Electric Fields and Electric Potential: Study Notes

Chapter 16: Electric Fields

Introduction to Electric Fields

Electric fields are fundamental concepts in physics, describing the influence that electric charges exert on each other at a distance. The electric field at a point in space is defined as the force per unit positive test charge placed at that point.

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

• Direction: The direction of the electric field is the direction of the force on a positive test charge.

• Formula: $\vec{E} = \frac{\vec{F}}{q_0}$, where $\vec{F}$ is the force on a test charge $q_0$.

Electric Field Lines

Electric field lines, or lines of force, provide a visual map of the electric field in the space surrounding electric charges.

• Properties of Field Lines:

  • Begin on positive charges and end on negative charges.

  • Never cross each other.

  • The number of lines leaving or entering a charge is proportional to the magnitude of the charge.

  • The density of lines indicates the strength of the field.

• Uniform Electric Field: Represented by equally spaced, parallel lines (e.g., between parallel plates).

• Example: The field around a single positive charge radiates outward; for a dipole, lines emerge from the positive and enter the negative charge.

Conceptual Example: Drawing Electric Field Lines

• Common mistakes include field lines not starting/ending on charges, crossing lines, or incorrect proportionality to charge magnitude.

The Electric Field Inside a Conductor: Shielding

Conductors behave uniquely under electrostatic conditions due to the mobility of their charges.

• Key Properties:

  • Excess charge resides on the surface of a conductor.

  • The electric field inside a conductor is zero at equilibrium.

  • Conductors shield their interiors from external electric fields (Faraday cage effect).

  • The electric field just outside a conductor is perpendicular to the surface.

• Example: A charge placed at the center of a hollow conductor induces an equal and opposite charge on the inner surface and an equal charge of the same sign on the outer surface.

Chapter 17: Electric Potential

Potential Energy in Electric and Gravitational Fields

Potential energy is the energy stored due to the position of an object in a force field. The concept is analogous for gravitational and electric fields.

• Gravitational Potential Energy: $W_{AB} = mgh_A - mgh_B = GPE_A - GPE_B$

• Electric Potential Energy: $W_{AB} = EPE_A - EPE_B$

• Both gravitational and electric forces are conservative, so the work done is path-independent.

The Electric Potential Difference

The electric potential at a point is the electric potential energy per unit charge at that point. The potential difference (voltage) between two points is the work done per unit charge to move a test charge between those points.

• Definition: $V = \frac{EPE}{q_0}$

• Potential Difference: $V_{BA} = V_B - V_A = \frac{EPE_B - EPE_A}{q_0} = -\frac{W_{AB}}{q_0}$

• SI Unit: Volt (V), where $1\,\text{V} = 1\,\text{J}/\text{C}$

Common Usages of Electric Potential

Device	Typical Voltage (V)
Flashlight battery	1.5
Car battery	12
Electrical outlet (US)	120
Electrical outlet (Europe)	220
Small Van de Graaff generator	100,000
Large Van de Graaff generator	5,000,000
Medium lightning bolt	35,000,000

Van de Graaff Generator

A Van de Graaff generator is a device that uses a moving belt to accumulate very high voltages on a hollow metal sphere. It is often used as a particle accelerator in physics experiments.

• Key Components: Hollow metallic sphere, moving belt, rollers, electrodes.

• Operation: Charges are transferred to the sphere, creating a large potential difference.

Work, Potential Energy, and Electric Potential: Example

• Given: $q_0 = +2.0 \times 10^{-6}\,\text{C}$, $W_{AB} = +5.0 \times 10^{-5}\,\text{J}$

• Find: $EPE_B - EPE_A = -5.0 \times 10^{-5}\,\text{J}$

• Potential difference: $V_B - V_A = \frac{-5.0 \times 10^{-5}}{2.0 \times 10^{-6}} = -25\,\text{V}$

Acceleration of Charges in Electric Fields

• A positive charge accelerates from higher to lower potential.

• A negative charge accelerates from lower to higher potential.

Electric Potential Energy in Total Energy

• Total energy can include translational, rotational, gravitational, elastic, and electric potential energies:

$E = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 + mgh + \frac{1}{2}kx^2 + EPE$

• Electron Volt (eV): $1\,\text{eV} = 1.60 \times 10^{-19}\,\text{J}$ (not an SI unit, but commonly used in atomic and particle physics).

Van de Graaff Example Calculation

• Change in electric potential energy: $EPE_B - EPE_A = q(V_B - V_A)$

• Kinetic energy gained: $KE_B = EPE_A - EPE_B$

• Final speed: $v_B = \sqrt{2KE_B/m_e}$

• Gravitational potential energy is negligible compared to electric potential energy for electrons.

Potential Difference in a Parallel-Plate Capacitor

• For a uniform field: $\Delta V = \frac{Qd}{A\varepsilon_0}$

• Work done: $W_{AB} = Fd = q_0Ed$

Electric Potential Difference Created by Point Charges

• Potential due to a point charge: $V = \frac{kq}{r}$

• Potential difference between two points: $V_B - V_A = kq\left(\frac{1}{r_B} - \frac{1}{r_A}\right)$

• For multiple charges, potentials add algebraically.

Example: Potential of a Point Charge

• For $q = 4.0 \times 10^{-9}\,\text{C}$ at $r = 1.2\,\text{m}$: $V = +300\,\text{V}$

• For $q = -4.0 \times 10^{-9}\,\text{C}$ at $r = 1.2\,\text{m}$: $V = -300\,\text{V}$

Example: Total Electric Potential from Multiple Charges

• Sum the potentials from each charge at the point of interest.

• Example: For two charges $+8.0 \times 10^{-9}\,\text{C}$ and $-8.0 \times 10^{-9}\,\text{C}$ separated by $0.40\,\text{m}$, the potential at the midpoint is $0\,\text{V}$.

Potential Zero Points for Dipoles

• On the perpendicular bisector (mid-plane) of an electric dipole, the potential is zero everywhere.

• For charges $+2q$ and $-q$, the total potential is zero at two points along the line joining the charges, found by solving $V_P = 0$.