📚 Electric Potential due to a Point Charge
The electric potential, often denoted as $V$, at a distance $r$ from a point charge $q$ is given by the formula:
$\qquad V = \frac{kq}{r}$
where $k$ is Coulomb's constant ($k \approx 8.99 \times 10^9 \text{ N m}^2/\text{C}^2$). From this equation, we can see that the electric potential $V$ is inversely proportional to the distance $r$ from the point charge.
📈 Graphing Potential vs. Distance
When graphing electric potential ($V$) against distance ($r$) from a point charge, we observe a hyperbolic relationship. Here’s what you need to know:
- 📉 Shape of the Graph: The graph is a hyperbola. As $r$ increases, $V$ decreases, approaching zero as $r$ approaches infinity.
- 📍 Important Points: Close to the charge (small $r$), the potential $V$ is very large. As you move away (large $r$), the potential $V$ becomes smaller.
- ➕/➖ Sign of the Charge: For a positive charge, the potential is positive; for a negative charge, the potential is negative. This affects which part of the hyperbola you see on the graph.
⚡ Electric Potential vs. Electric Field
It's important to distinguish between electric potential ($V$) and electric field ($E$). Here’s a comparison:
| Feature |
Electric Potential ($V$) |
Electric Field ($E$) |
| Definition |
The electric potential energy per unit charge at a point in space. |
The force per unit charge experienced by a test charge at a point in space. |
| Nature |
Scalar quantity (magnitude only) |
Vector quantity (magnitude and direction) |
| Formula (Point Charge) |
$V = \frac{kq}{r}$ |
$E = \frac{kq}{r^2}$ |
| Dependence on Distance |
Inversely proportional to distance ($V \propto \frac{1}{r}$) |
Inversely proportional to the square of the distance ($E \propto \frac{1}{r^2}$) |
| Graph (vs. Distance) |
Hyperbolic decay |
Quadratic decay |
🔑 Key Takeaways
- 📏 Inverse Relationship: Electric potential decreases as you move away from the charge.
- 📊 Graph Shape: The $V$ vs. $r$ graph is a hyperbola, reflecting the inverse relationship.
- ➕/➖ Sign Matters: The sign of the charge determines the sign of the potential.
- 💡 Scalar vs. Vector: Remember that potential is a scalar, while the electric field is a vector.