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Finding the Equivalent Resistance of a Circuit
Resistance has been defined to be the ratio of the voltage across a component to the current flowing through it. So, even if a component is not called "a resistor", we can still find a figure for its resistance.  
   
Any useful circuit will generally have multiple components connected together in various ways.
   
We will now find an expression to allow us to calculate the equivalent resistance of components when connected in series with each other. From now on, we will call them resistors but keep in mind that they could, in principle, be other components.
   
The Equivalent Resistance of Resistors in Series  
 
Here we see three resistors connected in series.  
As they are in series, the same current I flows though all of them.  
Also, if the voltages across the resistors are V1 V2 and V3 then as shown here, we can write  
 
From the definition of resistance we can therefore write  
 
 
If this single resistor RE is to be the equivalent of the three resistors above, we mean that it will allow the same current to flow for the same voltage. In other words, it would take the same current from a given battery as the three resistors together.  
   
Therefore, the current I is the same as in the previous circuit, so, again using the definition of resistance, we have  
 
Putting these together and dividing by I we have the (not altogether surprising) conclusion  
 
and, more generally, if there are N components in series the result can be expressed as  
 
   
The Equivalent Resistance of Resistors in Parallel  
Here we see three resistors connected in parallel.  
 
As they are in parallel, we know that they all have the same voltage applied to them.  
Also, we know from Kirchhoff's current rule  
 
and from the definition of resistance  
 
 
and, as in the series case, if RE is equivalent to these three resistors in parallel then it must take the same current as all of them from a given battery.  
Therefore, we have  
 
Putting these two equations together and dividing by V we have  
and, again, more generally, if there are N components in parallel the result can be expressed as  
 
To simplify calculations it is worth noting that if we have N identical resistors, each of resistance, R in parallel with each other, then their equivalent resistance is given by  
 
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