Capacitors and Electric Current: Key Concepts and Applications

Capacitors

Planar, Cylindrical, and Spherical Capacitors

Capacitors are devices used to store electric charge and energy in the electric field between two conductors. The geometry of the conductors determines the capacitance and the electric field distribution. The three main types of capacitors discussed here are planar (parallel plate), cylindrical, and spherical capacitors.

• Planar Capacitor: Consists of two parallel plates separated by a dielectric. The capacitance is given by: $C = \varepsilon_0 \frac{A}{d}$ where $A$ is the area of the plates and $d$ is the separation distance.

• Cylindrical Capacitor: Formed by two coaxial cylinders. The capacitance per unit length is: $C' = \frac{2\pi\varepsilon_0}{\ln(b/a)}$ where $a$ and $b$ are the radii of the inner and outer cylinders, respectively.

• Spherical Capacitor: Consists of two concentric spherical shells. The capacitance is: $C = 4\pi\varepsilon_0 \frac{ab}{b-a}$ where $a$ and $b$ are the radii of the inner and outer spheres.

Example Application: Spherical and cylindrical capacitors are commonly used in high-voltage applications and in situations where a uniform electric field is required over a curved surface.

Potential Difference vs. Potential Drop

Definitions and Usage

The potential difference between two points is defined as the difference in electric potential energy per unit charge between those points. It is calculated as:

• Potential Difference: $V = V_{\text{final}} - V_{\text{initial}}$

• Potential Drop: $V = V_{\text{initial}} - V_{\text{final}}$

Potential drop is often used when describing the decrease in potential as charge moves through a resistor or other circuit element.

Current and Current Density

Electric Current

Electric current ($I$) is the rate of flow of electric charge through a conductor or circuit element. It is measured in amperes (A):

• $I = \frac{\Delta Q}{\Delta t}$

where $\Delta Q$ is the charge passing through a cross-section in time $\Delta t$.

Current Density

Current density ($\vec{J}$) is the current per unit area flowing through a surface perpendicular to the direction of flow:

• $\vec{J} = \frac{I}{A}$

where $A$ is the cross-sectional area. The direction of $\vec{J}$ is the direction of positive charge flow.

Concept Check: Current Density in Cylindrical Conductors

For a cylindrical conductor with varying radii, the current density at different radii can be related as follows:

• $J_2 = \frac{r_1^2 J_1}{r_2^2}$

• $J_3 = \frac{r_1^2 J_1}{r_3^2}$

This relationship arises from the conservation of current, assuming the total current is constant and the current density is uniform over each cross-section.

Resistance and Temperature Dependence

Resistance

Resistance ($R$) is a measure of how much a material opposes the flow of electric current. It is given by:

• $R = \rho \frac{L}{A}$

where $\rho$ is the resistivity, $L$ is the length, and $A$ is the cross-sectional area of the conductor.

Temperature Dependence

The resistance of most conductors increases with temperature. The relationship is:

• $R = R_0 [1 + \alpha (T - T_0)]$

where $R_0$ is the resistance at reference temperature $T_0$, $T$ is the new temperature, and $\alpha$ is the temperature coefficient of resistance.

Formula Sheet: Capacitors

Type	Capacitance Formula
Parallel Plate	$C = \varepsilon_0 \frac{A}{d}$
Cylindrical	$C' = \frac{2\pi\varepsilon_0}{\ln(b/a)}$
Spherical	$C = 4\pi\varepsilon_0 \frac{ab}{b-a}$

Note: $\varepsilon_0$ is the permittivity of free space.

Summary Table: Current Density Relationships

Radius	Current Density
$r_1$	$J_1$
$r_2$	$J_2 = \frac{r_1^2 J_1}{r_2^2}$
$r_3$	$J_3 = \frac{r_1^2 J_1}{r_3^2}$