Electric Potential and Electric Potential Energy

Objectives

• Distinguish various electrical concepts
  • Electrical potential energy, electric potential, and potential difference.
• Problem-solving in electrical energy and potential difference.
• Energy conversions in batteries.

Mechanical Energy

• Forms of Mechanical Energy:
  • Kinetic Energy (KE): Energy of motion, calculated as:
  • KE = \frac{1}{2} mv^2 (where $m$ = mass, $v$ = velocity)
  • Potential Energy (PE): Stored energy due to position/configuration:
  • Gravitational PE: PE_{grav} = mgh (where $g$ = gravity, $h$ = height)
  • Elastic PE: PE_{elastic} = \frac{1}{2} kx^2 (where $k$ = spring constant, $x$ = displacement)
• Electric Potential Energy (PE electric): Component of mechanical energy, given by:
  • ME = KE + PE{grav} + PE{elastic} + PE_{electric}

Electrical Potential Energy

• Definition: Potential energy associated with a charge in an electric field.
• In a Uniform Electric Field:
  • PE_{electric} = -qEd (where $q$ = charge, $E$ = electric field strength, $d$ = displacement)
  • Negative sign indicates:
  • Increase in PE for negative charges
  • Decrease for positive charges
• SI Unit: Joule (J)

Potential Difference

• Electric Potential (V): Work done against electric forces to move charge:
  • V = \frac{PE_{electric}}{q}
• Potential Difference (ΔV): Change in electric potential energy per unit charge:
  • \Delta V = \frac{\Delta PE_{electric}}{q}
  • SI Unit: Volt (V)

Key Differences

• Electric Potential vs. Electric Potential Difference:
  • Electric Potential (V): Refers to one point; measured from infinity.
  • Potential Difference (ΔV): Involves two points; work done to move unit charge.

Formula for Potential Difference in Uniform Electric Fields

• \Delta V = -Ed (where $E$ = magnitude of electric field, $d$ = displacement)

Sample Problem: Potential Energy and Potential Difference

• Example: Charge moves 2.0 cm in a uniform electric field of 215 N/C, change in electrical potential energy = -6.9 \times 10^{-19} J.
• Finding charge (q):
  • Rearranging:
  • PE_{electric} = -qEd\Rightarrow -6.9 \times 10^{-19} = -q(215)(0.020)\Rightarrow q = 1.6 \times 10^{-19} C
• Finding potential difference (ΔV):
  • \Delta V = -Ed = -(215 N/C)(0.020 m) = -4.3 V

Batteries and Electric Work

• How Batteries Work:
  • Batteries provide constant potential difference (e.g., a 1.5 V battery):
  • Moves negative charges from positive to negative terminal.
  • Delivers energy as charges flow through a device:
    • Energy delivered = 1.5 J for every 1 C of charge.

Practice Problems

1. A particle moves 10.0 m along an electric field of strength 75 N/C, PE decreases by 4.8 \times 10^{-16} J. Find charge (q).
2. Potential difference between initial and final locations of the particle in Problem 1.
3. An electron moves 4.5 m in an electric field of strength 325 N/C; find change in PE.

Answers to Practice

1. 6.4 \times 10^{-19} C
2. -750 V
3. 2.3 \times 10^{-16} J

The Superposition Principle

• Use: Calculate total electric potential from multiple charges.
• Method: Sum the potentials from all individual charges (potentials are scalars). Keep signs in mind:
  • Positive near positive charges, negative near negative charges.

Review Questions

1. Difference between PE{electric} and \Delta PE{electric}.
2. Factors affecting electrical potential energy in uniform electric fields.
3. Conditions for conservation of mechanical energy.
4. Reference points for electrical potential energy measurements.
5. Change in electrical potential energy of a $12 \mu C$ charge in a 250 N/C field from $(0, 0)$ to $(20 cm, 50 cm)$.
6. Change in electrical potential energy for 35 C of charge in a 2.0 km drop with a uniform field of 1.0 \times 10^6 N/C.
7. Minimum potential difference in a gap of 0.060 cm with an electric field of 3.0 \times 10^6 V/m for a spark plug.