PART VI KNOWLEDGE STRUCTURE
Electricity and Magnetism

Key Findings

What are the overarching findings of Part VI?

Charge, like mass, is a fundamental property of matter.

There are two kinds of charge: positive and negative.
Charges interact via the electric field:
- Source charges create an electric field.
- A charge in the field experiences an electric force.
Moving charges (currents) interact with other moving charges via magnetic fields and forces.
Fields are also created when other fields change:
- A changing magnetic field induces an electric field.
- A changing electric field induces a magnetic field.

Laws

What laws of physics govern electricity and magnetism?

Maxwell’s equations are the formal statements of the laws of electricity and magnetism, but most problem solving is based on simpler versions of these equations:

Coulomb’s law	$\vec{E}_{point charge} = \frac{1}{4 π ε_{0}} \frac{q}{r^{2}} \hat{r}$	Gauss’s law	$\oint \vec{E} \cdot d \vec{A} = \frac{Q_{in}}{ε_{0}}$
Biot-Savart law	$\vec{B}_{point charge} = \frac{μ_{0}}{4 π} \frac{q \vec{v} \times \hat{r}}{r^{2}}$	Ampère’s law	$\oint \vec{B} \cdot d \vec{s} = μ_{0} I_{through}$
Lorentz force law	$\vec{F}_{on q} = q \vec{E} + q \vec{v} \times \vec{B}$
Faraday’s law	$ε = \| d Φ_{m} / d t \| I_{induced} = ε / R in the direction of Lenz ’ s law$

Models

What are the most important models of electricity and magnetism?

Charge model

Two types of charge: positive and negative.
- Like charges repel, opposite charges attract.
- Charge is conserved but can be transferred.
Two types of materials: conductors and insulators.
Neutral objects (no net charge) can be polarized.

The figure consists of two parts that illustrate the nature of force acting on two types of charges.

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Electric field model

Charges interact via the electric field.
- Source charges create an electric field.
- Charge $q$ experiences $\vec{F}_{on q} = q \vec{E}$ .
  
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Important electric field models
- Point charge
- Charged sphere
- Long charged wire
- Charged plane
  
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Magnetic field models

Moving charges and currents interact via the magnetic field.
- Current creates a magnetic field.
- Charge $q$ experiences $\vec{F}_{on q} = q \vec{v} \times \vec{B}$ .
- Current $I$ experiences $\vec{F}_{on I} = I \vec{l} \times \vec{B}$ .
  
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Important magnetic field models
- Straight, current-carrying wire
- Current loop
- Solenoid
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Electromagnetic waves

An electromagnetic wave is a self-sustaining electromagnetic field.
- $\vec{E} and \vec{B}$ are perpendicular to $\vec{v}$ .
- Wave speed $v_{em} = 1 / \sqrt{ϵ_{0} μ_{0}} = c$ .
  
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Tools

What are the most important tools introduced in Part VI?

Electric potential and potential energy

Electric interactions can also be described by an electric potential:

$V_{point charge} = q / 4 π ε_{0} r$
- A charge $q$ in a potential has potential energy $U = q V$ .
- Mechanical energy is conserved.
Field and potential are closely related.
- The field is perpendicular to equipotential surfaces.
- The field points “downhill” toward lower potential.
  
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Circuits

Batteries are sources of emf.
Current is the rate of flow of charge: $I = d Q / d t$ .

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Circuit elements:
- Capacitors: $Q = C Δ V$
- Resistors: $I = Δ V / R$ (Ohm’s law)
- Capacitors and resistors can be combined in series or in parallel.
Kirchhoff’s laws:
- Loop law: The sum of potential differences around a loop is zero.
- Junction law: The sum of currents at a junction is zero.

Uniform fields

A parallel-plate capacitor creates a uniform electric field.
A solenoid creates a uniform magnetic field.

The figure illustrates the electric field inside a parallel plate capacitor.

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Induced currents

Lenz’s law: An induced electric current flows in the direction such that the induced magnetic field opposes the change in the magnetic flux.

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