Electromagnetic Fields, Maxwell's Equations, and Fundamentals of Electric Circuits ch 28

Displacement Current and Maxwell's Equations

Displacement Current in a Capacitor

The displacement current is a concept introduced by Maxwell to account for the changing electric field in regions where there is no conduction current, such as between the plates of a charging capacitor. This allows Ampère's law to be consistent with the continuity of current in circuits containing capacitors.

• Conduction current ($I_{through}$) passes through wires and surfaces intersecting the wire, but not through the gap between capacitor plates.

• Displacement current ($I_{disp}$) is defined as $I_{disp} = rac{d ext{(electric flux)}}{dt} \ \ \ = \epsilon_0 \frac{d\Phi_E}{dt}$, where $\Phi_E$ is the electric flux.

• For a closed curve around the capacitor, the sum of conduction and displacement currents must be the same for all surfaces bounded by the curve: $I_{through,1} + I_{disp,1} = I_{through,2} + I_{disp,2}$.

• This ensures the magnetic field calculated from Ampère-Maxwell law is consistent.

Example: In a parallel plate capacitor, the conduction current passes through the wire (surface 2), but not through the gap (surface 1). The displacement current through the gap equals the conduction current through the wire.

Maxwell's Equations

Maxwell's equations provide a complete mathematical description of electric and magnetic fields and their interactions.

• Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = \frac{q_{in}}{\epsilon_0}$

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

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

• Ampère-Maxwell Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{through} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$

The Lorentz force law describes the force on a charged particle in 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:

• $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$

• $\nabla \cdot \vec{B} = 0$

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

• $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$

Physical interpretation: Divergence relates to sources (charges), curl relates to how fields circulate or change in time.

Production of Electromagnetic Waves

Electromagnetic (EM) waves are produced by accelerating charges, such as:

• Oscillating electric currents in antennas (radio, microwave)

• Transitions of electrons in atoms (visible, UV, X-ray)

• Collisions of charged particles (infrared, thermal radiation)

• Accelerating electrons in synchrotrons (gamma rays)

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

Electric Flux and Magnetic Field in Capacitors

Electric Flux in a Capacitor

In a parallel plate capacitor, the electric flux through a surface is given by:

• $\Phi_E = \vec{E} \cdot \vec{A}$, where $\vec{E}$ is the electric field and $\vec{A}$ is the area vector.

• For concentric circular surfaces within the plates, the flux is the same for surfaces that enclose the same field lines.

• For surfaces smaller than the plate, the flux is proportional to the area: $\Phi_E = E \cdot \pi r^2$.

Example: For three concentric circles of radii $R/2$, $R$, and $3R/2$, the flux through the largest surface (C) is greatest, while the flux through A and B (if they enclose all field lines) is the same.

Magnetic Field in a Capacitor

The magnetic field between the plates of a charging capacitor is due to the displacement current. By symmetry, the field is tangential to circles centered on the axis.

• From Ampère-Maxwell law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$

• For a radius $r$ inside the plates: $B(2\pi r) = \mu_0 \epsilon_0 \frac{d}{dt}(E \cdot \pi r^2)$

• Thus, $B(r) = \frac{\mu_0 \epsilon_0}{2} r \frac{dE}{dt}$

• At $r = R/2$, $B_1 = \frac{1}{2} B_2$ (where $B_2$ is at $r = R$)

Example: The magnetic field at half the radius is half the field at the edge.

Fundamentals of Electric Circuits

Circuit Elements and Diagrams

Electric circuits are composed of various elements, each with a specific function:

• Batteries: Provide emf (electromotive force) by chemical reactions.

• Wires: Conduct current with negligible resistance.

• Resistors: Impede current, dissipate energy as heat.

• Bulbs: Convert electrical energy to light.

• Capacitors: Store charge and energy.

• 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.

Kirchhoff's Laws and Basic Circuits

Kirchhoff's junction rule: The sum of currents entering a junction equals the sum leaving: $\sum I_{in} = \sum I_{out}$.

Kirchhoff's loop rule: The sum of potential differences around any closed loop is zero: $\sum \Delta V = 0$.

• Potential increases across a battery (from - to + terminal).

• Potential decreases across a resistor in the direction of current: $\Delta V = -IR$.

A complete circuit forms a continuous path for current between the terminals of an emf source.

Potential Difference

The potential difference across circuit elements determines the flow of current and energy transfer.

• For a battery with emf $\mathcal{E}$ and internal resistance $r$, $\Delta V_{bat} = \mathcal{E} - Ir$.

• In a loop, the sum of all potential changes must be zero.

Example: In a circuit with resistors and batteries, use the loop rule to solve for unknown emf or current.

Energy and Power in Circuits

Power quantifies the rate of energy transfer in a circuit.

• Power delivered by a source: $P_{bat} = I\mathcal{E}$

• Power dissipated by a resistor: $P_R = \Delta V \cdot I = I^2 R = \frac{(\Delta V)^2}{R}$

Power Dissipated in Wires

When wires have resistance, they dissipate power as heat. The wire with greater resistance dissipates more power for the same current.

• $P = I^2 R$

Material	Resistivity (Ω·m)
Cu	1.7 × 10-8
Fe	9.7 × 10-8
Nichrome	1.0 × 10-6

Series and Parallel Resistors

Series: Resistors in series have the same current; total resistance is the sum:

• $R_{eq} = R_1 + R_2 + R_3 + \ldots$

Parallel: Resistors in parallel have the same potential difference; total resistance is:

• $R_{eq} = \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots \right)^{-1}$

Three Bulbs in Series and Parallel

For identical bulbs:

• Series: Current is the same through each bulb; potential difference divides equally.

• Parallel: Potential difference is the same across each bulb; current divides equally.

Example: In series, $V_{bulb} = \frac{\mathcal{E}_{bat}}{3}$, $I_{bulb} = I_{bat}$; in parallel, $V_{bulb} = \mathcal{E}_{bat}$, $I_{bat} = 3I_{bulb}$.

Bulb Brightness in Series vs. Parallel

Brightness is proportional to power dissipated. For identical bulbs and batteries:

• In series: $P_{bulb,series} = \frac{(\mathcal{E}_{bat}/3)^2}{R_{bulb}}$

• In parallel: $P_{bulb,parallel} = \frac{\mathcal{E}_{bat}^2}{R_{bulb}}$

• Ratio: $\frac{P_{bulb,parallel}}{P_{bulb,series}} = 9$

Bulbs in parallel are much brighter than bulbs in series.

Real Batteries and Short Circuits

Real batteries have internal resistance $r$, causing the terminal voltage to be less than the emf:

• $\Delta V_{bat} = \mathcal{E} - Ir$

A short circuit occurs when a low-resistance path connects the terminals, causing high current and power dissipation:

• $P = \frac{\mathcal{E}^2}{r}$

Example: For a battery with $\mathcal{E} = 6.2\ V$ and $r = 0.7\ \Omega$, $P = 55\ W$.

Summary Table: Series vs. Parallel Bulbs

Configuration	Current through each bulb	Potential across each bulb	Power per bulb
Series	$I_{bat}$	$\mathcal{E}_{bat}/3$	$\frac{(\mathcal{E}_{bat}/3)^2}{R_{bulb}}$
Parallel	$I_{bat}/3$	$\mathcal{E}_{bat}$	$\frac{\mathcal{E}_{bat}^2}{R_{bulb}}$

Additional info: These notes expand on the original slides by providing definitions, equations, and examples for each concept, ensuring a self-contained study guide for exam preparation.