the divergence theorem.
Gauss’s law has some limitations but is a powerful tool in solving electrostatic problems. slide 10:

References

Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons. slide 11:

Electric field due to a charged conducting sphere

Consider a uniformly charged conducting sphere of radius R and charge Q.
The electric field inside the conducting sphere is zero.
The electric field outside the sphere is the same as that of a point charge located at the center of the sphere: E = k * Q / r^2.
The charge Q on the conducting sphere is distributed uniformly over the surface. slide 12:

Electric field due to an infinite line of charge

Consider an infinite line of charge with charge density λ.
The electric field at a distance r from the line is given by E = 2 * k * λ / r.
The direction of the electric field is radially outward or inward, depending on the sign of the charge. slide 13:

Electric field due to an infinite sheet of charge

Consider an infinite sheet of charge with charge density σ.
The electric field above the sheet is given by E = σ / (2 * ε₀).
The electric field below the sheet is given by E = -σ / (2 * ε₀).
The electric field is perpendicular to the sheet and equal in magnitude on both sides. slide 14:

Electric field due to a dipole

A dipole consists of two equal and opposite charges separated by a fixed distance.
The electric field at a point along the axial line of the dipole is given by E = 2 * k * p / r^3.
The electric field at a point along the equatorial line of the dipole is given by E = k * p / r^3. slide 15:

Electric potential due to a point charge

The electric potential at a distance r from a point charge Q is given by V = k * Q / r.
The electric potential decreases as the distance from the charge increases.
The electric potential is a scalar quantity. slide 16:

Electric potential due to a system of point charges

The total electric potential at a point due to a system of point charges is the algebraic sum of the individual potentials due to each charge.
Mathematically, V_total = V₁ + V₂ + V₃ + …
Electric potential is a scalar quantity. slide 17:

Electric potential due to a continuous charge distribution

For a continuous charge distribution, the electric potential at a point is given by the integral of the electric potential due to each infinitesimally small charge element.
Mathematically, V_total = ∫ k * dq / r, where dq is the charge element and r is the distance between the charge element and the point of interest. slide 18:

Equipotential surfaces

An equipotential surface is a surface in which every point has the same electric potential.
Equipotential surfaces are perpendicular to the electric field lines.
The work done in moving a charge along an equipotential surface is zero.
Electric field lines are always perpendicular to the equipotential surfaces. slide 19:

Potential difference

The potential difference between two points in an electric field is defined as the work done in moving a unit positive charge from one point to another.
Mathematically, potential difference ΔV = V₂ - V₁.
Potential difference is a scalar quantity and is measured in volts (V). slide 20:

Capacitors and capacitance

Capacitors are devices used to store electric charge.
Capacitance is a measure of a capacitor’s ability to store charge and is defined as the ratio of the stored charge to the potential difference across the capacitor.
Mathematically, capacitance C = Q / ΔV.
The unit of capacitance is the farad (F). slide 21:

Electric potential energy

Electric potential energy is the energy associated with the position of a charged object in an electric field.
The electric potential energy of a point charge in an electric field is given by U = k * Q1 * Q2 / r.
The electric potential energy increases when like charges repel and decreases when opposite charges attract.
The unit of electric potential energy is the joule (J). slide 22:

Electric field due to a charged parallel plate capacitor

A parallel plate capacitor consists of two parallel conducting plates with opposite charges.
The electric field between the plates of a parallel plate capacitor is uniform and given by E = V / d, where V is the potential difference between the plates and d is the distance between the plates.
The electric field outside the capacitor is zero. slide 23:

Capacitance of a parallel plate capacitor

The capacitance of a parallel plate capacitor is given by C = ε₀ * A / d, where ε₀ is the permittivity of free space, A is the area of each plate, and d is the distance between the plates.
The larger the area of the plates and the smaller the distance between them, the greater the capacitance.
The unit of capacitance is the farad (F). slide 24:

Capacitors in series

Capacitors connected in series have the same charge on each capacitor.
The reciprocal of the total capacitance of capacitors in series is equal to the sum of the reciprocals of the individual capacitances: 1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + …
The total potential difference across capacitors in series is equal to the sum of the potential differences across the individual capacitors. slide 25:

Capacitors in parallel

Capacitors connected in parallel have the same potential difference across each capacitor.
The total capacitance of capacitors in parallel is equal to the sum of the individual capacitances: C_total = C₁ + C₂ + C₃ + …
The total charge stored in capacitors in parallel is equal to the sum of the charges stored in the individual capacitors. slide 26:

RC circuits - Charging

An RC circuit consists of a resistor (R) and a capacitor (C) connected in series to a voltage source.
During charging in an RC circuit, the capacitor charges up to the potential difference of the voltage source through the resistor.
The time constant (τ) of an RC circuit is given by τ = R * C, where R is the resistance and C is the capacitance.
The charging process is exponential in nature, and the capacitor reaches approximately 63% of its maximum charge after one time constant. slide 27:

RC circuits - Discharging

During discharging in an RC circuit, the capacitor releases its charge through the resistor.
The time constant (τ) of the RC circuit determines the rate at which the capacitor discharges.
The discharging process is also exponential in nature, and the capacitor reaches approximately 37% of its initial charge after one time constant.
The time constant is used to calculate the time it takes for the charge to decrease to a certain percentage of its initial value. slide 28:

Magnetic field due to a straight current-carrying conductor

Ampere’s law states that the magnetic field around a closed loop is directly proportional to the current enclosed by the loop.
The magnetic field at a distance r from a long straight conductor carrying current I is given by B = μ₀ * I / (2πr), where μ₀ is the permeability of free space.
The magnetic field lines around a current-carrying conductor form concentric circles with the conductor as their axis. slide 29:

Magnetic field due to a current in a solenoid

A solenoid is a long coil of wire with many turns.
The magnetic field inside a solenoid is nearly uniform and strong.
The magnetic field inside an ideal solenoid is given by B = μ₀ * N * I, where N is the number of turns per unit length and I is the current flowing through the solenoid.
The magnetic field outside the solenoid is very weak. slide 30:

Faraday’s law of electromagnetic induction

Faraday’s law of electromagnetic induction states that a changing magnetic field induces an electromotive force (emf) in a circuit.
The magnitude of the emf is given by ΔV = -N * ΔΦ / Δt, where N is the number of loops in the circuit, ΔΦ is the change in magnetic flux, and Δt is the change in time.
The negative sign indicates the direction of the induced emf, which opposes the change in magnetic flux.