Coulomb's Law: Describes the force between two charges.

\[ F = k \frac{|q_1 q_2|}{r^2} \]

Where:

• \( k = 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \)
• \( q_1, q_2 \) are the magnitudes of the charges.
• \( r \) is the distance between the charges.

Electric Field (E): A field around a charged object where other charges experience a force.

Formula: The electric field due to a point charge is given by:

\[ \vec{E} = k_e \frac{q}{r^2} \hat{r} \]

Where:

• \( \vec{E} \) is the electric field vector.
• \( k_e \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)).
• \( q \) is the source charge.
• \( r \) is the distance from the charge.
• \( \hat{r} \) is the unit vector in the direction from the charge to the point of interest.

The net electric field caused by multiple charges is the vector sum of the electric fields due to individual charges:

\[ \vec{E}_{net} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \dots \]

Electric Potential (V): The work done per unit charge to move a test charge from infinity to a point in space.

Formula: For a point charge:

\[ V = k_e \frac{q}{r} \]

Where:

• \( V \) is the electric potential.
• \( k_e \) is Coulomb's constant.
• \( q \) is the source charge.
• \( r \) is the distance from the charge.

The electric field is the negative gradient of the electric potential:

\[ \vec{E} = -\nabla V \]

This indicates that the electric field points in the direction of decreasing potential.

Capacitance (C): The ability of a system to store charge per unit voltage.

Formula: Capacitance is defined as:

\[ C = \frac{Q}{V} \]

Where:

• \( C \) is the capacitance.
• \( Q \) is the charge stored on the capacitor.
• \( V \) is the voltage across the capacitor.

The capacitance of a parallel plate capacitor is given by:

\[ C = \frac{\epsilon_0 A}{d} \]

Where:

• \( \epsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \, \text{F/m} \)).
• \( A \) is the area of one of the plates.
• \( d \) is the separation between the plates.

The energy \( U \) stored in a capacitor is given by:

\[ U = \frac{1}{2} C V^2 \]

Electric Current (I): The flow of electric charge through a conductor.

Formula: Current is defined as:

\[ I = \frac{dQ}{dt} \]

Where:

• \( I \) is the current (in amperes).
• \( Q \) is the charge.
• \( t \) is the time.

Ohm's Law relates the voltage \( V \), current \( I \), and resistance \( R \) in a circuit:

\[ V = IR \]

The resistance of a material is given by:

\[ R = \rho \frac{L}{A} \]

Where:

• \( R \) is the resistance.
• \( \rho \) is the resistivity of the material.
• \( L \) is the length of the conductor.
• \( A \) is the cross-sectional area of the conductor.

The power dissipated in a resistor is given by:

\[ P = IV = I^2R = \frac{V^2}{R} \]

DC circuits consist of resistors, capacitors, and batteries. The current flows in a single direction.

For resistors in series, the equivalent resistance \( R_{eq} \) is the sum of the individual resistances:

\[ R_{eq} = R_1 + R_2 + R_3 + \dots \]

For resistors in parallel, the reciprocal of the equivalent resistance \( R_{eq} \) is the sum of the reciprocals of the individual resistances:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \]

For capacitors in series:

\[ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots \]

For capacitors in parallel:

\[ C_{eq} = C_1 + C_2 + C_3 + \dots \]

Magnetic Field (\( \vec{B} \)): A field that exerts a force on moving charges and current-carrying wires.

Units: Tesla (T), where 1 T = 1 N/(A·m).

The magnetic force \( \vec{F}_B \) on a charge \( q \) moving with velocity \( \vec{v} \) in a magnetic field \( \vec{B} \) is given by:

\[ \vec{F}_B = q\vec{v} \times \vec{B} \]

The direction of the force is given by the right-hand rule.

The force on a straight current-carrying wire of length \( L \) in a uniform magnetic field is given by:

\[ \vec{F} = I\vec{L} \times \vec{B} \]

Where:

• \( I \) is the current in the wire.
• \( L \) is the length of the wire.
• \( \vec{B} \) is the magnetic field.

The Biot-Savart Law describes the magnetic field generated by a steady current:

\[ d\vec{B} = \frac{\mu_0 I d\vec{L} \times \hat{r}}{4\pi r^2} \]

Where:

• \( d\vec{B} \) is the infinitesimal magnetic field.
• \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T·m/A} \)).
• \( I \) is the current.
• \( d\vec{L} \) is the infinitesimal length element of the current-carrying wire.
• \( \hat{r} \) is the unit vector from the current element to the point of interest.
• \( r \) is the distance from the current element to the point of interest.

