Comprehensive Physics Review: Electricity, Magnetism, and Electromagnetic Waves

Electric Fields and Gauss's Law

Electric Field Due to a Long Straight Wire

The electric field E at a distance r from a very long straight wire carrying a uniform linear charge density λ is given by:

$E = \frac{1}{2\pi\varepsilon_0} \frac{\lambda}{r}$

• λ (linear charge density): $\lambda = \frac{q}{L}$, where q is charge and L is length.

• Finding q: $q = \lambda L$

• Effect of changing parameters: If r is doubled, E is halved. If E is doubled, r is halved.

Electric Field at a Surface (Conductors)

The electric field just outside a charged conductor is related to the surface charge density σ by:

$E = \frac{\sigma}{\varepsilon_0}$

• If electric potential V and separation d between plates are given: $E = \frac{V}{d}$

• Equipotential Surface: A surface on which the electric potential is the same at every point.

Current, Resistance, and Circuits

Resistance and Ohm's Law

Resistance R of a wire depends on its resistivity ρ, length L, and cross-sectional area A:

$R = \rho \frac{L}{A}$

• Ohm's Law: $V = IR$ or $I = \frac{V}{R}$

• If the diameter of a wire is doubled, the area increases by a factor of 4 (since $A \propto d^2$), so resistance decreases by a factor of 4.

• If the original resistance is given, you can find the new resistance for different diameters and lengths.

Series and Parallel Resistors

• Series: $R_s = R_1 + R_2 + R_3 + \ldots$

• Parallel: $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$

• Current in Parallel: $I = I_1 + I_2 + I_3 + \ldots$

Example: If $V = 20\,\text{V}$, $I = 5\,\text{A}$, and all resistors have resistance $R$, calculate $R$. Use $V = IR$ to find $R = \frac{V}{I} = 4\,\Omega$.

Capacitors and Dielectrics

• Dielectric: An insulating material placed between the plates of a capacitor.

• When a dielectric is inserted, the capacitance increases, voltage V decreases (if charge is constant), and electric field E decreases.

Magnetic Fields and Forces

Magnetic Field Due to Currents

• For two long wires carrying currents in the same direction, the magnetic fields add between the wires and subtract outside.

• For opposite directions, the fields subtract between the wires and add outside.

• The magnetic field is zero at points where the fields due to each wire cancel.

Magnetic Force on a Current-Carrying Wire

The force on a wire of length L carrying current I in a magnetic field B at angle θ:

$F = I L B \sin \theta$

• If any two quantities are known, the third can be found.

Solenoids and Magnetic Field

• If the distance from the axis of a solenoid, electric field, and magnetic field are given, you can find the radius of the solenoid using Ampère's Law.

Energy Density in a Magnetic Field

The energy density u in a magnetic field is:

$u = \frac{B^2}{2\mu_0}$

For a long straight wire carrying current I at distance r:

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

Electromagnetic Induction

Faraday's Law and Lenz's Law

• Changing magnetic flux through a loop induces an emf (Faraday's Law):

$\varepsilon = -\frac{d\Phi_B}{dt}$

• Direction of induced emf opposes the change in flux (Lenz's Law).

Inductance and Induced emf

• For an inductor, the induced emf is:

$\varepsilon = -L \frac{di}{dt}$

• If current decreases, $\frac{di}{dt} < 0$, so emf is positive in the direction opposing the decrease.

Electromagnetic Waves

Properties of Electromagnetic Waves

• Both electric and magnetic fields are perpendicular to the direction of propagation and to each other.

• The ratio of their magnitudes is $E = cB$.

• Electromagnetic waves do not require a medium and travel at speed $c$ in vacuum.

• They are transverse waves.

Electromagnetic Spectrum

• Gamma rays have the highest frequency; radio waves have the lowest.

• Visible light falls between ultraviolet and infrared in the spectrum.

Direction of Propagation

• If the directions of E and B are given, the direction of propagation is given by the vector product E × B.

Poynting Vector

• The Poynting vector S represents the energy flux (power per unit area) of an electromagnetic wave:

$\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$

• If $E = 0.2\,\text{V/m}$, use $B = \frac{E}{c}$ to find $S$.

Geometric Optics

Index of Refraction

• Index of refraction n is given by:

$n = \frac{c}{v}$

• Where c is the speed of light in vacuum, v is the speed in the material.

• When light passes from a material with higher n to lower n, it bends away from the normal.

Snell's Law

Describes refraction at an interface:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

Total Internal Reflection

• Occurs when light attempts to move from a medium with higher n to lower n at an angle greater than the critical angle.

• Critical angle $\theta_c$ is given by:

$\sin \theta_c = \frac{n_2}{n_1}$

• For total internal reflection, angle of incidence $> \theta_c$.

Polarization

• Light can be polarized by reflection at a specific angle (Brewster's angle).

Additional info: The notes cover key concepts from chapters on electric fields, circuits, magnetism, electromagnetic waves, and optics, providing formulas, definitions, and example applications for each topic.