Introduction to Electricity and Magnetism

Electricity and magnetism are two fundamental aspects of electromagnetism, one of the four fundamental forces of nature. This guide covers the essential concepts, laws, and equations you need to understand E&M physics.

1. Electric Charge and Coulomb's Law

Electric Charge

Electric charge is a fundamental property of matter. There are two types: positive (+) and negative (-). Like charges repel, opposite charges attract. Charge is quantized in units of the elementary charge:

\[e = 1.602 \times 10^{-19} \text{ C}\]

where: \(e\) = elementary charge (charge of a proton or electron)

Coulomb's Law

The force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

\[\vec{F} = k_e \frac{q_1 q_2}{r^2} \hat{r}\] \[k_e = \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2\]

where:

\(F\) = electrostatic force (N)
\(q_1, q_2\) = charges (C)
\(r\) = distance between charges (m)
\(k_e\) = Coulomb's constant
\(\epsilon_0 = 8.85 \times 10^{-12}\) C²/(N·m²) = permittivity of free space

Tip: If the force is positive, charges repel. If negative, they attract.

2. Electric Field

The electric field is a vector field that describes the force per unit charge at any point in space.

Electric Field Definition

\[\vec{E} = \frac{\vec{F}}{q}\]

Electric Field from a Point Charge

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

where: \(E\) = electric field strength (N/C or V/m)

Electric Field from Multiple Charges (Superposition)

\[\vec{E}_{total} = \sum_{i=1}^{n} \vec{E}_i = k_e \sum_{i=1}^{n} \frac{q_i}{r_i^2} \hat{r}_i\]

Electric Field Lines

Field lines point in the direction a positive test charge would move
Field lines originate on positive charges and terminate on negative charges
Density of lines indicates field strength
Field lines never cross

3. Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface.

Gauss's Law (Integral Form)

\[\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}\]

where:

\(\oint\) = closed surface integral
\(\vec{E}\) = electric field
\(d\vec{A}\) = differential area element (points outward)
\(Q_{enclosed}\) = total charge inside the surface

Applications of Gauss's Law

Electric Field of an Infinite Line Charge:

\[E = \frac{\lambda}{2\pi\epsilon_0 r}\]

where \(\lambda\) = linear charge density (C/m)

Electric Field of an Infinite Sheet:

\[E = \frac{\sigma}{2\epsilon_0}\]

where \(\sigma\) = surface charge density (C/m²)

4. Electric Potential

Electric potential is the electric potential energy per unit charge. It's a scalar quantity.

Electric Potential Difference

\[\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{l}\]

Electric Potential from a Point Charge

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

where: \(V\) = electric potential (V = J/C)

Relationship Between Field and Potential

\[\vec{E} = -\nabla V = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)\]

Note: Electric field points in the direction of decreasing potential. The negative sign indicates this relationship.

Electric Potential Energy

\[U = qV = k_e \frac{q_1 q_2}{r}\]

5. Capacitance and Capacitors

A capacitor is a device that stores electric charge and energy in an electric field.

Capacitance Definition

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

where:

\(C\) = capacitance (F = Farads)
\(Q\) = charge stored (C)
\(V\) = voltage across capacitor (V)

Parallel Plate Capacitor

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

where \(A\) = plate area (m²), \(d\) = separation distance (m)

Energy Stored in a Capacitor

\[U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}\]

Capacitors in Circuits

Series Combination:

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

Parallel Combination:

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

6. Electric Current and Resistance

Electric Current

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

where: \(I\) = current (A = Amperes = C/s)

Current Density

\[\vec{J} = \frac{I}{A} = nq\vec{v}_d\]

where \(n\) = charge carrier density, \(\vec{v}_d\) = drift velocity

Ohm's Law

\[V = IR\]

where:

\(V\) = voltage (V)
\(I\) = current (A)
\(R\) = resistance (Ω = Ohms)

Resistance and Resistivity

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

where \(\rho\) = resistivity (Ω·m), \(L\) = length, \(A\) = cross-sectional area

Power Dissipation

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

where: \(P\) = power (W = Watts = J/s)

Resistors in Circuits

Series Combination:

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

Parallel Combination:

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

7. Kirchhoff's Circuit Laws

Kirchhoff's Current Law (KCL)

The sum of currents entering a junction equals the sum of currents leaving.

\[\sum I_{in} = \sum I_{out}\]

Kirchhoff's Voltage Law (KVL)

The sum of voltage changes around any closed loop is zero.

\[\sum V = 0\]

8. RC Circuits

Charging Capacitor

\[Q(t) = Q_{max}(1 - e^{-t/RC})\] \[V_C(t) = V_{max}(1 - e^{-t/RC})\] \[I(t) = I_0 e^{-t/RC}\]

Discharging Capacitor

\[Q(t) = Q_0 e^{-t/RC}\] \[V_C(t) = V_0 e^{-t/RC}\] \[I(t) = -I_0 e^{-t/RC}\]

Time Constant

\[\tau = RC\]

After time \(\tau\), the capacitor charges to 63% of maximum or discharges to 37% of initial value.

