DC Circuit

The flow of direct current is governed by Ohm’s Law, which states that the current (I) flowing through a conductor is directly proportional to the voltage (V) applied across it and inversely proportional to the resistance (R) of the conductor. This relationship can be represented by the equation:

I = V/R

In a DC circuit, components or “loads” can be connected in two fundamental ways: series and parallel. Each type of connection affects the behavior of current and voltage differently.

Series Circuit

In a series circuit, all components are connected end-to-end, forming a single path for the current to flow. If one component fails or is disconnected, the entire circuit is interrupted, and the current stops flowing.

Current

The same current flows through all components in the series. It means the current remains constant, regardless of the number of components:

\[ I_{total} = I_1 = I_2 = I_3 = \dots  \]

Voltage 

The total voltage in the circuit is the sum of the voltage drops across each component. It follows from Kirchhoff’s Voltage Law (KVL):

\[ V_{total} = V_1 + V_2 + V_3 + \dots  \]

A detailed description of Kirchhoff’s Law can be found here.

Resistance

The total resistance in a series circuit is the sum of the individual resistances and is given by:

\[ R_{total} = R_1 + R_2 + R_3 + \dots \]

Example Problem: Consider three resistors connected in series with a 12 V battery: R₁ = 2 Ω, R₂ = 3 Ω, and R₃ = 5 Ω. What is the current flowing in the circuit?

Solution

Given

V = 12 V, R₁ = 2 Ω, R₂ = 3 Ω, and R₃ = 5 Ω

The total resistance is:

R_total = 2 + 3 + 5 = 10 Ω

Using Ohm’s Law, the total current is:

\[ I = \frac{V_{total}}{R_{total}} = \frac{12 \, V}{10 \, \Omega} = 1.2 \, A \]

The current of 1.2 A flows through each resistor.

Parallel Circuits

In a parallel circuit, components are connected across the same two points, creating multiple paths for the current to flow. If one component fails or is disconnected, the other paths remain active, allowing current to continue flowing through the remaining components.

Voltage

The voltage across each component in a parallel circuit is the same and equal to the total voltage applied by the source:

\[ V_{total} = V_1 = V_2 = V_3 = \dots \]

Current

The total current in the circuit is the sum of the currents through each parallel branch. It follows from Kirchhoff’s Current Law (KCL):

\[ I_{total} = I_1 + I_2 + I_3 + \dots \]

Resistance

The total resistance of a parallel circuit is less than the smallest individual resistance. The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances:

\[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \]

Example Problem: Consider three resistors connected in parallel with a 12 V battery: R₁ ​= 2 Ω, R₂ = 3 Ω and R₃ = 6 Ω. What is the total current in the circuit?

Solution

Given

V = 12 V, R₁ = 2 Ω, R₂ = 3 Ω, and R₃ = 6 Ω

The total resistance is:

\[ \frac{1}{R_{total}} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1 \, \Omega

Using Ohm’s Law, the total current is:
\[ I_{total} = \frac{V_{total}}{R_{total}} = \frac{12 \, V}{1 \, \Omega} = 12 \,A

The current of 12 A divides among the branches, with more current flowing through paths of lower resistance.