Electric Current: Connecting Resistors in Parallel and Serial, and Kirchhoff’s Circuit Laws

A micro-course in electric current: Connecting resistors in parallel and serial, and Kirchhoff’s circuit laws for high school students.

 

Connecting Resistors in Parallel

 

                   Resistors with resistance of R₁ , R₂ , R₃ , ..., R_n , connected in parallel.

When resistors with resistance of R₁, R₂ , R₃ , ..., R_n, are connected in parallel, the electric potential difference between two terminals of them, is 'v'; so the electric current on them are as follows:

The equivalent resistance, R_eq , between points A and B, is given by:

where 'I' is the sum of electric currents passing through resistors:

So we have:

Connecting Resistors in Serial

 

             Resistors with resistance of R₁ , R₂ , R₃ , ..., R_n, connected in serial.

When resistors with resistance of R₁, R₂ , R₃ , ..., R_n, are connected in serial, the electric potential difference between two terminals of them, are v₁, v₂ , v₃ , ..., v_n, and electric current passing through of them is 'I'. So we have,

 

The equivalent resistance, R_eq, between points A and B, is given by,

Where 'V' is the sum of electric potential drops on resistors, i.e.,

So we have,

 

Kirchhoff’s Circuit Laws

Kirchhoff’s circuit laws are two laws:
  

1. Kirchhoff’s Current Law:

This law is the consequence of conservation law of electric charge; and it is also called nodal, or junction, or point rule. According to this law, at any given point of the electric circuit, the sum of currents flowing into the point is equal to the sum of currents flowing out of the point. So the algebraic sum of the electric currents entering and exiting a given point (node) in an electric circuit is zero, i.e.,

In this equation 'N' is the number of branches connected to the node, and In is the electric current of the n’th branch.

Convention: current entering the node is positive, and current leaving the node is negative.


2. Kirchhoff’s Voltage Law:


This law comes from conservation law of energy; and it is also called Kirchhoff’s loop (or mesh) rule. According to this law, around the electric loop, the sum of voltage drops is equal to the sum of voltage rises, i.e.:

In this equation 'N' is the number of voltages (or number of electric components) in the loop, and V_n is the n’th voltage (voltage drop or rise in n’th component).

Convention: voltage drop is negative, and voltage rise is positive.