[<< wikibooks] FHSST Physics/Electricity/Simple Parallel Circuits
= Simple parallel circuits =
Let's start with a parallel circuit consisting of three resistors and a single battery:
Fhsst electricity78.png
The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is
because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points
must always be the same at any given time. Therefore, in the above circuit, the voltage across 
  
    
      
        
          R
          
            1
          
        
      
    
    {\displaystyle R_{1}}
   is equal to the voltage across

  
    
      
        
          R
          
            2
          
        
      
    
    {\displaystyle R_{2}}
   which is equal to the voltage across 
  
    
      
        
          R
          
            3
          
        
      
    
    {\displaystyle R_{3}}
   which is equal to the voltage across the battery. This equality of voltages can be
represented in another table for our starting values:
Fhsst electricity79.png
Just as in the case of series circuits, the same caveat for Ohm's Law applies: values for voltage, current, and resistance must be in the
same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm's
Law to each resistor to find its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor:
Fhsst electricity80.png
Fhsst electricity81.png
At this point we still don't know what the total current or total resistance for this parallel circuit is, so we can't apply Ohm's Law
to the rightmost ("Total") column. However, if we think carefully about what is happening it should become apparent that the total
current must equal the sum of all individual resistor ("branch") currents:
Fhsst electricity82.png
As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the flow splits off at
point 7 to go up through 
  
    
      
        
          R
          
            1
          
        
      
    
    {\displaystyle R_{1}}
  , some more splits off at point 6 to go up through 
  
    
      
        
          R
          
            2
          
        
      
    
    {\displaystyle R_{2}}
  , and the remainder goes up through 
  
    
      
        
          R
          
            3
          
        
      
    
    {\displaystyle R_{3}}
  .
Like a river branching into several smaller streams, the combined flow rates of all streams must equal the flow rate of the whole river. The
same thing is encountered where the currents through 
  
    
      
        
          R
          
            1
          
        
      
    
    {\displaystyle R_{1}}
  , 
  
    
      
        
          R
          
            2
          
        
      
    
    {\displaystyle R_{2}}
  , and 
  
    
      
        
          R
          
            3
          
        
      
    
    {\displaystyle R_{3}}
   join to flow back to the positive terminal of the battery
(+) toward point 1: the flow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through 
  
    
      
        
          R
          
            1
          
        
      
    
    {\displaystyle R_{1}}
  , 
  
    
      
        
          R
          
            2
          
        
      
    
    {\displaystyle R_{2}}
  , and

  
    
      
        
          R
          
            3
          
        
      
    
    {\displaystyle R_{3}}
  .
This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using
this principle, we can fill in the 
  
    
      
        
          I
          
            T
          
        
      
    
    {\displaystyle I_{T}}
   spot on our table with the sum of 
  
    
      
        
          I
          
            R
            1
          
        
      
    
    {\displaystyle I_{R1}}
  , 
  
    
      
        
          I
          
            R
            2
          
        
      
    
    {\displaystyle I_{R2}}
  , and 
  
    
      
        
          I
          
            R
            3
          
        
      
    
    {\displaystyle I_{R3}}
  :
Fhsst electricity83.png
Finally, applying Ohm's Law to the rightmost ("Total") column, we can calculate the total circuit resistance:
Fhsst electricity84.png
Please note something very important here. The total circuit resistance is only 625 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  : less than any one of the
individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was
bound to be greater than any one of the resistors individually. Here in the parallel circuit, however, the opposite is
true: we say that the individual resistances diminish rather than add to make the total. This principle completes our
triad of "rules" for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance.
Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this:
Fhsst electricity85.png
The same basic form of equation works for any number of resistors connected together in parallel, just add as many 1/R terms
on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit.