Electric Energy and Potential: Field, Potential, and Applications

Electric Field and Coulomb's Law

Definition and Formula

The electric field is a vector field that describes the force per unit charge exerted on a test charge at any point in space. It is fundamental to understanding electrostatic interactions.

• Coulomb's Law: The electric field due to a point charge is given by: $\vec{E} = k_e \frac{q}{r^2} \hat{r}$ where $k_e$ is Coulomb's constant, $q$ is the charge, $r$ is the distance from the charge, and $\hat{r}$ is the unit vector pointing from the charge to the field point.

• Superposition Principle: The net electric field from multiple charges is the vector sum of the fields from each charge: $\vec{E}_{net} = \sum \vec{E}$

• Coulomb's Constant: $k_e = \frac{1}{4\pi\varepsilon_0} = 8.988 \times 10^9 \text{ N} \cdot \text{m}^2/\text{C}^2$

• Permittivity of Free Space: $\varepsilon_0 = 8.854 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2$

Vector Notation and Magnitude

• Position vector: $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$

• Magnitude: $r = |\vec{r}| = \sqrt{x^2 + y^2 + z^2}$

• Unit vector: $\hat{r} = \frac{\vec{r}}{|\vec{r}|}$

Force on a Charge

• The force on a charge $q$ in an electric field $\vec{E}$ is: $\vec{F}_E = q\vec{E}$

Physical Constants and S.I. Prefixes

• Elementary charge: $e = 1.602 \times 10^{-19}$ C

• Electron mass: $m_e = 9.109 \times 10^{-31}$ kg

• Proton mass: $m_p = 1.673 \times 10^{-27}$ kg

• Neutron mass: $m_n = 1.675 \times 10^{-27}$ kg

S.I. Prefix	Value
T (tera)	$10^{12}$
G (giga)	$10^9$
M (mega)	$10^6$
k (kilo)	$10^3$
p (pico)	$10^{-12}$
n (nano)	$10^{-9}$
μ (micro)	$10^{-6}$
m (milli)	$10^{-3}$

Electric Potential and Energy

Potential Energy and Work

The electric potential energy of a charge in an electric field is related to the work done by or against the field.

• Kinetic Energy: $K = \frac{1}{2}mv^2$

• Work-Energy Principle: $\Delta K = W$

• External Work: $\Delta K + \Delta U = W_{ext}$

• Change in Potential Energy: $\Delta U = q \Delta V$

Work and Potential Difference

• Work done by the field: $W = \vec{F} \cdot \Delta \vec{l}$

• Work as an integral: $W = \int_{path} \vec{F} \cdot d\vec{l}$

• Potential difference: $\Delta V = -\vec{E} \cdot \Delta \vec{l}$

• Potential difference as an integral: $\Delta V = -\int_{path} \vec{E} \cdot d\vec{l}$

Electric Field from Potential

• For a field varying only along $x$: $E_x = -\frac{dV}{dx}$

• General case: $\vec{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right)$

Potential and Energy Formulas

• Potential due to a point charge: $V = k_e \frac{q}{r}$

• Net potential: $V_{net} = \sum V$

• Potential energy of two point charges: $U = k_e \frac{q_1 q_2}{r_{12}}$

• Potential energy for a system of charges: $U = \frac{1}{2} k_e \sum_{i \neq j} \frac{q_i q_j}{r_{ij}}$

Field from Potential: Calculus Connection

Fundamental Theorem of Calculus

The relationship between electric field and potential is rooted in calculus. The electric field is the negative gradient of the electric potential.

• For one dimension: $E_x = -\frac{dV}{dx}$

• For three dimensions: $\vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right)$

• When the field is uniform, the derivative is the same as the "slope": $E_x = -\frac{\Delta V}{\Delta x}$

Units of Electric Field

• The electric field can be expressed in units of V/m (volts per meter) or N/C (newtons per coulomb). These are equivalent: $1 \text{ V/m} = 1 \text{ N/C}$

Graphical Analysis: Electric Potential and Field

Plotting $V(x)$ and $E_x(x)$

Given a plot of electric potential $V$ as a function of position $x$, the electric field $E_x$ can be found as the negative slope (derivative) of $V(x)$.

• Where $V(x)$ is constant, $E_x = 0$.

• Where $V(x)$ changes linearly, $E_x$ is constant and equal to the negative slope.

• Where $V(x)$ changes nonlinearly, $E_x$ varies accordingly.

Example: Piecewise Linear $V(x)$

• If $V(x)$ decreases linearly, $E_x$ is a constant negative value.

• If $V(x)$ has a sharp change (e.g., a 'V' shape), $E_x$ will be large in regions of steep slope and zero where $V(x)$ is flat.

Example: Sinusoidal $V(x)$

• If $V(x)$ is sinusoidal, $E_x(x)$ will be the negative derivative, resulting in a cosine-shaped field.

Special Case: Potential Varies with |x|

Absolute Value Potential

If the electric potential varies only with position along the x-axis as $V(x, y, z) = -2\pi k_e q |x|$, the electric field is found by differentiating with respect to $x$:

• $E_x(x) = -\frac{dV}{dx} = 2\pi k_e q \cdot \text{sgn}(x)$, where $\text{sgn}(x)$ is the sign function.

• Qualitatively, $V(x)$ is a 'V' shape, and $E_x(x)$ is a step function, changing sign at $x=0$.

Summary Table: Electric Field and Potential Relationships

Quantity	Formula	Units
Electric Field (point charge)	$\vec{E} = k_e \frac{q}{r^2} \hat{r}$	N/C or V/m
Electric Potential (point charge)	$V = k_e \frac{q}{r}$	V
Potential Energy (two charges)	$U = k_e \frac{q_1 q_2}{r_{12}}$	J
Field from Potential	$\vec{E} = -\nabla V$	N/C or V/m
Work by Field	$W = q \Delta V$	J

Key Concepts and Applications

• Electric field describes the force per unit charge at a point in space.

• Electric potential is the energy per unit charge due to the field.

• The field is the negative gradient of the potential.

• Graphical analysis of $V(x)$ allows determination of $E_x(x)$ via the slope.

• Units of electric field (N/C and V/m) are equivalent in SI.

Example Application:

• Given a plot of $V(x)$, calculate $E_x(x)$ by finding the negative slope at each point.

• For a system of charges, use superposition to find net field and potential.

Additional info: Graphs and exercises in the original notes illustrate how to extract the electric field from potential plots, including piecewise, sinusoidal, and absolute value functions.