Electromagnetism is a branch of physics that deals with the study of the electromagnetic force, which includes both electric and magnetic fields. It is described by Maxwell's equations, a set of fundamental equations that govern the behavior of electric and magnetic fields.
Here are some important formulas and theorems related to electromagnetism:
Coulomb's Law:
Coulomb's law describes the force between two charged particles.
Formula: F = k * (q1 * q2) / r^2
Where:
F is the electrostatic force between the charges,
k is the electrostatic constant (k = 8.988 × 10^9 N m^2/C^2),
q1 and q2 are the magnitudes of the charges,
r is the distance between the charges.Gauss's Law:
Gauss's law relates the electric flux through a closed surface to the charge enclosed by that surface.
Formula: ∮ E · dA = (1/ε₀) * Q_enclosed
Where:
∮ E · dA represents the electric flux through a closed surface,
ε₀ is the electric constant (ε₀ = 8.854 × 10^-12 C^2/(N m^2)),
Q_enclosed is the charge enclosed by the surface.Ampere's Law:
Ampere's law relates the magnetic field around a closed loop to the electric current passing through that loop.
Formula: ∮ B · dl = μ₀ * I
Where:
∮ B · dl represents the magnetic field integral around a closed loop,
μ₀ is the magnetic constant (μ₀ = 4π × 10^-7 T m/A),
I is the electric current passing through the loop.Faraday's Law of Electromagnetic Induction:
Faraday's law describes how a changing magnetic field induces an electromotive force (EMF) in a conductor.
Formula: ε = -dΦ/dt
Where:
ε is the induced EMF,
dΦ/dt is the rate of change of magnetic flux through a loop.Maxwell's Equations:
Maxwell's equations combine the above laws and form the foundation of classical electrodynamics. They consist of four
equations: Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law, and Ampere's law with Maxwell's addition.Gauss's law for electric fields:
∮ E · dA = (1/ε₀) * Q_enclosedGauss's law for magnetic fields:
∮ B · dA = 0Faraday's law:
∮ E · dl = -dΦ/dtAmpere's law with Maxwell's addition:
∮ B · dl = μ₀ * (I + ε₀ * dΦ/dt)Lorentz Force Law:
The Lorentz force law describes the force experienced by a charged particle moving in an electric and magnetic field.
Formula: F = q * (E + v x B)
Where:
F is the force on the particle,
q is the charge of the particle,
E is the electric field,
v is the velocity of the particle,
B is the magnetic field.Ohm's Law:
Ohm's law relates the current flowing through a conductor to the voltage across it and its resistance.
Formula: V = I * R
Where:
V is the voltage,
I is the current,
R is the resistance.Lenz's Law:
Lenz's law states that the direction of an induced current in a closed loop is such that it opposes the change that caused it.
Formula (conceptual): The induced current opposes the change in magnetic flux.Biot-Savart Law:
The Biot-Savart law describes the magnetic field created by a steady current in a wire.
Formula: d𝐵 = (μ₀ / 4π) * (I * d𝐿 x ẑ) / r²
Where:
d𝐵 is the magnetic field vector,
μ₀ is the magnetic constant,
I is the current flowing through the wire segment d𝐿,
d𝐿 is the vector element along the wire,
r is the distance between the wire element and the point where the magnetic field is calculated.Maxwell's
Displacement Current:
Maxwell's addition to Ampere's law introduced the concept of displacement current, which accounts for the changing electric
field producing magnetic fields.
Formula: ∮ E · dl = -dΦ/dt + μ₀ * ε₀ * d(∮ B · dA)/dtPoynting's Theorem:
Poynting's theorem relates the flow of electromagnetic energy in a region to the electromagnetic fields present.
Formula: ∇ · S = -∂u/∂t - J · E
Where:
∇ · S is the divergence of the Poynting vector (energy flow),
∂u/∂t is the rate of change of energy density,
J is the current density,
E is the electric field.Maxwell's Equations in Differential Form:
Maxwell's equations can also be expressed in their differential form, which provides a more detailed and precise description
of electromagnetic phenomena. The four equations are:Gauss's law for electric fields:
∇ · E = (1/ε₀) * ρGauss's law for magnetic fields:
∇ · B = 0Faraday's law:
∇ × E = -∂B/∂tAmpere's law with Maxwell's addition:
∇ × B = μ₀ * (J + ε₀ * ∂E/∂t)Where:
∇ is the del operator (vector differential operator),
E is the electric field,
B is the magnetic field,
ρ is the charge density,
J is the current density.Electromagnetic Waves:
Electromagnetic waves are a fundamental aspect of electromagnetism, representing the propagation of combined electric and magnetic fields through space. They are described by the wave equation and have various properties, such as wavelength, frequency, and propagation speed.Wave equation: ∇²E - (1/c²) * ∂²E/∂t² = 0
Where:
∇² is the Laplacian operator,
E is the electric field,
c is the speed of light in vacuum (c ≈ 3 × 10^8 m/s).Snell's Law of Refraction:
Snell's law describes the relationship between the angles of incidence and refraction when light passes through a boundary between two media with different refractive indices.
Formula: n₁ * sin(θ₁) = n₂ * sin(θ₂)
Where:
n₁ and n₂ are the refractive indices of the media,
θ₁ and θ₂ are the angles of incidence and refraction, respectively.Larmor Formula:
The Larmor formula calculates the power radiated by an accelerated charged particle.
Formula: P = (2/3) * (q² * a²) / (4πε₀ * c³)
Where:
P is the radiated power,
q is the charge of the particle,
a is the particle's acceleration,
ε₀ is the electric constant,
c is the speed of light in vacuum.
These formulas and theorems provide a basic understanding of electromagnetism and its mathematical description. However, the field is vast and encompasses many more concepts and equations, including electromagnetic waves, electromagnetic radiation, and electromagnetic potentials, among others.