Notes on Electric Potential and Electric Energy

The Coulomb force is a conservative force similar to gravitational force, which allows for potential energy association.
Work done by a system is defined as:
- W = -\Delta PE
- Work can also be expressed as:
- W = Fd
Equating these two provides:
- Fd = -\Delta PE

The Coulomb force is defined as a force that does work on charged particles.
The magnitude of the Coulomb force can be expressed in two forms:
- F = \dfrac{k \cdot |q1||q2|}{r^2}
- Where:\n - k = Coulomb's constant
  - q1, q2 = magnitudes of the charges
  - r = distance between the charges
Changes in electric potential energy due to this force can be expressed:
- -\Delta PE = Fd

For a uniform electric field, we can express the change in electric potential energy as:
- -\Delta PE = qEd
- Definitions:
- q = charge of the particle
- E = magnitude of the electric field
- d = distance traveled by the particle
If all quantities (q, E, and d) are positive, then \Delta PE is negative, indicating that the potential energy decreases as a positive charge moves away from the source of the electric field.
- Intuition: Similar to how an object falling under gravity moves to a lower gravitational potential energy state.

Potential difference is expressed as:
- \Delta V = -\dfrac{\Delta PE}{q} = -E \cdot d
In electrical terms, it’s more useful to think in terms of potential per unit charge:
- V = \dfrac{PE}{q}
Substituting gives:
- PEb - PEa = -qEd
Denoted as the potential difference V_{ba}:
- V{ba} = Vb - V_a

Both potential and potential difference are measured in Volts (V), named after Alessandro Volta.
Defined as:
- V = 1 J/C
Choosing a zero reference point (e.g., ground or infinity) is arbitrary as only differences in potential matter.

At point (a), a positive charge is closer to another positive charge, resulting in high potential energy due to repulsion.
At point (b), it is near a negative charge, resulting in lower potential energy due to attraction.
The relationship between change in potential energy and movement through an electric field remains consistent:
- \Delta PE = -qEd

Example (a): Both rocks at the same height have different potential energies due to mass.
Example (b): Charge with magnitude 2Q possesses twice the potential energy as charge Q due to scaling in \Delta PE = -qEd.
Both charges experience the same potential difference V{ba} regardless of their magnitudes because V{ba} = Ed is independent of the charge.

The relationship can be expressed in several ways:
- \Delta PE = q\Delta V
If a charge q moves through a potential difference V_{ba}, its potential energy changes by:
- \Delta PE = qV_{ba}
Example: For two parallel plates with a voltage difference of V_{ba} = +8 V and a charge of +2 C:
- \Delta PE = (8 V)(2 C) = 16 J
Conversely, moving from b to a:
- \Delta PE = (-8 V)(2 C) = -16 J

Given an electron accelerating through a potential difference of 5000 V:
- Determine the changes in potential energy and final kinetic energy, and the velocity of the electron.

The electric field E can be defined as:
- E = -\dfrac{\Delta V}{\Delta d}
- Where units of E are Newtons per Coulomb (N/C) and can be converted to Volts per meter (V/m).
Example: For parallel charged plates separated 5 cm apart with a voltage of 50 V:
- E = -\dfrac{V_{ba}}{d} = -\dfrac{50 V}{0.05 m} = 1000 V/m

An electron volt (eV) is defined as the energy gained by a single electron moving across a potential difference of 1 V.
- Defined mathematically:
- -\Delta PE = qV
- Thus 1 eV = 1.6 \times 10^{-19} J
Converting retroactively between eV and Joules is necessary in calculations regarding velocities or energy loss.

Define voltage at infinity where potential goes to zero far from the charge, leading to the potential expression from a point charge.
Voltage due to a single point charge is derived using calculus.
The general relationship becomes:
- V = \dfrac{k q}{r}

Voltage can also be described in terms of electric field as follows:
- V = \dfrac{k q}{r}
Observing patterns:
- Positive charge - Voltage increases as distance decreases.
- Negative charge - Voltage decreases as distance increases, both leading to similar voltages depending on proximity.

A capacitor stores electric charge, consisting typically of two conductive plates that are separated and do not touch.
Capacitors serve various functions, including charge storage for devices and memory in digital electronics.

The battery connected to a capacitor generates a potential difference (e.g. 12 V).
The exact voltage is irrelevant; only the difference matters.
Commonly represented as V (capital italics V) denoting the potential difference.

The charge Q on a capacitor is proportional to the potential difference V across it:
- Q = CV
- Where:
- C = capacitance in farads (F)
Typical capacitance values range from picofarads (pF) to millifarads (mF).
For a 1.0 nC charge at 12 V, the capacitance would yield:
- C = \dfrac{10^{-9} C}{12 V} = 8.3 \times 10^{-11} F = 83 pF

Capacitance depends on physical characteristics such as size, shape, and distance between plates, not the charge Q or the voltage V:
- For a parallel-plate capacitor, it is given by:
- C = \dfrac{\epsilon_0 A}{d}
- Where:
- \epsilon_0 = permittivity of free space (8.85 x 10^{-12} C²/N m²)

For a parallel-plate capacitor with given dimensions:
- C = \dfrac{\epsilon_0 A}{d} = \dfrac{(8.85 \times 10^{-12} C²/N m²)(0.20 m)(0.030 m)}{1.0 \times 10^{-3} m} = 5.3 \times 10^{-11} F
For a 12 V connection, the charge accumulated would be:
- Q = CV = (5.3 \times 10^{-12} F)(12 V) = 6.4 \times 10^{-10} C
Electric field between plates can be calculated as:
- E = \dfrac{V}{d} = \dfrac{12V}{1.0 \times 10^{-3} m} = 1.2 \times 10^4 V/m

Dielectrics are insulating materials placed between capacitor plates that enhance charge storage by preventing charge flow.
Dielectrics allow plates to be placed closer together while increasing capacitance from C = \epsilon .

The equation for parallel-plate capacitors is adjusted for dielectrics to reflect different material permittivities:
- C = K \cdot \epsilon_0A/d
- Where:
- K = dielectric constant, specific to each material

A capacitor stores energy as separated positive and negative charges.
Energy stored is equivalent to work done, represented as:
- W = qV
The incremental work \Delta W needed to move a small charge is expressed as:
- \Delta W = V \Delta q
The total work for charge Q involves the average voltage across the charging process, giving rise to the formula for stored energy in capacitors:
- PE = \frac{1}{2}QV
Rewriting in various useful forms gives similar forms such as:
- PE = \frac{1}{2}C V^2
- PE = \frac{1}{2} \frac{Q^2}{C}

The relationship between energy stored in electric fields (e.g., across matched capacitor plates) can be derived similarly, linking volume to energy density as:
- \text{Energy Density} = \frac{1}{2}\epsilon E^2
Indicates that energy storage per unit volume is proportional to the square of the electric field magnitude, valid across various electric field scenarios.