5.3 Maxwell’s Equations

Introduction

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields in space. They were first proposed by James Clerk Maxwell in the 19th century and are still widely used today in various fields of physics, including electromagnetism, optics, and quantum mechanics.

In this article, we will discuss Maxwell's equations, which includes an additional equation that describes the behavior of electric charges in space.

Key Vocabulary

• Electromagnetic field: A physical field that arises from the presence of electrically charged particles and affects the behavior of those particles.
• Electric field: A region of space around an electrically charged particle or group of particles, in which other electrically charged particles experience a force.
• Magnetic field: A region of space around a magnet or moving electric charge, in which other magnets or moving electric charges experience a force.
• Electromagnetic waves: A type of wave that consists of oscillating electric and magnetic fields, which propagate through space at the speed of light.
• Coulomb’s Law: A fundamental law of electrostatics that describes the interaction between electric charges.
• Gauss’s Law: A fundamental law of electromagnetism that relates the electric flux through a closed surface to the charge enclosed within that surface.
• Faraday’s Law: A fundamental law of electromagnetism that describes how a changing magnetic field induces an electric field.
• Ampere’s Law: A fundamental law of electromagnetism that relates the magnetic field around a closed loop to the electric current flowing through that loop.
• Maxwell’s Equations: A set of four partial differential equations that describe the behavior of electromagnetic fields and their interactions with charged particles.

Maxwell's Equations

Maxwell's equations are a set of four partial differential equations that relate the behavior of electric and magnetic fields to their sources. They are:

1. Gauss's Law for Electric Fields:

∇ · E = ρ / ε0

where E is the electric field, ρ is the charge density, and ε0 is the electric constant (also known as the vacuum permittivity).

This equation relates the divergence of the electric field to the charge density at a point in space. It states that the electric field lines originate from positive charges and terminate on negative charges.

2. Gauss's Law for Magnetic Fields:

∇ · B = 0

where B is the magnetic field.

This equation relates the divergence of the magnetic field to the absence of magnetic charges, which are also known as magnetic monopoles. It states that the magnetic field lines always form closed loops, and there are no isolated magnetic charges.

3. Faraday's Law of Electromagnetic Induction:

∇ × E = - ∂B / ∂t

This equation relates the curl of the electric field to the time rate of change of the magnetic field. It states that a changing magnetic field generates an electric field in the surrounding space.

4. Ampere's Law with Maxwell's Correction:

∇ × B = μ0(J + ε0∂E / ∂t)

where B is the magnetic field, J is the current density, μ0 is the magnetic constant (also known as the vacuum permeability), and ε0 is the electric constant.

This equation relates the curl of the magnetic field to the current density and the time rate of change of the electric field. It states that a current generates a magnetic field, and a changing electric field generates a magnetic field.

5. Ampere-Maxwell Law:

∇ × E = - μ0∂B / ∂t - J / ε0

This equation relates the curl of the electric field to the time rate of change of the magnetic field and the charge density. It states that a changing magnetic field generates an electric field and a charge density generates an electric field.

Practice Problems

1. A charge density of 2 nC/m3 is distributed uniformly throughout a sphere of radius 10 cm. Find the electric field at a distance of 5 cm from the center of the sphere.

Solution:

Using Gauss's law for electric fields, we have:

∇ · E = ρ / ε0

Since the charge density is uniform, we can find the total charge enclosed by a sphere of radius r as:

Q = ρ(4/3)πr3

Thus, the electric field at a distance of r from the center of the sphere is given by:

E = Q / (4πε0r2)

Substituting r = 5 cm and Q = 2 nC, we get:

E = (2 × 10^-9 C) / (4πε0(0.05 m)^2) ≈ 1.8 × 10^6 N/C

2. A wire loop of radius 5 cm lies in the x-y plane and carries a current of 2 A in the clockwise direction when viewed from the positive z-axis. Find the magnetic field at the center of the loop.

Solution:

Using Ampere's law, we have:

∇ × B = μ0J

Since the loop is circular and lies in the x-y plane, we can choose a circular Amperian loop centered on the z-axis and lying in the same plane as the loop. The current density J is given by:

J = I / (πr^2)

where I is the current in the loop and r is the radius of the Amperian loop. The direction of J is clockwise when viewed from the positive z-axis.

The Amperian loop has a circumference of 2πr, so we have:

∫B·dl = μ0I / πr^2 * πr^2

where B is the magnetic field and dl is a small element of the Amperian loop. The integral is evaluated over the entire circumference of the loop.

Since the magnetic field is constant along the circumference of the loop, we have:

B * 2πr = μ0I

Solving for B, we get:

B = μ0I / (2πr) = μ0I / (2π * 0.05 m) ≈ 3.2 × 10^-5 T

3. A plane electromagnetic wave is traveling in free space in the z-direction. The electric field of the wave is given by E = E0sin(kz - ωt), where E0 = 5 V/m, k = 2π/λ, and ω = 2πf. Find the magnetic field of the wave.

Solution:

Using Faraday's law of electromagnetic induction, we have:

∇ × E = - ∂B / ∂t

Since the wave is traveling in the z-direction, we have:

∂/∂x = ∂/∂y = 0

Taking the curl of E, we get:

∇ × E = (∂Ey/∂z - ∂Ez/∂y)i + (∂Ez/∂x - ∂Ex/∂z)j + (∂Ex/∂y - ∂Ey/∂x)k

Since E only has a z-component, we have:

∂Ey/∂z = ∂Ez/∂y = ∂Ex/∂z = ∂Ez/∂x = 0

and

∂Ex/∂y = E0kcos(kz - ωt)

∂Ey/∂x = -E0kcos(kz - ωt)

Substituting into the curl expression, we get:

∇ × E = E0k(cos(kz - ωt)j - sin(kz - ωt)i)

Comparing with the expression for ∂B/∂t, we have:

B = -E0/(ω/c)sin(kz - ωt)k = -E0/cos(θ)sin(kz - ωt)k

where θ is the angle between the electric and magnetic fields, which is π/2 in this case. Substituting the given values, we get:

B = -5/(3 × 10^8)sin(2π/λ z - 2πft)k