Consider the two circuits shown below. 





In the circuit on the left, there is a d.c. supply. 

This supply delivers a constant strength current,
I_{dc} 

In the circuit on the right we have an a.c.
supply delivering a current which is constantly changing its
magnitude and sense. 

We imagine that we have found two identical light bulbs
and that the a.c. supply has been carefully adjusted so as to
illuminate the bulb to exactly the same brightness as in
the other circuit. 



It would then seem reasonable to suggest that the current,
i_{ac} has the same
effective value as the current I_{dc
} 

Note that, when referring to alternating
currents we usually use lower case letters for the
instantaneous values of current (or voltage) and capitals
for the maximum values. 







However, if the current in the circuit varies as shown above
then the simple mathematical average is equal to zero, not
a very suitable figure to give for the effective value of the current! 

We get out of this embarrassing situation
as follows: 

1. remember that the power dissipated in
the bulb (or in any component possessing resistance) is
proportional to the current squared (see here for proof) 

2. remember that the square of a negative
number has a positive value 

With these ideas in mind, first draw a
graph of the current squared. All the values are positive. 







Now find the average value of the current
squared. 

As the variation is symmetrical, this is
obviously half of the maximum value, as shown in the next graph. 







Now we take the square root of that average
value to give us a meaningful effective value for the
current called, rather logically, the root mean square value or
r.m.s. 







Therefore, the effective or root mean square
value of the current is given by 



and, although current and voltage are, of
course, totally different quantities, as the discussion above is
purely mathematical, we can use the same logic for an alternating
voltage, so 



where V is the maximum voltage. 



Note that these equations apply when the
variation is sinusoidal as shown in the graphs above. 

In situation where the variation is as
shown below, 





the r.m.s value is equal to the maximum value as there
is virtually zero time when the current has any other magnitude than
I. 
