Electric Potential and Potential Difference

Electric Potential Energy and Potential Difference

Introduction

Electric potential energy and electric potential difference are fundamental concepts in electrostatics, describing the energy changes associated with moving charges in electric fields. These concepts are essential for understanding electric circuits, fields, and the behavior of charges in various physical situations.

Potential Energy Difference

Work and Energy in Electric Fields

• Potential Energy Change (ΔUAB): The change in potential energy when moving an object from point A to point B against a force F is given by: $\Delta U_{AB} = U_B - U_A = -W_{AB} = - \int_A^B \vec{F} \cdot d\vec{r}$

• For a Charge in an Electric Field: The force on a charge q in an electric field E is $\vec{F} = q\vec{E}$, so the energy change is: $\Delta U = -q\vec{E} \cdot \Delta \vec{r}$

• Uniform Electric Field: For a uniform field and straight path, $\Delta U = -qE \Delta r \cos \theta$

• Example: Moving an electron through a uniform field of 300 kN/C over 0.10 m results in a potential difference of 30 kV and an energy change of $-4.8 \times 10^{-15}$ J.

Electric Potential and Potential Difference

Definition and Calculation

• Electric Potential (V): The electric potential at a point is the potential energy per unit charge at that point. $V = \frac{U}{q}$

• Potential Difference (ΔV): The difference in electric potential between two points A and B is: $\Delta V = V_B - V_A = \frac{\Delta U_{AB}}{q}$

• General Formula: $\Delta V = - \int_A^B \vec{E} \cdot d\vec{r}$

• Uniform Field: $\Delta V_{AB} = -\vec{E} \cdot \Delta \vec{r}$

• Non-uniform Field: Integration is required if the field or path varies.

• Dot Product: The dot product $\vec{E} \cdot d\vec{r}$ is $E dr \cos \theta$.

Units and Physical Meaning

• Unit of Potential Difference: Volt (V), where $1\ \text{V} = 1\ \text{J}/\text{C}$

• Interpretation: A 1 V potential difference means 1 joule of energy is gained or lost per coulomb of charge moved between two points.

• Example: A battery supplying 9 J to every coulomb of charge has a potential difference of 9 V.

• Electron Example: An electron moving through 1 V potential difference changes its energy by $1.6 \times 10^{-19}$ J.

Table: Key Quantities and Units

Symbol/Equation	Quantity	Units
$\vec{E} = \frac{\vec{F}}{q}$	Force per unit charge	N/C
$\Delta U = -\int_A^B \vec{F} \cdot d\vec{r}$	Potential energy difference	J
$\Delta V = \frac{\Delta U}{q}$	Potential difference	V (J/C)
$\Delta V = -\int_A^B \vec{E} \cdot d\vec{r}$	Potential difference (general)	V
$\Delta V_{AB} = -\vec{E} \cdot \Delta \vec{r}$	Potential difference (uniform field)	V

Potential Difference in Practice

Effects of Changing Parameters

• Doubling Electric Field Strength (E): Potential difference doubles.

• Doubling Distance (Δr): Potential difference doubles.

• Moving Perpendicular to Field: Potential difference becomes zero.

World Map of Mains Voltage

• Application: Different countries use different standard voltages (e.g., 110 V, 220 V) for household electricity, as shown in the world map.

Potential Difference in Non-uniform Fields

Integration Along a Path

• Non-uniform Field: The electric field and/or the path may vary, requiring integration: $\Delta V = - \int_A^B \vec{E} \cdot d\vec{r}$

• Path Dependence: The value of the integral depends on the path taken between points A and B if the field is not uniform.

• Example: The potential difference is largest where the field is strongest and the path is longest in the direction of the field.

Electric Potential Due to Point Charges

Superposition Principle

• Potential at a Point (P): The total electric potential at a point due to multiple point charges is the sum of the potentials from each charge: $V(P) = \sum_i \frac{kq_i}{r_i}$ where $k$ is Coulomb's constant, $q_i$ is the i-th charge, and $r_i$ is the distance from $q_i$ to point P.

Potential Difference Between Two Points

• Between rA and rB from a Point Charge: $\Delta V_{AB} = -kq \left[ \frac{1}{r} \right]_{r_A}^{r_B} = kq \left( \frac{1}{r_B} - \frac{1}{r_A} \right)$ As $r_A \to \infty$, $V(r_B) = 0$ at infinity.

Relationship Between Electric Field and Potential

Gradient and Derivatives

• Electric Field as Gradient of Potential: $\vec{E} = - \left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right)$ This expresses the electric field as the negative gradient of the electric potential.

• Physical Meaning: The electric field points in the direction of greatest decrease of potential.

Summary Table: Key Concepts

Concept	Electric Field (E)	Electric Potential (V)
Definition	Force per unit charge	Potential energy per unit charge
Vector/Scalar	Vector	Scalar
Units	N/C or V/m	V (J/C)
Formula	$\vec{E} = -\nabla V$	$V = -\int \vec{E} \cdot d\vec{r}$

Additional info:

• These notes cover the core concepts of electric potential, potential difference, and their relationship to electric fields, as typically presented in a college-level introductory physics course.