Notes on Electric Potential and Electric Energy

Chapter 17: Electric Potential and Electric Energy

Electric Potential 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:

  • $Fd = -\Delta PE$

Coulomb Force and Work

• 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:

  • $F = \dfrac{k \cdot |q<em>1||q</em>2|}{r^2}$

  • Where:\n - $k$ = Coulomb's constant

    • $q<em>1$, $q</em>2$ = magnitudes of the charges

    • $r$ = distance between the charges

• Changes in electric potential energy due to this force can be expressed:

  • $-\Delta PE = Fd$

Electric Field and Potential Energy Change

• 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

• 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:

  • $PE<em>b - PE</em>a = -qEd$

• Denoted as the potential difference $V_{ba}$:

  • $V<em>{ba} = V</em>b - V_a$

Units of Electric Potential

• 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.

Conceptual Understanding of Electric Potential Energy

• 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:

  • $\Delta PE = -qEd$

Caution: Differentiating Potential from Potential Energy

• 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 $V<em>{ba}$ regardless of their magnitudes because $V</em>{ba} = Ed$ is independent of the charge.

Relationship Between Potential and Potential Energy

• 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:

  • $\Delta PE = (-8 V)(2 C) = -16 J$

Example: Electron Acceleration

• 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.

Electric Field and Voltage Relationship

• 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:

  • $E = -\dfrac{V_{ba}}{d} = -\dfrac{50 V}{0.05 m} = 1000 V/m$

Electron Volt as a Unit of Energy

• 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.

Electric Potential from Point Charges

• 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:

  • $V = \dfrac{k q}{r}$

Relationship Between Voltage and Electric Field

• 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.

Capacitance

• 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.

Potential Difference of Capacitor

• 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.

Charge and Capacitance Relationship

• 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:

  • $C = \dfrac{10^{-9} C}{12 V} = 8.3 \times 10^{-11} F = 83 pF$

Factors Affecting Capacitance

• 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)

Example: Capacitance Calculation

• 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

• 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 $C = \epsilon$ .

Generalizing for Dielectric Materials

• 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

Energy Storage in capacitors

• 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:

  • $PE = \frac{1}{2}C V^2$

  • $PE = \frac{1}{2} \frac{Q^2}{C}$

Electric Field Energy Density

• 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.