Electric Potential: Concepts, Calculations, and Applications

Electric Potential

Definition of Electric Potential

Electric potential at a point is defined as the electric potential energy (EPE) per unit charge at that point. It is a scalar quantity and provides a measure of the work done by the electric field in moving a test charge from infinity to that point.

• Formula: $V = \frac{EPE}{q_0}$

• SI Unit: Joule/Coulomb = Volt (V)

• Potential Difference: Only differences in electric potential matter physically. $V_{BA} = V_B - V_A = \frac{EPE_B - EPE_A}{q_0} = -\frac{W_{BA}}{q_0}$

Total Energy Including Electric Potential Energy

Electric potential energy is included as part of the total energy an object can have:

• Formula: $E = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 + mgh + \frac{1}{2}kx^2 + EPE$

• Energy Types: Translational KE, Rotational KE, Gravitational PE, Elastic PE, Electric PE

• Electron Volt (eV): $1\ eV = 1.60 \times 10^{-19}\ \text{Joule}$ The eV is not an SI unit but is useful for subatomic particle energies (keV, MeV, GeV).

Example: Van de Graaff Generator as a Particle Accelerator

An electron is accelerated across a +100,000 V potential difference. The change in electric potential energy, kinetic energy gained, and final speed are calculated.

• Change in Electric Potential Energy: $EPE_B - EPE_A = q(V_B - V_A) = (-1.6 \times 10^{-19})(10^5 - 0) = -1.6 \times 10^{-14}\ \text{J}$

• Kinetic Energy Gained: $KE_B = KE_A + (EPE_A - EPE_B)$ $KE_B = 0 - (-1.6 \times 10^{-14}) = 1.6 \times 10^{-14}\ \text{J} = 100\ \text{keV}$

• Final Speed: $v_B = \sqrt{\frac{2KE_B}{m_e}} = \sqrt{\frac{2(1.6 \times 10^{-14})}{9.11 \times 10^{-31}}} = 1.87 \times 10^8\ \text{m/s}$

• Comparison with Gravitational PE: $GPE_B - GPE_A = m_e g (h_B - h_A) = (9.11 \times 10^{-31})(9.8)(1) = 8.93 \times 10^{-30}\ \text{J}$ Gravitational PE is negligible compared to electric PE for electrons.

Electric Potential Difference in Capacitors

The potential difference between the plates of a parallel-plate capacitor can be calculated using charge, area, and separation.

• Formula: $V_{BA} = \Delta V = \frac{\Delta (EPE)}{q_0} = -\frac{W_{BA}}{q_0}$

• For a Parallel-Plate Capacitor: $\Delta V = \frac{Qd}{A\varepsilon_0}$

• Example Calculation: $\Delta V = \frac{(4.0 \times 10^{-9})(0.001)}{(0.01)(8.85 \times 10^{-12})} = -45\ \text{V}$

Electric Potential Difference Created by Point Charges

The potential difference between two points due to a point charge is determined by the work done by the electric field.

• Work Done: $W_{BA} = \int_A^B \vec{F} \cdot d\vec{s}$

• Potential Difference: $V_B - V_A = -\frac{W_{BA}}{q_0} = \frac{kq}{r_B} - \frac{kq}{r_A}$

• Potential of a Point Charge: $V = \frac{kq}{r}$, for $r > 0$

• Example: For $q = 4.0 \times 10^{-9}\ \text{C}$ at $r = 1.2\ \text{m}$, $V = +300\ \text{V}$ (positive charge), $V = -300\ \text{V}$ (negative charge)

Total Electric Potential from Multiple Charges

When multiple charges are present, the total potential at a point is the algebraic sum of the potentials due to each charge.

• Formula: $V_{total} = \sum_i \frac{kq_i}{r_i}$

• Example: For two charges $+8.0 \times 10^{-9}\ \text{C}$ and $-8.0 \times 10^{-9}\ \text{C}$ separated by 0.4 m, at point A: $V_A = \frac{(8.99 \times 10^9)(8.0 \times 10^{-9})}{0.20} + \frac{(8.99 \times 10^9)(-8.0 \times 10^{-9})}{0.60} = +240\ \text{V}$ At point B: $V_B = 0\ \text{V}$

Electric Dipoles and Zero Potential

For an electric dipole, the potential at the mid-plane between the charges is zero due to symmetry.

• Formula: $V_p = \frac{kq}{r} + \frac{k(-q)}{r} = 0$

• Conceptual Example: For charges $+2q$ and $-q$ separated by distance $d$, there are two points on the line where the total potential is zero, found by solving: $\frac{2q}{|r_1|} - \frac{q}{|r_2|} = 0$

Equipotential Surfaces and Their Relation to the Electric Field

An equipotential surface is a surface on which the electric potential is the same everywhere. The net electric force does no work on a charge as it moves on an equipotential surface.

• Formula: $V = \frac{kq}{r}$ (for a point charge)

• Relation to Electric Field: The electric field is always perpendicular to equipotential surfaces and points in the direction of decreasing potential.

• In Conductors: In equilibrium, conductors are always equipotential surfaces.

• Field Lines and Equipotentials: For a point charge, field lines are perpendicular to equipotentials. For a parallel-plate capacitor, field lines are uniform and equipotential spacing is constant.

Electric Field and Potential Gradient

The electric field can be related to the change in electric potential over a distance.

• Formula: $E = -\frac{\Delta V}{\Delta s}$ $\Delta V/\Delta s$ is called the potential gradient.

• For Parallel Plate Capacitor: $E = \frac{V}{d}$ or $V = Ed$

• Example: For plates separated by $0.032\ \text{m}$ and $V_B - V_A = -64\ \text{V}$, $E = \frac{-64}{0.032} = 2.0 \times 10^3\ \text{V/m}$ For a $3.0\ \text{V}$ difference, spacing $\Delta s = \frac{-3.0}{2.0 \times 10^3} = 1.5 \times 10^{-3}\ \text{m}$

Summary Table: Key Equations and Concepts

Concept	Equation	Unit
Electric Potential	$V = \frac{EPE}{q_0}$	Volt (V)
Potential Difference	$\Delta V = \frac{\Delta (EPE)}{q_0}$	Volt (V)
Point Charge Potential	$V = \frac{kq}{r}$	Volt (V)
Parallel Plate Capacitor	$\Delta V = \frac{Qd}{A\varepsilon_0}$	Volt (V)
Electric Field (Potential Gradient)	$E = -\frac{\Delta V}{\Delta s}$	V/m
Electron Volt	$1\ eV = 1.60 \times 10^{-19}\ \text{J}$	Joule (J)

Additional info: These notes cover the main concepts, formulas, and applications of electric potential, including its relation to electric fields, capacitors, point charges, and equipotential surfaces, as presented in a college-level physics course.