Electric Current, Current Densities, and Continuity Equation Notes

Fundamentals of Electric Current

Definition: Electric current is defined as the rate of flow of net charge through a conductor with respect to time.
Direction of Flow: Conventionally, the direction of electric current is defined to be in the direction of the flow of positive charges.
Mathematical Expression: If a net charge $Q$ flows through any cross-sectional area of a conductor in time $t$ , the electric current $I$ is given by: \n I = \frac{Q}{t}\n
Units of Measurement:
- The SI unit of electric charge is the Coulomb, denoted as $C$ .
- The SI unit of time is the second, denoted as $s$ .
- The SI unit of electric current is Coulomb per second ( $C/s$ ), which is defined as the Ampere ( $A$ ).

Current Density and Conduction Current Density

Current Density ( $\mathbf{J}$ ):
- Definition: Current density is defined as the ratio of the current to the surface area whose plane is normal to the direction of charge motion.
- Vector Nature: Current density is a vector quantity denoted by $\mathbf{J}$ , pointing in the direction of charge motion.
- Differential Relation: The relationship between current $I$ and surface area element $d\mathbf{s}$ is given by: \n \mathbf{J} = \frac{dI}{ds} \quad \text{or} \quad dI = \mathbf{J} \cdot d\mathbf{s}\n
Conduction Current Density ( $\mathbf{J}_1$ ):
- Definition: The current density produced specifically by the motion of conduction electrons inside a conducting medium is known as the conduction current density.
- Physical Model of Current Distribution:
  - Grey Cylinder: Represents the conducting wire.
  - Blue Circle at the Base: Represents the cross-sectional area $A$ , which is oriented perpendicular to the direction of current flow.
  - Red Arrows Along the Axis: Represent the total current $I$ , corresponding to the net flow of charges through the wire.
  - Green Arrows on the Base: Represent current density vectors $\vec{J} = \frac{I}{A}$ , illustrating current per unit cross-sectional area.
  - Distribution Principle: Current flows through the volume of the conductor, while current density describes how this current is distributed across its cross-sectional area.
Derivation of Conduction Current Density Relation:
- By Ohm's Law, the potential difference $V$ across a conductor having resistance $R$ carrying current $I$ is: \n V = I R \quad \text{--- (1)}\n
- For a conductor of length $l$ subject to electric field intensity $E$ , the potential difference $V$ is: \n V = E l \quad \text{--- (2)}\n
- Equating equation (1) and equation (2): \n I R = E l \quad \text{--- (3)}\n
- Resistance $R$ expressed in terms of electrical resistivity $\rho$ and electrical conductivity $\sigma$ (where $\rho = \frac{1}{\sigma}$ ): \n R = \rho \frac{l}{A} = \left(\frac{1}{\sigma}\right)\left(\frac{l}{A}\right) \quad \text{--- (4)}\n
- Substituting equation (4) into equation (3): \n I \left(\frac{1}{\sigma}\right)\left(\frac{l}{A}\right) = E l\n
- Simplifying by canceling $l$ on both sides: \n \frac{I}{\sigma A} = E \implies \left(\frac{I}{A}\right) = \sigma E\n
- Since conduction current density $\vec{J}_1 = \frac{I}{A}$ , the final expression is: \n \vec{J}_1 = \sigma \vec{E} \quad \text{--- (5)}\n
- Proportionality: Conduction current density $\vec{J}_1$ is directly proportional to the electric field intensity $\vec{E}$ .

