Steady Currents and Ohmic Conduction

The Microscopic View of Current

Imagine a metal wire as a lattice of fixed positive ions with free electrons moving through it. In the absence of an electric field, these electrons move randomly due to thermal energy, resulting in no net flow of charge. When a potential difference is applied across the wire, an electric field \(\mathbf{E}\) is established inside the conductor. This field exerts a force on the free electrons, accelerating them in the direction opposite to the field. However, electrons do not accelerate indefinitely; they frequently collide with the lattice ions and other electrons. These collisions randomize their motion, limiting their average velocity to a small, constant value known as the drift velocity \(\mathbf{v}_d\).

The current \(I\) is the rate at which charge passes through a cross-section of the wire. If the wire has a cross-sectional area \(A\) and the electron number density is \(n\), the current is given by \(I = n e A v_d\), where \(e\) is the elementary charge. The current density \(\mathbf{J}\) is defined as the current per unit area, \(\mathbf{J} = I/A\). In a steady state, the charge density \(\rho\) inside the conductor is constant in time, meaning \(\partial \rho / \partial t = 0\). According to the continuity equation, this implies that the divergence of the current density must be zero: \(\nabla \cdot \mathbf{J} = 0\). This condition ensures that charge does not accumulate at any point within the conductor; whatever current flows into a small volume must flow out of it.

Ohm's Law and Resistivity

Experimentally, it is observed that for many materials, the current density is proportional to the electric field. This linear relationship is known as Ohm's law in its local form:

Here, \(\sigma\) is the conductivity of the material. The inverse of conductivity is the resistivity \(\rho\), such that \(\sigma = 1/\rho\). Substituting this into the expression for current, we find that the total current \(I\) through a uniform conductor of length \(L\) and cross-sectional area \(A\) is related to the potential difference \(V\) by:

This macroscopic Ohm's law shows that resistance \(R\) depends on the geometry of the conductor and the material property \(\rho\). The power dissipated as heat in the resistor is given by \(P = IV = I^2 R = V^2/R\). This energy loss is due to the work done by the electric field on the electrons, which is transferred to the lattice via collisions, increasing the thermal energy of the material.

Kirchhoff's Rules and Circuit Analysis

When multiple resistors and voltage sources are connected in a network, we use Kirchhoff's rules to determine the currents and potentials. These rules are direct consequences of charge conservation and energy conservation.

Kirchhoff's Junction Rule: The sum of currents entering a junction equals the sum of currents leaving it. This is a direct application of the continuity equation for steady currents (\(\nabla \cdot \mathbf{J} = 0\)) applied to the node. Charge cannot accumulate at a junction, so the net flow must be zero.

Kirchhoff's Loop Rule: The algebraic sum of potential differences around any closed loop in a circuit is zero. This reflects the conservative nature of the electrostatic field in steady-state conditions. The work done by the electric field in moving a charge around a closed path is zero, implying that the sum of voltage rises (from sources) equals the sum of voltage drops (across resistors).

These rules allow us to set up a system of linear equations to solve for unknown currents and node potentials in complex DC networks.

Energy Dissipation and the RC Circuit

In a purely resistive circuit, energy is continuously supplied by the battery and dissipated as heat. However, when capacitors are involved, energy can be stored in the electric field. Consider a series RC circuit with a battery of emf \(\mathcal{E}\), a resistor \(R\), and a capacitor \(C\). When the switch is closed, the capacitor begins to charge. The current \(I(t)\) is not constant but decays exponentially as the capacitor voltage \(V_C(t)\) increases.

The governing differential equation is derived from Kirchhoff's loop rule: \(\mathcal{E} - I(t)R - V_C(t) = 0\). Since \(I(t) = dQ/dt\) and \(V_C(t) = Q(t)/C\), we have:

Solving this first-order linear differential equation with the initial condition \(Q(0) = 0\) yields the charge on the capacitor:

The time constant \(\tau\) determines the rate of charging. After one time constant, the capacitor is charged to approximately 63% of its maximum value. The current is \(I(t) = \frac{\mathcal{E}}{R}e^{-t/\tau}\). The energy stored in the capacitor at time \(t\) is \(U_C(t) = \frac{1}{2} C V_C(t)^2\), while the energy dissipated in the resistor is the integral of \(I^2 R\) over time. In the limit \(t \to \infty\), the total energy supplied by the battery is \(C\mathcal{E}^2\), half of which is stored in the capacitor and half is dissipated as heat in the resistor.