Study Notes: Electricity and Magnetism Fundamentals

Electricity and Magnetism Fundamentals

Electric Forces and Potential

This section covers the basic concepts of electric forces between charges, the calculation of electric fields and potentials, and the energy associated with charge configurations.

• Force on One of the Vector Charges: The force between two point charges is given by Coulomb's Law:

$F = k_e \frac{|q_1 q_2|}{r^2}$

• Potential Energy of a Configuration (Scalar): The potential energy of two point charges separated by a distance $r$ is:

$U = k_e \frac{q_1 q_2}{r}$

• E-Field at Location: The electric field due to a point charge $q$ at a distance $r$ is:

$E = k_e \frac{q}{r^2}$

• Potential at Location: The electric potential at a distance $r$ from a point charge $q$ is:

$V = k_e \frac{q}{r}$

Circuits

This section introduces the analysis of electric circuits, including resistors, capacitors, and the rules governing their behavior.

• RLC Circuits: Circuits containing resistors (R), inductors (L), and capacitors (C). The general differential equation for an RLC circuit is:

$L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0$

• Resistor-Only Circuits (Ohm's Law): The current through a resistor is proportional to the voltage across it:

$V = IR$

• Internal Resistance of a Battery: The terminal voltage $V_{term}$ of a battery with internal resistance $r$ is:

$V_{term} = \mathcal{E} - Ir$

• Capacitor Only on a Circuit: When a capacitor is connected to a voltage source, it charges up to the source voltage.

• RC Circuits: Circuits containing resistors and capacitors. The charging and discharging of a capacitor in an RC circuit follows exponential behavior.

Charging:

$Q(t) = Q_{max}(1 - e^{-t/RC})$

Discharging:

$Q(t) = Q_0 e^{-t/RC}$

• Time Constant: The time constant $\tau$ for an RC circuit is $\tau = RC$.

Magnetism

This section discusses the magnetic field produced by currents, the force on moving charges, and the application of the right-hand rule.

• B-Field Produced by Currents: The magnetic field around a long straight wire is given by:

$B = \frac{\mu_0 I}{2\pi r}$

• Motional EMF: A conductor of length $l$ moving at velocity $v$ perpendicular to a magnetic field $B$ experiences an emf:

$\mathcal{E} = Blv$

• Force on Moving Charges in B-Field (Second Law and Faraday's Law): The force on a charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ is:

$\vec{F} = q\vec{v} \times \vec{B}$

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is especially useful for non-uniform charge distributions.

$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$

• E-Field Due to Non-Uniform Charge Distribution: Use symmetry and Gauss's Law to calculate the electric field for complex charge distributions.

Right-Hand Rule

The right-hand rule is a mnemonic for determining the direction of the magnetic force, field, or current in electromagnetic situations.

• Application: Point your thumb in the direction of current (or velocity for a positive charge), and your fingers in the direction of the magnetic field; your palm points in the direction of the force.