Displacement Current Density

Concept and Mechanism:
- Conduction current resulting from physical movement of electrons cannot cross the space between parallel capacitor plates, as the plates are separated by a dielectric medium.
- To allow continuity of charge exchange between plates without physical charge movement across the dielectric, an equivalent process occurs in the dielectric.
- The effective current associated with this time-varying process in the dielectric medium is called the displacement current.
- Displacement Current Density ( $\vec{J}_2$ ): Defined as the displacement current per unit area.
- Physical Meaning: Displacement current density is not an actual physical flow of free charges through the dielectric, but an equivalent current density arising from a time-varying electric field.
Classification of Electric Fields:
- An electric field $\vec{E}$ is generated whenever an electric charge or potential difference (voltage) exists.
- Static Electric Field: An electric field that remains constant in both magnitude and direction over time.
- Time-Varying Electric Field: An electric field whose strength, direction, or both change over time.
Visualization in a Parallel Plate Capacitor:
Derivation of Displacement Current Density ( $\vec{J}_2$ ):
- In a capacitor, current $I_c$ is defined by: \n I_c = \frac{dQ}{dt} = \frac{d(C V)}{dt} = C \frac{dV}{dt} \quad \text{--- (1)}\n where $Q$ is the charge across the plates, $C$ is the capacitance, and $V$ is the potential difference across the plates.
- For a parallel plate capacitor, capacitance $C$ is: \n C = \frac{\varepsilon A}{d} \quad \text{--- (2)}\n where $\varepsilon$ is the electric permittivity, $A$ is the plate area, and $d$ is the distance between capacitor plates.
- Substituting equation (2) into equation (1): \n I_c = \left(\frac{\varepsilon A}{d}\right) \frac{dV}{dt} \implies \frac{I_c}{A} = \frac{\varepsilon}{d} \frac{dV}{dt}\n
- Displacement current density $J_2$ is given by: \n J_2 = \frac{I_c}{A} = \varepsilon \left[\frac{d}{dt}\left(\frac{V}{d}\right)\right]\n
- Since electric field $E = \frac{V}{d}$ , substituting $E$ yields: \n J_2 = \varepsilon \frac{dE}{dt} = \frac{d(\varepsilon E)}{dt}\n
- Using the electric displacement vector definition $\vec{D} = \varepsilon \vec{E}$ , the vector form is: \n \vec{J}_2 = \frac{d\vec{D}}{dt}\n
Justification of the Term "Displacement": This current does not pass directly through the dielectric medium of the capacitor; it is an apparent current representing the rate of charge flow occurring from electrode to electrode through the external circuit, justifying the term "displacement current."

Continuity Equation Derivation and Physical Significance

Mathematical Model:
- Consider a closed surface $S$ enclosing a volume $V$ containing a net electric charge.
- The total current $i$ flowing outward through closed surface $S$ is: \n i = \oint_S \mathbf{J} \cdot d\mathbf{s}\n where $\mathbf{J}$ is the current density vector over differential surface area element $d\mathbf{s}$ .
- If $\rho$ represents the volume charge density inside volume $V$ at any given instant, total charge $q$ enclosed is: \n q = \int_V \rho \, dv\n
Derivation Steps:
- Outward current $i$ corresponds to the rate of decrease of charge within volume $V$ : \n i = -\frac{dq}{dt} = -\frac{d}{dt} \int_V \rho \, dv = -\int_V \left(\frac{\partial \rho}{\partial t}\right) dv\n
- Applying Gauss's Divergence Theorem to convert the closed surface flux integral into a volume integral: \n \oint_S \mathbf{J} \cdot d\mathbf{s} = \int_V (\nabla \cdot \mathbf{J}) \, dv\n
- Equating the two expressions for outward current $i$ : \n \int_V (\nabla \cdot \mathbf{J}) \, dv = -\int_V \left(\frac{\partial \rho}{\partial t}\right) dv\n
- Combining into a single volume integral: \n \int_V \left(\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t}\right) dv = 0\n
- Since the volume $V$ chosen is completely arbitrary, the integrand must be zero everywhere: \n \nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0\n
Physical Significance:
- The differential equation $\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0$ is called the Equation of Continuity.
- It represents the fundamental mathematical statement of the Law of Conservation of Charge.
Steady-State Condition:
- In steady-state conditions, charge density at any given point is constant over time, meaning $\frac{\partial \rho}{\partial t} = 0$
- For steady currents, the continuity equation reduces to: \n \nabla \cdot \mathbf{J} = 0\n