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 V_{1} V_{2}
and V_{3} then as shown here, we can write 



From the definition of resistance we can therefore write 





If this single resistor R_{E} 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 R_{E} 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 


