Chapter 20: Electric Potential and Electric Potential Energy

Review of Gravitational Potential Energy

• Definition: Gravitational potential energy (GPE) is energy associated with an object's height that is available for conversion to kinetic energy.
• Redefinition of Symbol: GPE denotes the gravitational potential energy.
• Work Done by the Gravitational Force:
  • The work done by the gravitational force when moving from point A to point B is represented as:
  • $W<em>{AB} = mgh</em>A - mgh<em>B = GPE</em>A – GPE_B = - riangle GPE$
• Conservative Nature: The work done does not depend on the path taken between points A and B.

Electric Potential Energy (EPE)

• Definition: Electric potential energy is defined as the work done on a test charge when placed in an electric field.
• Notable Characteristics:
  • A test charge in an electric field feels a force.
  • As the charge is pushed by the electric field towards the lower potential plate, work is done on the charge.
  • Work Done Formula:
  • $W<em>{AB} = U</em>A - U_B = - riangle U$
  • The units of electric potential energy are consistent with energy units.
• Conservative Work: Similar to gravitational forces, the electric force is conservative, meaning the work done is independent of the path taken.

LQ #1: Electric Potential Energy and Force

• Scenario: A proton and an electron exist in a constant electric field created by oppositely charged plates.
  • Question: Which feels the larger electric force?
  • a) proton
  • b) electron
  • c) both feel the same force
  • d) they feel the same magnitude force but in opposite directions

LQ #2: Electric Potential Energy and Acceleration

• Scenario: In a constant electric field created by oppositely charged plates, a proton is released from the positive side and an electron from the negative side.
  • Question: Which has the larger acceleration?
  • a) proton
  • b) electron
  • c) both feel the same acceleration
  • d) they feel the same magnitude acceleration but in opposite directions

Work and Electric Potential Energy

• If the electric force is conservative, there must be an associated potential energy.
• Work required to move a charge perpendicular to an electric field is connected to changes in electric potential energy, represented as:
  • $W = - riangle U$

Work Formula and Electric Field Relationship

• Work done by an electric force when moving through a distance can be expressed as:
  • $W = F imes d = F d imes ext{cos}( heta)$
  • When plugging in for electric fields, this can be related to electric potential as:
  • $W = ext{-}Eq d ext{cos}( heta)$

Electric Potential Difference

• Definition: Electric potential difference (denoted by $riangle V$) is the electric potential energy per unit charge.
• Symbols and Units:
  • Denoted by symbol $V$, also known as potential or voltage, with units of $J/C$ (Joules per Coulomb).
• Mathematical Expression:
  • $V = \frac{U}{q_0}$
  • Electric potential difference between two points A and B is given by:
  • $riangle V = V<em>B - V</em>A = \frac{U<em>B}{q</em>0} - \frac{U<em>A}{q</em>0} = -\frac{W<em>{AB}}{q</em>0} = \frac{ riangle U}{q_0}$

Electric Potential in Terms of Electric Potential Energy

• It is useful to define the electric potential independently from electric potential energy.
• Electrons and Kinetic Energy:
  • LQ #3 Scenario: In an electric field created by charged plates, upon release when the charges reach opposite plates, the quiz examines which charge carries more kinetic energy.

Energy Relationships and Electric Forces

• Energy Conservation for Charges: For charges moving from point A to B due to conservative forces, the total energy formula may be expressed as:
  • $E = \frac{1}{2} mv^2 + mgh + \frac{1}{2} kx^2 + riangle U$

Electric Potential for Point Charges

• The force on a charge due to a point charge is not constant because it varies with distance. To find the work done on a charge by the electric force, calculus is necessary. The integral approach yields:
  • $W = k \frac{q<em>1 q</em>2}{r_{AB}}$

Electric Potential and Charge

• Potential Difference for Point Charges:
• If point B is at infinity, then the work done leads to
  • $V<em>B - V</em>A = -\frac{W<em>{AB}}{q</em>0}$
  • Hence, for point charges, the potential difference formula is given by:
  • $V = k \frac{q}{r}$

Electric Potential of Multiple Point Charges

• The total electric potential of a group of point charges is the algebraic sum of the potentials from each individual charge.

Equipotential Surfaces

• Definition: Equipotential surfaces are defined as surfaces where electric potential remains constant.
• Characteristics:
  • Moving a charge along an equipotential surface requires no work:
  • $W<em>{AB} = 0$ if $V</em>A = V_B$
• Electric fields are perpendicular to equipotential surfaces, meaning no work is done when moving charge along them.

Electric Field and Equipotential Relationship

• Equipotential surfaces and the electric field share a fundamental relationship, where lines of constant potential indicate the direction and strength of the electric field.

Learning Queries and Problems

• Various learning queries posed throughout the chapters relate to counterintuitive aspects of electric fields and potentials, including comparisons between charges, energy calculations, and properties of capacitors.

Capacitance and Capacitors

• A capacitor consists of two conducting plates separated by a distance.
• Capacitance Relation:
  • $C = \frac{Q}{V}$ where Q is charge and V is potential difference.
• Unit of Capacitance: Farad (F), defined as coulomb per volt.
• Influences on Capacitance: Capacitance increases with larger plate areas and decreases with increasing separation distance.

Dielectrics and Electric Fields

• Dielectrics reduce the potential difference between capacitor plates with the same charge, enhancing capacitance and affecting the electric field inside the dielectric.

Electrical Energy Storage and Applications

• Energy stored within capacitors can be utilized in various technologies, including cameras, pacemakers, and medical devices.