Electric Circuits, Magnetic Fields, and Capacitors: Exam Solutions and Study Guide

Electric Circuits

Ohm's Law and Current in Circuits

Ohm's Law describes the relationship between voltage, current, and resistance in an electric circuit. It is fundamental for analyzing simple and complex circuits.

• Ohm's Law: The current $I$ through a resistor is given by $I = \frac{V}{R}$, where $V$ is the voltage across the resistor and $R$ is the resistance.

• Changing Voltage: Increasing the voltage across a resistor increases the current proportionally.

• Changing Resistance: Increasing resistance decreases the current for a fixed voltage.

• Example: Tripling the voltage will triple the current if resistance remains constant.

Resistors in Series and Parallel

Resistors can be combined in series or parallel to form more complex circuits. The total resistance depends on the configuration.

• Series: $R_{\text{eq}} = R_1 + R_2 + \ldots$

• Parallel: $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

• Example: Two bulbs in series have $R_{\text{eq}} = 2R$; in parallel, $R_{\text{eq}} = \frac{R}{2}$.

Power in Electric Circuits

Power dissipated by a resistor is related to the current and voltage.

• Power Formula: $P = I^2 R$ or $P = \frac{V^2}{R}$

• Example: If current doubles, power increases by a factor of four.

Magnetic Fields and Charged Particle Motion

Force on a Charged Particle in a Magnetic Field

A charged particle moving in a magnetic field experiences a force perpendicular to both its velocity and the field.

• Lorentz Force: $\vec{F} = q\vec{v} \times \vec{B}$

• Direction: Use the right-hand rule to determine the direction of the force.

• Example: A positive particle moving left in a downward field experiences a force out of the page.

Circular Motion in Magnetic Fields

Charged particles can move in circles under the influence of a magnetic field.

• Centripetal Force: $qvB = \frac{mv^2}{r}$

• Radius of Path: $r = \frac{mv}{qB}$

• Example: Calculating the radius for an electron in a given field and velocity.

Balancing Electric and Magnetic Forces

In some setups, electric and magnetic forces can be balanced so that a particle passes through undeflected.

• Balance Condition: $qE = qvB \implies E = vB$

• Example: For a velocity selector, only particles with $v = \frac{E}{B}$ pass through undeflected.

Capacitors and Dielectrics

Capacitance and Energy Storage

Capacitors store electric charge and energy. Their properties depend on geometry and materials.

• Capacitance Formula: $C = \varepsilon_0 \frac{A}{d}$, where $A$ is plate area, $d$ is separation, and $\varepsilon_0$ is the permittivity of free space.

• Charge Stored: $Q = CV$

• Energy Stored: $U = \frac{1}{2}CV^2$ or $U = \frac{Q^2}{2C}$

• Example: For $A = 1\,\text{m}^2$, $d = 1\,\text{mm}$, $V = 120\,\text{V}$, $C = 8.85 \times 10^{-9}\,\text{F}$, $Q = 1.06 \times 10^{-6}\,\text{C}$, $U = 6.37 \times 10^{-5}\,\text{J}$.

Effect of Dielectrics

Inserting a dielectric increases the capacitance by a factor equal to the dielectric constant $K$.

• New Capacitance: $C' = KC$

• Voltage Change: If charge is constant, $V' = \frac{V}{K}$

• Energy Change: $U' = \frac{Q^2}{2C'}$

• Example: If $K = 3$, $C' = 3C$, $V' = 40\,\text{V}$ (for $V = 120\,\text{V}$ initially).

Additional info:

• Some answers and steps are inferred from context and standard physics problem-solving methods.

• All equations are provided in LaTeX format for clarity.

• Topics covered are typical for a college-level introductory physics course (electricity and magnetism).