Electromagnetic Fields, Maxwell's Equations, and Fundamentals of Circuits ch 31

Chapter 30: Electromagnetic Induction

30.6 Induced Fields

A changing magnetic field creates an induced electric field. These induced electric fields form closed loops and do not start or end on charges. The electromotive force (emf) around a closed curve is given by:

• Faraday's Law:

$\varepsilon = \oint \vec{E} \cdot d\vec{l}$

• Lenz's Law: The direction of the induced current (and thus the induced electric field) opposes the change in magnetic flux that produced it.

• Application: Induced electric fields are fundamental in the operation of generators and transformers.

Change in Magnetic Flux

When the magnetic flux through a region changes, an electric field is induced. For a uniform magnetic field directed out of the page, increasing at a steady rate, the induced electric field forms concentric circles around the axis of the changing field.

• Direction: Determined by Lenz's Law; the induced field opposes the change in flux.

• Example: If the magnetic field is increasing out of the page, the induced electric field at a point on the circle is tangent to the circle, in the direction that would produce a current opposing the increase.

Chapter 31: Changing Electric Fields

31.1 E or B? It Depends on Your Perspective

The observed electric and magnetic fields depend on the observer's frame of reference. If observer B moves with velocity $v_{BA}$ relative to observer A, the fields transform as:

• $\vec{E}_B = \vec{E}_A + v_{BA} \times \vec{B}_A$

• $\vec{B}_B = \vec{B}_A - \frac{1}{c^2} v_{BA} \times \vec{E}_A$

where $c$ is the speed of light. These transformations are valid for $v_{BA} \ll c$.

• Example: A stationary observer sees only an electric field from a stationary charge, while a moving observer sees both electric and magnetic fields.

31.3 The Displacement Current

The displacement current is defined as the rate of change of electric flux through a surface:

$I_{disp} = \varepsilon_0 \frac{d\Phi_E}{dt}$

To account for both conduction current and changing electric flux, Ampère's law is generalized as the Ampère-Maxwell law:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 (I_{through} + I_{disp})$

• Induced Magnetic Field: A changing electric field creates an induced magnetic field.

Displacement Current in a Capacitor

In a parallel plate capacitor, the conduction current passes through the wires but not through the gap between plates. The displacement current accounts for the changing electric field in the gap, ensuring continuity in Ampère-Maxwell law.

• Key Points:

  • Conduction current $I_{through,1}$ through surface 1 (between plates) is zero; $I_{through,2}$ (through wire) is nonzero.

  • Displacement current $I_{disp,1}$ through surface 1 equals conduction current $I_{through,2}$ through surface 2.

  • Both surfaces connect to the same curve, so $\oint \vec{B} \cdot d\vec{l}$ must be the same for both.

• Example: The magnetic field direction can be determined from the current direction using the right-hand rule.

Electric Flux in a Capacitor

For a circular parallel plate capacitor with radius $R$, the electric flux through concentric surfaces depends on the area enclosed:

• For $r < R$, the electric field is uniform and perpendicular to the surface, so $\Phi_E = E \pi r^2$.

• All field lines pass through surfaces A and B, so $\Phi_{E,A} = \Phi_{E,B}$.

• For surface C ($r > R$), no field lines pass through, so $\Phi_{E,C} = 0$.

Surface	Electric Flux $\Phi_E$
A	$E \pi R^2$
B	$E \pi R^2$
C	0

Magnetic Field in a Capacitor

The strength of the magnetic field at a radius $r$ from the center of a charging capacitor is proportional to $r$:

• At $r = R$, $B_1$ is the reference value.

• At $r = R/2$, $B_2 = \frac{1}{2} B_1$.

Derived from Ampère-Maxwell law:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$

• Example: The magnetic field inside the capacitor increases linearly with distance from the center.

31.4 Maxwell's Equations

Maxwell's equations provide a complete mathematical description of electric and magnetic fields:

• Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{in}}{\varepsilon_0}$

• Gauss's Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$

• Faraday's Law: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

• Ampère-Maxwell Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{through} + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$

The Lorentz force describes how a charged particle interacts with electric and magnetic fields:

$\vec{F}_{on\ q} = q(\vec{E} + \vec{v} \times \vec{B})$

Alternative Forms of Maxwell's Equations

Maxwell's equations can also be written in differential form using the divergence and curl theorems:

• $\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$

• $\nabla \cdot \vec{B} = 0$

• $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$

• $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$

Divergence theorem: Relates the flux of a vector field through a surface to the divergence inside the volume.

Curl theorem: Relates the circulation of a vector field around a loop to the curl inside the surface.

Production of Electromagnetic Waves

Electromagnetic waves are produced by accelerating charges. Examples include:

• Oscillating electric currents in antennas (radio waves)

• Transitions of electrons in atoms (visible light, ultraviolet, x-rays)

• Acceleration of charges in collisions (infrared, microwave)

• Acceleration of electrons in synchrotrons (gamma rays)

Electromagnetic waves span a wide range of frequencies and wavelengths, from radio waves to gamma rays.

Chapter 28: Fundamentals of Circuits

28.1 Circuit Elements and Diagrams

Basic circuit elements include:

• Batteries: Chemical reactions create charge separation.

• Wires: Conduct electrons with negligible resistance.

• Resistors: Convert electrical energy to heat, opposing current flow.

• Bulbs: Emit light when current passes through.

• Capacitors: Store charge by separating positive and negative charges.

• Junctions: Points where current can split or combine.

• Switches: Open or close circuits.

Circuit diagrams use standardized symbols to represent these elements and their connections.

28.2 Kirchoff's Laws and the Basic Circuit

Kirchoff's junction rule: The sum of currents entering a junction equals the sum leaving it:

$\sum I_{in} = \sum I_{out}$

Kirchoff's loop rule: The sum of potential differences around any closed loop is zero:

$\sum \Delta V = 0$

• Potential difference across elements:

  • Batteries: $\Delta V = +\varepsilon$ (from negative to positive terminal)

  • Resistors: $\Delta V = -IR$ (in direction of current)

• Source: An element that produces emf, such as a battery.

• Complete circuit: Forms a continuous path between terminals of an emf source.

Potential Difference in Circuits

The emf of a battery in a circuit can be determined by applying Kirchoff's loop rule and summing the potential differences across all elements in the loop.

• Example: For a circuit with resistors and batteries, sum the voltage drops and rises to solve for the unknown emf.

Demo: Generator and Light

When a generator is connected to a load (such as a light bulb), it is harder to turn because energy is being converted to thermal energy in the bulb. This demonstrates conservation of energy in electrical circuits.

• Application: Car alternators use generators to provide electrical power.

*Additional info: The notes cover topics from Chapters 28, 30, and 31, including electromagnetic induction, Maxwell's equations, displacement current, electric and magnetic fields in capacitors, circuit elements, Kirchoff's laws, and practical applications such as generators and electromagnetic wave production. These are all core topics in a college-level Physics course.*