Electric Forces, Fields, Potential, and Circuits: College Physics Study Guide

Electric Forces and Coulomb's Law

Forces Between Point Charges

Electric forces arise from the interaction between charged particles. The magnitude of the force between two point charges is given by Coulomb's Law:

• Formula: $|F| = k_e \frac{|Q_1 Q_2|}{r^2}$ where $k_e = 8.99 \times 10^9\ \mathrm{N\cdot m^2/C^2}$ is Coulomb's constant, $Q_1$ and $Q_2$ are the charges, and $r$ is the separation distance.

• Direction: The force is attractive if the charges are opposite, and repulsive if they are the same sign.

• Example: Two charges, $+16\ \mu C$ and $-20\ \mu C$, separated by $20\ \mathrm{cm}$, experience a force calculated using the above formula.

Forces in a Three-Charge System

When more than two charges are present, the net force on any charge is the vector sum of the forces due to each of the other charges.

• Superposition Principle: Forces from multiple charges add vectorially.

• Example: Given three charges on the x-axis, calculate the net force on a third charge using Coulomb's Law for each pair and summing the results.

Electric Field

Definition and Calculation

The electric field at a point in space is defined as the force per unit charge experienced by a small positive test charge placed at that point.

• Formula: $\vec{E} = \frac{\vec{F}}{q}$ For a point charge: $E = k_e \frac{|Q|}{r^2}$

• Units: Newtons per coulomb (N/C).

• Direction: Away from positive charges, toward negative charges.

• Example: Calculate the electric field at $(x, y) = (5.0\ \mathrm{cm}, 0)$ due to a $20\ \mu C$ charge.

Electric Field and Weight of Charged Particles

To balance the weight of a charged particle (such as a proton or electron) in an electric field:

• Formula: $E = \frac{mg}{q}$ where $m$ is mass, $g$ is acceleration due to gravity, and $q$ is the charge.

• Direction: For a proton, the field must point upward; for an electron, downward.

Electric Potential and Potential Energy

Electric Potential

The electric potential at a point is the electric potential energy per unit charge at that point.

• Formula: $V = k_e \frac{Q}{r}$

• Potential Difference: $\Delta V = V_B - V_A$

• Example: The potential at a point halfway between two identical charges is the average of their potentials.

Electric Potential Energy

When a charge moves in an electric field, its potential energy changes:

• Formula: $\Delta U = q \Delta V$

• Work Done by Electric Force: $W = -\Delta U$

• Example: A charge moving from point A to B through a potential difference experiences a change in potential energy.

Capacitance and Capacitors

Definition and Calculation

A capacitor stores electric charge and energy. Its capacitance is a measure of its ability to store charge per unit potential difference.

• Formula: $C = \frac{Q}{V}$ where $C$ is capacitance (farads), $Q$ is charge (coulombs), and $V$ is potential difference (volts).

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

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

Capacitors in Circuits

• Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$

• Parallel: $C_{eq} = C_1 + C_2 + \cdots$

Current, Resistance, and Circuits

Electric Current

Electric current is the rate of flow of electric charge.

• Formula: $I = \frac{Q}{t}$ where $I$ is current (amperes), $Q$ is charge (coulombs), and $t$ is time (seconds).

• Example: A lightning strike transfers $2.5\ \mathrm{C}$ in $0.20\ \mathrm{ms}$; calculate the current.

Resistance and Ohm's Law

Resistance is a measure of how much a material opposes the flow of electric current.

• Formula: $R = \frac{V}{I}$ where $R$ is resistance (ohms), $V$ is voltage (volts), and $I$ is current (amperes).

• Resistivity: $R = \rho \frac{L}{A}$ where $\rho$ is resistivity, $L$ is length, and $A$ is cross-sectional area.

• Example: Calculate the resistance of a hair dryer or electric blanket given power and voltage.

Series and Parallel Circuits

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

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

• Example: Combining resistors to achieve a desired equivalent resistance.

Electric Field Lines and Conductors

Field Lines and Neutral Conductors

Electric field lines show the direction and strength of the electric field. A neutral conducting sphere placed between charged plates will polarize, with field lines bending around it.

• Field lines: Point from positive to negative charges.

• Conductors: Free charges move to cancel internal fields; field inside a conductor is zero in electrostatic equilibrium.

Additional Info

• Some problems reference the hydrogen atom and circular motion, relating to the quantization of angular momentum and centripetal force provided by electrostatic attraction.

• Problems include multiple choice and diagram selection, testing conceptual understanding and circuit analysis.

Key Equations Table

Concept	Equation (LaTeX)	Description
Coulomb's Law	$|F| = k_e \frac{|Q_1 Q_2|}{r^2}$	Force between two point charges
Electric Field (point charge)	$E = k_e \frac{|Q|}{r^2}$	Field due to a point charge
Electric Potential	$V = k_e \frac{Q}{r}$	Potential at distance $r$ from charge $Q$
Current	$I = \frac{Q}{t}$	Charge flow per unit time
Ohm's Law	$V = IR$	Relationship between voltage, current, resistance
Capacitance	$C = \frac{Q}{V}$	Charge stored per unit voltage
Energy in Capacitor	$U = \frac{1}{2}CV^2$	Energy stored in a capacitor