9. Magnetic Fields

Moving charges create magnetic fields, and magnetic fields exert forces on moving charges.

Magnetic Force on a Moving Charge

\[\vec{F} = q\vec{v} \times \vec{B}\] \[F = qvB\sin\theta\]

where:

\(\vec{B}\) = magnetic field (T = Tesla)
\(\vec{v}\) = velocity of charge
\(\theta\) = angle between \(\vec{v}\) and \(\vec{B}\)

Tip: Use the right-hand rule: fingers point in direction of \(\vec{v}\), curl toward \(\vec{B}\), thumb points in direction of \(\vec{F}\) for positive charges.

Magnetic Force on a Current-Carrying Wire

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

Circular Motion in Magnetic Field

A charged particle entering perpendicular to a uniform magnetic field follows a circular path:

\[r = \frac{mv}{qB}\] \[T = \frac{2\pi m}{qB}\] \[f = \frac{qB}{2\pi m}\]

10. Sources of Magnetic Field

Biot-Savart Law

Gives the magnetic field produced by a current element:

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

where \(\mu_0 = 4\pi \times 10^{-7}\) T·m/A = permeability of free space

Magnetic Field of a Long Straight Wire

\[B = \frac{\mu_0 I}{2\pi r}\]

Magnetic Field at Center of Circular Loop

\[B = \frac{\mu_0 I}{2R}\]

where \(R\) = radius of loop

Magnetic Field Inside a Solenoid

\[B = \mu_0 nI\]

where \(n = N/L\) = number of turns per unit length

Ampère's Law

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

11. Electromagnetic Induction

Magnetic Flux

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

where: \(\Phi_B\) = magnetic flux (Wb = Weber = T·m²)

Faraday's Law of Induction

A changing magnetic flux induces an EMF:

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

For \(N\) turns of a coil:

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

Lenz's Law

The induced current flows in a direction to oppose the change in magnetic flux that produced it (hence the negative sign in Faraday's law).

Motional EMF

For a conductor moving through a magnetic field:

\[\mathcal{E} = BLv\]

where \(L\) = length of conductor, \(v\) = velocity perpendicular to \(B\)

12. Inductance

Self-Inductance

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

where: \(L\) = inductance (H = Henry)

Inductance of a Solenoid

\[L = \mu_0 n^2 A l\]

where \(n\) = turns per unit length, \(A\) = cross-sectional area, \(l\) = length

Energy Stored in an Inductor

\[U_L = \frac{1}{2}LI^2\]

RL Circuits

Current Growth:

\[I(t) = I_{max}(1 - e^{-t/\tau})\]

Current Decay:

\[I(t) = I_0 e^{-t/\tau}\]

Time Constant:

\[\tau = \frac{L}{R}\]

13. Maxwell's Equations

The four fundamental equations that describe all of electromagnetism:

Maxwell's Equations (Integral Form)

1. Gauss's Law for Electricity:

\[\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}\]

2. Gauss's Law for Magnetism:

\[\oint \vec{B} \cdot d\vec{A} = 0\]

(No magnetic monopoles exist)

3. Faraday's Law:

\[\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}\]

4. Ampère-Maxwell Law:

\[\oint \vec{B} \cdot d\vec{l} = \mu_0 I + \mu_0\epsilon_0\frac{d\Phi_E}{dt}\]

14. Electromagnetic Waves

Changing electric and magnetic fields propagate through space as electromagnetic waves.

Speed of Light

\[c = \frac{1}{\sqrt{\mu_0\epsilon_0}} = 3.00 \times 10^8 \text{ m/s}\]

Wave Properties

\[c = f\lambda\] \[E = cB\]

Energy and Intensity

\[I = \frac{P}{A} = \frac{E_0 B_0}{2\mu_0} = \frac{cB_0^2}{2\mu_0} = \frac{c\epsilon_0 E_0^2}{2}\]

Important Constants Summary

Constant	Symbol	Value
Elementary charge	\(e\)	\(1.602 \times 10^{-19}\) C
Coulomb's constant	\(k_e\)	\(8.99 \times 10^9\) N·m²/C²
Permittivity of free space	\(\epsilon_0\)	\(8.85 \times 10^{-12}\) C²/(N·m²)
Permeability of free space	\(\mu_0\)	\(4\pi \times 10^{-7}\) T·m/A
Speed of light	\(c\)	\(3.00 \times 10^8\) m/s

Key Takeaways

Electric forces arise from charges and follow Coulomb's Law
Electric fields describe the force per unit charge in space
Gauss's Law relates electric flux to enclosed charge
Electric potential is energy per unit charge (scalar)
Capacitors store charge and energy in electric fields
Ohm's Law relates voltage, current, and resistance
Magnetic fields arise from moving charges and currents
Faraday's Law shows changing magnetic flux induces EMF
Maxwell's Equations unify electricity and magnetism
Electromagnetic waves travel at the speed of light

Important: Always use consistent units (SI system) and pay attention to vector directions when solving E&M problems!