Ampère's Law relates the magnetic field around a closed loop to the total current passing through the loop:

\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} \]

Where \( I_{\text{enc}} \) is the total current enclosed by the loop.

Faraday's Law of Electromagnetic Induction: A changing magnetic field induces an electromotive force (EMF) in a circuit:

\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]

Where:

• \( \mathcal{E} \) is the induced EMF.
• \( \Phi_B \) is the magnetic flux.

Magnetic flux through a surface is given by:

\[ \Phi_B = \int \vec{B} \cdot d\vec{A} \]

Where:

• \( \vec{B} \) is the magnetic field.
• \( d\vec{A} \) is the area element vector perpendicular to the surface.

Lenz's Law states that the direction of the induced EMF and current will oppose the change in magnetic flux that produced it.

Inductance (L): The property of a circuit element that opposes changes in current flowing through it.

Self-Inductance: The induced EMF in a coil is proportional to the rate of change of current through the coil:

\[ \mathcal{E} = -L \frac{dI}{dt} \]

The inductance of a solenoid is given by:

\[ L = \frac{\mu_0 N^2 A}{l} \]

Where:

• \( \mu_0 \) is the permeability of free space.
• \( N \) is the number of turns of the solenoid.
• \( A \) is the cross-sectional area of the solenoid.
• \( l \) is the length of the solenoid.

The energy \( U \) stored in an inductor is given by:

\[ U = \frac{1}{2} L I^2 \]

Alternating Current (AC): Current that varies sinusoidally with time.

AC Voltage: The voltage in an AC circuit is given by:

\[ V(t) = V_0 \sin(\omega t) \]

Where:

• \( V_0 \) is the peak voltage.
• \( \omega \) is the angular frequency (\( \omega = 2\pi f \)).
• \( t \) is the time.

For a resistor in an AC circuit, the current and voltage are in phase. The voltage across the resistor is:

\[ V_R(t) = I_0 R \sin(\omega t) \]

For an inductor in an AC circuit, the current lags the voltage by 90 degrees. The voltage across the inductor is:

\[ V_L(t) = I_0 \omega L \cos(\omega t) \]

For a capacitor in an AC circuit, the current leads the voltage by 90 degrees. The voltage across the capacitor is:

\[ V_C(t) = \frac{I_0}{\omega C} \cos(\omega t) \]

Impedance (Z): The total opposition to current in an AC circuit, combining resistance, inductive reactance, and capacitive reactance.

Formula:

\[ Z = \sqrt{R^2 + \left( \omega L - \frac{1}{\omega C} \right)^2} \]

The average power delivered to an AC circuit is given by:

\[ P_{avg} = \frac{1}{2} V_0 I_0 \cos(\phi) \]

Where \( \phi \) is the phase angle between the voltage and current.

Electromagnetic Waves: Waves of electric and magnetic fields that propagate through space. They are transverse waves and travel at the speed of light in a vacuum.

Speed of Light: The speed of electromagnetic waves in a vacuum is:

\[ c = 3.00 \times 10^8 \, \text{m/s} \]

The electric and magnetic fields in an electromagnetic wave satisfy the wave equation:

\[ \frac{\partial^2 E}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 E}{\partial t^2} \]

And similarly for the magnetic field \( B \).

The energy density \( u \) of an electromagnetic wave is given by:

\[ u = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2} \frac{B^2}{\mu_0} \]

Where \( \epsilon_0 \) is the permittivity of free space and \( \mu_0 \) is the permeability of free space.

The intensity \( I \) of an electromagnetic wave is the power per unit area:

\[ I = \frac{P}{A} = \frac{c \epsilon_0 E_0^2}{2} \]

Where \( E_0 \) is the peak electric field.

Reflection: The bouncing back of light when it hits a surface.

Law of Reflection: The angle of incidence equals the angle of reflection (\( \theta_i = \theta_r \)).

Snell's Law: The relationship between the angles of incidence and refraction when light passes from one medium to another:

\[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]

Where \( n_1 \) and \( n_2 \) are the indices of refraction of the two media.

The lens and mirror equation relates the object distance (\( d_o \)), the image distance (\( d_i \)), and the focal length (\( f \)):

\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]

Magnification \( M \) is the ratio of the height of the image (\( h_i \)) to the height of the object (\( h_o \)):

\[ M = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \]