Electromagnetism and Circuits: Exam Study Guide

Electromagnetic Induction and Circuits

Induced EMF in Moving Loops

When a conducting loop moves through a magnetic field, an electromotive force (EMF) is induced according to Faraday's Law. This principle is fundamental in understanding how electric currents are generated in circuits exposed to changing magnetic environments.

• Faraday's Law of Induction: The induced EMF ($\mathcal{E}$) in a loop is proportional to the rate of change of magnetic flux ($\Phi_B$) through the loop.

• Formula: $\mathcal{E} = -\frac{d\Phi_B}{dt}$

• Magnetic Flux: $\Phi_B = B \cdot A$, where $B$ is the magnetic field strength and $A$ is the area perpendicular to the field.

• Direction of Induced Current: Determined by Lenz's Law, which states that the induced current will flow in a direction that opposes the change in magnetic flux.

• Example: A loop moving into a region with a uniform magnetic field will experience an increasing flux, inducing a current that creates a magnetic field opposing the increase.

RL Circuits and Current Establishment

RL circuits consist of a resistor (R) and an inductor (L) in series. When a voltage is applied, the current does not immediately reach its maximum value due to the inductor's opposition to changes in current.

• Time Constant ($\tau$): $\tau = \frac{L}{R}$, determines how quickly the current approaches its steady-state value.

• Current Growth: $I(t) = \frac{\mathcal{E}}{R} \left(1 - e^{-t/\tau}\right)$

• Final Value: As $t \to \infty$, $I(t) \to \frac{\mathcal{E}}{R}$.

• Example: If $L = 0.03$ H, $R = 0.04$ Ω, and $\mathcal{E} = 100$ V, then $\tau = 0.75$ s and the final current is $2500$ A.

Current Direction and Magnitude

The direction of the induced current in a loop depends on the relative motion of the loop and the magnetic field, as well as the orientation of the field.

• Clockwise or Counterclockwise: Use the right-hand rule and Lenz's Law to determine the direction.

• Magnitude: The magnitude of the current can be plotted as a function of position or time as the loop moves through the field.

• Example: If the loop moves at $10$m/s and the field region is $0.4$ m wide, the time to traverse is $0.04$ s.

Electromagnetic Waves

Properties of Sinusoidal Radio Waves

Electromagnetic waves consist of oscillating electric and magnetic fields that propagate through space. The strength and direction of these fields are related by Maxwell's equations.

• Wave Equation: $E = E_0 \cos(kx - \omega t)$, $B = B_0 \cos(kx - \omega t)$

• Relationship: $B_0 = \frac{E_0}{c}$, where $c$ is the speed of light.

• Direction: The electric field ($\vec{E}$), magnetic field ($\vec{B}$), and wave vector ($\vec{k}$) are mutually perpendicular.

• Example: For $E_0 = 100$ V/m and $f = 100$ MHz, $B_0 = 3.33 \times 10^{-7}$ T.

Maximum Induced Voltage in a Loop

The orientation of a loop in an electromagnetic wave affects the induced voltage. Maximum amplitude is achieved when the plane of the loop is perpendicular to the magnetic field vector.

• Induced EMF: $\mathcal{E}_{max} = A B_0 \omega$

• Example: For $A = 1$ cm$^2$, $B_0 = 3.33 \times 10^{-7}$ T, $\omega = 2\pi f$, $\mathcal{E}_{max} \approx 2.1 \times 10^{-4}$ V

Poynting Vector and Energy Density

The Poynting vector represents the rate of energy transfer per unit area in an electromagnetic wave.

• Poynting Vector: $S = \frac{E_0 B_0}{\mu_0}$

• Energy Density: $u = \frac{S}{c} = \frac{E_0^2}{\mu_0 c}$

• Example: For $E_0 = 100$ V/m, $u \approx 8.85 \times 10^{-5}$ J/m$^3$

Capacitors and Electric Fields

Parallel Plate Capacitor Connected to a Battery

A parallel plate capacitor stores energy in the electric field between its plates. When connected to a battery, the voltage across the plates remains constant as long as the battery is connected.

• Electric Field: $E = \frac{V}{d}$, where $V$ is the voltage and $d$ is the plate separation.

• Current: When the plate separation changes, a current flows as the charge on the plates adjusts to maintain the voltage.

• Example: For $V = 10$ V, $d = 0.01$ m, $E = 1.0 \times 10^3$ V/m.

Magnetic Field in a Capacitor

When the electric field between the plates changes with time, a displacement current is created, which produces a magnetic field.

• Displacement Current: $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$

• Magnetic Field: $B = \frac{\mu_0 I_d}{2\pi r}$, where $r$ is the distance from the axis.

• Example: If the plate separation increases at $0.60$ mm/s, calculate $dE/dt$ and then $B$ at $r = 5$ cm.

Changing Plate Separation and Effects

When the battery is disconnected and the plate separation changes, the charge on the plates remains constant, but the voltage and electric field change.

• Constant Charge: $Q = C V$ remains constant.

• Electric Field: $E = \frac{Q}{\epsilon_0 A}$

• Magnetic Field: If $E$ is not changing, $B = 0$.

Capacitor Circuits and Dielectrics

Series Capacitor Circuits

When capacitors are connected in series, the total voltage is divided among them according to their capacitances.

• Voltage Across Capacitor: $V_i = \frac{Q}{C_i}$

• Charge: In series, all capacitors have the same charge.

• Example: For $C_1 = 2$ μF, $C_2 = 1.34$ μF, $V = 30$ V, $Q = 2.68 \times 10^{-5}$ C.

Charging and Discharging Capacitors

When a switch is closed in a circuit with capacitors and resistors, the charge on the capacitor changes exponentially with time.

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

• Time Constant: $\tau = RC$

• Example: For $R = 50$ Ω, $C = 2$ μF, $\tau = 0.1$ ms.

Dielectrics in Capacitors

Inserting a dielectric material between the plates of a capacitor increases its capacitance by a factor equal to the dielectric constant $k$.

• Capacitance with Dielectric: $C' = kC$

• Voltage and Charge: If the capacitor is isolated, the charge remains constant and the voltage decreases. If connected to a battery, the voltage remains constant and the charge increases.

• Example: For $k = 5$, the capacitance increases fivefold.

Summary Table: Key Formulas and Concepts

Concept	Formula	Key Points
Faraday's Law	$\mathcal{E} = -\frac{d\Phi_B}{dt}$	Induced EMF opposes change in flux
RL Circuit Current	$I(t) = \frac{\mathcal{E}}{R} (1 - e^{-t/\tau})$	Current grows exponentially
EM Wave Magnetic Field	$B_0 = \frac{E_0}{c}$	Fields are perpendicular
Poynting Vector	$S = \frac{E_0 B_0}{\mu_0}$	Energy flow per unit area
Capacitor Electric Field	$E = \frac{V}{d}$	Uniform between plates
Capacitance with Dielectric	$C' = kC$	Dielectric increases capacitance

Additional info: Some explanations and formulas have been expanded for clarity and completeness, and example values have been inferred for illustration.