Week 3 Lec. 3

Electric Potential and Electric Field

Electric Potential Outside and Inside a Charged Sphere

The electric potential V at a point in space due to a charged object is a measure of the potential energy per unit charge at that point. For a uniformly charged insulating sphere of radius a and total charge Q:

• Outside the Sphere (r > a): The electric field is radial and given by Gauss's law:

$E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$

• The potential at a distance r from the center (taking V = 0 at infinity):

$V(r) = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$

• At the Surface (r = a):

$V(a) = \frac{1}{4\pi\epsilon_0} \frac{Q}{a}$

• Inside the Sphere (r < a): The electric field is:

$E = \frac{1}{4\pi\epsilon_0} \frac{Q r}{a^3}$

• The potential at a point inside is:

$V(r) = \frac{1}{4\pi\epsilon_0} \frac{Q}{2a} \left(3 - \frac{r^2}{a^2}\right)$

• Key Point: The potential is continuous at the surface and decreases smoothly from the center outward.

Electric Potential and Electric Field Relationship

Finding E from V

The electric field E is related to the electric potential V by the negative gradient:

$\vec{E} = -\nabla V$

• In one dimension: $E = -\frac{dV}{ds}$

• In three dimensions: $E_x = -\frac{\partial V}{\partial x}$, $E_y = -\frac{\partial V}{\partial y}$, $E_z = -\frac{\partial V}{\partial z}$

Example: If $V = 3xy^2 - 2yz$, then:

• $E_x = -\frac{\partial V}{\partial x} = -3y^2$

• $E_y = -\frac{\partial V}{\partial y} = -6xy + 2z$

• $E_z = -\frac{\partial V}{\partial z} = 2y$

Note: Electric potential is a scalar field, while the electric field is a vector field.

Capacitance

Definition and Basic Properties

Capacitance (C) is the ability of a system to store electric charge per unit potential difference. For two conductors with equal and opposite charges:

$C = \frac{Q}{\Delta V}$

• The SI unit is the farad (F), where $1\,\mathrm{F} = 1\,\mathrm{C}/\mathrm{V}$.

• Capacitance depends only on the geometry and the medium between the conductors.

Parallel Plate Capacitor

A parallel plate capacitor consists of two plates of area A separated by distance d:

$C = \epsilon_0 \frac{A}{d}$

• Where $\epsilon_0$ is the vacuum permittivity.

• The electric field between the plates is uniform if $d \ll \sqrt{A}$.

Cylindrical Capacitor

For a cylindrical capacitor with inner radius a, outer radius b, and length l:

$C = \frac{2\pi\epsilon_0 l}{\ln(b/a)}$

• Derived using Gauss's law and integrating the electric field between the cylinders.

Dielectrics

Role and Effect of Dielectrics

Dielectrics are insulating materials that increase the capacitance of a capacitor by reducing the effective electric field. When a dielectric is inserted between the plates:

• The capacitance increases by a factor called the dielectric constant ($\kappa$):

$C = \kappa C_0$

• Where $C_0$ is the capacitance without the dielectric.

• The electric field inside the capacitor is reduced: $E = E_0/\kappa$

Combinations of Capacitors

Parallel and Series Configurations

• Parallel: The equivalent capacitance is the sum of individual capacitances:

$C_{eq} = C_1 + C_2 + \ldots$

• Voltage across each capacitor is the same.

• Series: The reciprocal of the equivalent capacitance is the sum of reciprocals:

$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$

• Charge on each capacitor is the same.

Energy Storage in Capacitors

Energy Stored

The energy U stored in a capacitor is given by:

$U = \frac{1}{2} C (\Delta V)^2$

• This energy is stored in the electric field between the plates.

Summary Table: Capacitance Formulas

Capacitor Type	Capacitance Formula
Parallel Plate	$C = \epsilon_0 \frac{A}{d}$
Cylindrical	$C = \frac{2\pi\epsilon_0 l}{\ln(b/a)}$
With Dielectric	$C = \kappa C_0$
Parallel Combination	$C_{eq} = C_1 + C_2 + \ldots$
Series Combination	$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$

Additional info: The notes above include expanded explanations, definitions, and examples to ensure completeness and clarity for exam preparation. All images included are directly relevant to the adjacent explanations and reinforce the concepts discussed.