Study Notes on Electric Potential

Chapter 17: Electric Potential

Contents of Chapter 17

• Electric Potential Energy and Potential Difference

• Relation between Electric Potential and Electric Field

• Equipotential Lines and Surfaces

• The Electron Volt, a Unit of Energy

• Electric Potential Due to Point Charges

• Capacitance

• Dielectrics

• Storage of Electric Energy

• Applications Digital; Binary Numbers; Signal Voltage

• TV and Computer Monitors: CRT, Flat Screens

• Electrocardiogram (ECG or EKG)

Work-Potential Energy (Conservative Forces)

• Work done by conservative forces on an object is equal to the negative change in potential energy of the object.

  • Formula: $W = - riangle PE$

• When a rock is dropped (supported by gravitational field):

  • Work done by gravity = decrease in gravitational potential energy = gain in kinetic energy as it falls.

• Moving a rock against the gravitational field:

  • Work done by gravity = increase in gravitational potential energy = decrease in kinetic energy as it moves upward.

Electric Potential Energy (EPE)

• Definition: The energy needed by a charged particle to move against an electric field.

• Note: The electric force is conservative, hence the work done is independent of the path taken.

17.1 Electric Potential Energy and Potential Difference

1. Work done by electric force on a positive charge particle to move from location “a” to “b”:

   • $W = - riangle PE$

   • Point a is high potential (positively charged plate) and point b is low potential (negatively charged plate).

   • Work done as charge moves through electric field can be expressed in terms of potential difference:

   • $V<em>{ab} = V</em>a - V_b$

   • The electric potential at a location is given by:\n $V = rac{PE}{q}$ (17−2)

2. Explanation:

   • To move 1 C of charge across a 1 V potential difference, this charge requires 1 J of energy.

Changes in Potential

• Only changes in potential can be measured.

• Work done on a charged particle moving from “a” to “b”:
  $W = - riangle V = - (V<em>a - V</em>b)$ (1)

• According to the potential energy theorem:
  $W = PE<em>2 - PE</em>1$ (2)
  Combining equations yields:
  $-V<em>{ab} = - rac{(PE</em>2 - PE_1)}{q}$ (3)

Electric Field and Electric Potential Relationship

• Electric potential can either increase or decrease depending on the direction of location relative to the field:
  $V = - rac{dV}{dx}$ (17-4b)

• The electric field at a point in space represents the rate at which electric potential decreases over distance.

17.2 Equipotential Lines and Surfaces

• Definition: An equipotential line/surface is where the potential remains constant.

• Electric field lines are perpendicular to equipotential lines or surfaces.

• Example: A positively charged left plate is at twenty volts and the negatively charged right plate is at zero volts.

17.3 Electric Potential Dipole

• Definition: An electric dipole consists of two equal but opposite charges separated by a small distance.

• Equipotential lines around a dipole (shown in green) and electric field lines (shown in red).

17.4 The Electron Volt (eV)

• In molecular, atomic, and nuclear systems, energy is often measured in electron-volts.

  • 1 eV = energy gained by an electron moving through a potential difference of 1 V.

  • $1 eV = 1.6022 \times 10^{-19} J$.

• Example: X-ray photons range from 100 eV to 100 keV.

Application of Electron Volts in Medicine

• High-energy electrons (4 MeV to 20 MeV) can destroy cancerous tumors via collisions transferring energy to tumor atoms, particularly effective for superficial tumors due to limited penetration (a few centimeters).

17.5 Electric Potential Due to Point Charges

• The electric potential (coulomb potential) at a distance from a single point charge Q:

  • $V = \frac{kQ}{r}$

  • where k is Coulomb’s constant.

• The electric field magnitude at distance r:
  $E = \frac{|Q|}{4\pi \epsilon_0 r^2}$

• Electric potential can be either positive or negative, depending on the sign of the charge; it is a scalar quantity, while electric field is a vector.

Plotting Electric Potential due to a Charge

• Positive charge: Potential increases as you move towards the charge.

• Negative charge: Potential decreases as you move towards the charge.

Potential Energy of Two Charges

• The potential energy of two point charges is the work required to bring them from infinite distance to a given distance r:
  $PE = k \frac{q<em>1q</em>2}{r}$.

17.6 Electric Dipole Potential

• Electric dipole potential is calculated as the sum of individual potentials:

  • For a point P far away:
    $V = \frac{p \cos(\theta)}{r^2}$ (17-6b)

• Dipole moment defined as: $p = Ql$, where Q is charge and l is separation between charges.

17.7 Capacitance

• A capacitor consists of two conductive plates close together but not touching. When a voltage is applied, charge builds on plates.

  • Capacitor example with terminal voltage V: $Q = C \cdot V$ (17-7), where C is capacitance.

  • Unit for capacitance: Farad (F).

  • Capacitance depends on geometry/material. For parallel-plate capacitor:
    $C = \frac{\epsilon_0 A}{d}$ (17-8), where A is the area of plates, d is the distance between them.

17.8 Dielectrics

• Dielectrics are non-conductive materials between capacitor plates, characterized by dielectric constant K.

  • Importance: Increases maximum potential difference without breakdown and increases capacitance.

  • Example dielectric constants provided in TABLE 17.3:

  • Vacuum: K = 1.0000

  • Air: K = 1.0006

  • Paraffin: K = 2.2

  • Glass: K = 5

17.9 Storage of Electric Energy

• Energy stored in a charged capacitor equals work done to charge it:
  $E = \frac{1}{2} C V^2$ (17-10).

• Energy density (energy per unit volume) remains constant across the electric field:
  $u = \frac{1}{2} \epsilon E^2$ (17-11).

• Sudden electric discharge can be dangerous; heart defibrillators use this principle to restore normal heartbeat.

17.10 Digital; Binary Numbers; Signal Voltage

1. Analog voltages vary continuously.

2. Conversion of analog signal to digital requires sampling: Higher sampling rate means better representation.

3. Digital signal is less sensitive to noise compared to analog signals.

17.11 TV and Computer Monitors: CRTs, Flat Screens

• CRTs work by emitting electrons which are steered to produce images on the screen; the display ranges from low resolution to HD (1080 x 1920).

17.12 Electrocardiogram (ECG or EKG)

• An ECG measures changes in potential on the surface of the heart to detect heart defects.

Summary of Chapter 17

1. Electric potential definition as potential energy per unit charge: $U = qV$ (17-2a)

2. Electric potential difference equated to work done moving charge.

3. Equipotential lines depict constant potential area; electric dipole potentials decrease with distance.

4. Capacitance reflects the relationship of charge and voltage in capacitors.

5. Dielectric materials are essential in capacitors to enhance performance.

6. Energy density and potential storage principles relevant in various applications including medical devices and electronics.