CHPTR 23 Notes: Electric Potential Energy

Overview of Potential Energy and Electric Potential

Conservative Forces

• The concept of potential energy is associated with all conservative forces.

• By utilizing potential energy along with the conservation of energy principle, problems involving varying forces can be solved.

  • Conservation of Energy: The principle stating that energy cannot be created or destroyed, only transformed.

  • Potential Energy: Energy stored in a system based on its positions and interactions.

Electric Potential Energy

• Definition: When a charge is moved within an electric field, the change in potential energy of the charge (system) is related to the path-independent line integral.

  • Equation for Work Done:
    W_{i o f} = qE ullet d = - riangle U

• Work is calculated as:
  $W = - riangle U = riangle (U<em>f - U</em>i) = <br>ightarrow <br>eg <br>abla <br>ightarrow$

Change in Potential Energy

• Movement of Positive Charges: Positive charges lose electrical potential energy and gain kinetic energy when moving in the direction of a constant electric field (E).

• Movement of Negative Charges: Conversely, negative charges gain electric potential energy when moving in the direction of E.

  • Formula:
    riangle U = -qE ullet s

Potential Energy of Charged Particles

• The potential energy of a system consisting of two charged particles is defined as: $U = k rac{q<em>1 q</em>2}{r}$ Where:

  • “k” is Coulomb's constant,

  • “r” is the distance between the charges,

  • Significance: The absence of absolute value indicates that the signs of charges (
    q1 and q2) play a crucial role in determining potential energy.

System of Charges

• The total potential energy of a system of charges equals the sum of potential energies of each pair of charges:
  U = k imes igg( rac{q1 q2}{r{12}} + rac{q1 q3}{r{13}} + rac{q2 q3}{r_{23}} igg)

Electric Potential of a Point Charge

• Definition: The electric potential (V) is the potential energy per unit charge, expressed as: $V = rac{U}{q}$

  • Measured in joules per coulomb (J/C) or volts (V).

  • Electric Potential Formula:
    $V = rac{kq}{r}$ where “r” is the distance from the charge.

  • Positive charges generate positive potential, while negative charges create negative potential.

Potential Due to Multiple Point Charges

• If multiple point charges exist, the total electric potential at a certain point is given by the sum of potentials due to individual charges:
  V = ext{Sum}i igg( k rac{qi}{r_i} igg) o ext{as} o V o 0 ext{ far from the charges.}

Continuous Charge Distribution

• The potential from a continuous charge distribution can be calculated using an integral:
  V = k imes igg( rac{1}{r} imes dq igg)

Electric Potential of Different Charge Geometries

• Formulae that apply for the electric potential of charged infinite lines, conducting cylinders, rings, and finite lines arise from the continuous charge distribution equation:
  $V = k imes ext{Integral} ext{ form}$

Conservation of Energy for a Moving Charge

• Conservation of energy can be expressed for a moving charge through a potential difference as: $K<em>o + U</em>o = K<em>f + U</em>f$

  • Thus, rewriting with potential voltage, we have:
    $K<em>o + qV</em>o = K<em>f + qV</em>f$

• Caution: Watch signs while applying formulas, especially in the context of energy transformations.

Electron Volt

• Definition: An electron volt (eV) is the energy gained by an electron (or proton) when it falls through a potential difference of 1 volt.

  • Conversion:
    $1 eV = 1.6 imes 10^{-19} J$
    and
    $1 MeV = 10^6 eV$

Electric Potential Differences

• The potential difference (Vf - Vi), can be calculated using the work done by the charge: $V<em>f - V</em>i = - rac{W}{q}$

  • Voltmeter Measurement: Measured potential differences (voltages) are what we observe in electrical circuits.

• Units:
  $1 V = 1 J/C$

Uniform Electric Fields

• The net potential difference in a uniform electric field, created by oppositely charged parallel plates, can be expressed as:

  • Formula:
    Vf - Vi = -E ullet d
    Where E denotes the electric field strength.

  • Directionality: Electric field lines indicate the direction of decreasing potential.

Equipotential Surfaces

• Definition: Equipotential surfaces are collections of points that share the same electric potential.

Relationship between Electric Field and Equipotential Surfaces

• Electric fields are inherently perpendicular to equipotential surfaces and indicate the direction in which potential decreases.

Charged Conductors in Equilibrium

• Key Principle: Every point on the surface of a charged conductor in equilibrium maintains the same electric potential.

• Mathematically, this is showcased in:
  $V<em>B - V</em>A = 0 o ext{which implies} ext{no E-field exists inside the conductor.}$

Conducting Objects and Electric Potential

• Equilibrium properties imply:

  • The surface of any charged conductor is an equipotential surface.

  • Inside the surface, the electric field (E) equals zero, and hence, the potential (V) remains constant throughout the interior, equal to the surface potential.

Induced Charges

• When a neutral conductor is placed within an external electric field, charge redistribution occurs in such a way that the internal electric field (E) results in zero.

Faraday's Cage Concept

• Description: When a charged conductor with a cavity is exposed to an external electric field, charge redistribution on the conductor occurs, resulting in zero electric field within the cavity.

  • Practical Application: Safe shelter during thunderstorms is provided inside vehicles (metal cages) due to this principle.

Electric Field and Potential Calculation

• For points separated by a small distance (ds), the potential difference is given by: dV = - E ullet ds = - (E dx, E dy, E dz)

  • This signifies the relationship between electric fields and potential gradients in three-dimensional space as:
    $<br>E<em>x = - rac{ riangle V}{ riangle x}, u</em>y = - rac{ riangle V}{ riangle y}, <br>u_z = - rac{ riangle V}{ riangle z}.<br>$