Comprehensive Study Notes: Electrostatic Potential Due to a Point Charge

Electrostatic Potential Due to a Point Charge

• Definition of Setup:

  • A source charge, denoted as $+Q$, is placed at a specific origin point.
  • The goal is to determine the electrostatic potential at a point $P$ located at a distance $r$ from the source charge $+Q$.
  • The potential at a point is defined by the work done in bringing a unit positive test charge from infinity ($\infty$) to that point $P$ against the electrostatic forces.

• Fundamental Relationship:

  • The relationship between the electric potential ($V$) and the electric field ($E$) over a distance ($dx$) is expressed by the integral formula:     $V = -\int_{\infty}^{r} E \, dx$

Mathematical Derivation and Calculus Process

• Electric Field Intensity ($E$):

  • At any intermediate distance $x$ from the source charge $+Q$, the magnitude of the electric field ($E$) is given by Coulomb's Law:     $E = \frac{kQ}{x^2}$
  • Here, $k$ is the electrostatic constant (where $k = \frac{1}{4\pi\epsilon_0}$).

• Setting up the Integral:

  • Substituting the expression for the electric field into the potential formula:     $V = -\int_{\infty}^{r} \frac{kQ}{x^2} \, dx$
  • Since $k$ and $Q$ are constants with respect to the distance $x$, they are taken outside the integral:     $V = -kQ \int_{\infty}^{r} x^{-2} \, dx$

• Application of the Integration Power Rule:

  • The general rule for integration is stated as:     $\int x^n \, dx = \frac{x^{n+1}}{n+1}$
  • Applying this to the term $x^{-2}$, we find the antiderivative:     $\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} = \frac{x^{-1}}{-1} = -\frac{1}{x}$

• Evaluating the Definite Integral:

  • Applying the antiderivative back into the potential equation with limits from $\infty$ to $r$:     $V = -kQ \left[ -\frac{1}{x} \right]_{\infty}^{r}$
  • The two negative signs (one from the formula and one from the antiderivative) cancel each other out, resulting in a positive expression:     $V = kQ \left[ \frac{1}{x} \right]_{\infty}^{r}$
  • Evaluating the limits (Upper Limit minus Lower Limit):     $V = kQ \left( \frac{1}{r} - \frac{1}{\infty} \right)$
  • Since the value of $\frac{1}{\infty}$ is mathematically defined as $0$, the expression simplifies significantly:     $V = kQ \left( \frac{1}{r} - 0 \right)$

Final Expression and Analytical Observations

• The Potential Formula:

  • The final resulting formula for the electrostatic potential due to a point charge $+Q$ at a distance $r$ is:     $V = \frac{kQ}{r}$

• Proportionality and Relationships:

  • Dependence on Charge ($Q$):
  • The potential is directly proportional to the magnitude of the source charge:       $V \propto Q$
  • Dependence on Distance ($r$):
  • Unlike the electric field which follows an inverse-square law ($\frac{1}{r^2}$), the electrostatic potential follows an inverse relationship with distance:       $V \propto \frac{1}{r}$
  • This implies that as the distance $r$ from the charge increases, the potential $V$ decreases linearly with the reciprocal of the distance.

Summary of Sidebar Identities and Rules

• Integration Identities Used:
  • Rule 1: $\int x^{n} \, dx = \frac{x^{n+1}}{n+1}$
  • Application: $\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} = -\frac{1}{x}$
• Mathematical Constants:
  • $\frac{1}{\infty} = 0$