Combining Ohm's Laws Formulas

Date: 04/12/2001 at 16:41:14
From: Tony
Subject: Algebraic formulas

I'm studying math for electronics, and we have two main formulas: 
V = IR and P = VI. My problem is to combine them to get one formula. 
I understand how to make one formula by dividing both sides by the 
same letter, like I = P/V, but I just can't grasp the idea of 
combining these separate formulas to make one. Is there an easy way to 
figure this out?

Can you help me please? 
Thanks, Dr. Math.

Date: 04/13/2001 at 13:09:32
From: Doctor TWE
Subject: Re: Algebraic formulas

Hi Tony - thanks for writing to Dr. Math.

The formulas E = I*R and P = I*E are called Ohm's laws, named after 
the person who first defined the relations, Georg Ohm. As you stated, 
we can divide each side of either equation by the same thing to solve 
for another value. Doing this, we get:

     E = I*R   and   P = I*E
     I = E/R         I = P/E
     R = E/I         E = P/I

The formulas on the left relate the values of voltage, current, and 
resistance, while the formulas on the right relate the values of 
power, current, and voltage. There are six more forms of Ohm's laws; 
three that relate power, current, and resistance; and three that 
relate power, voltage, and resistance. To get those formulas, we have 
to combine the two basic formulas above. The easiest way to do that is 
to use substitution.

Let's start with Ohm's power law, P = I*E. Now we know from Ohm's 
voltage law that E = I*R, so we'll substitute I*R into the first 
equation:

     P = I*E
           ^
         (I*R)

     P = I*(I*R)

See how I just took E out of the equation and replaced it with (I*R)? 
Now I can use the rules of algebra to simplify:

     P = I*(I*R)
       = I*I*R
       = I^2*R

So our first "new" equation is P = I^2*R. I can then solve that 
equation for I and R to get 2 more formulas:

       P = I^2*R   and     P = I^2*R

      P    I^2*R          P    I^2*R
     --- = -----         --- = -----
     I^2    I^2           R      R

       R = P/I^2         I^2 = P/R

                           I = sqrt(P/R)

So now we have the formulas R = P/I^2 and I = sqrt(P/R) as well.

Now go back to Ohm's power law, P = I*E. This time, substitute for I 
(using the form of Ohm's voltage law that states I = E/R). Replace I 
in the first equation with (E/R) and simplify. You'll have an equation 
with just P, E and R. You can then solve that equation for each of the 
three variables to get three more formulas. I'll leave that for you to 
do.

When you are finished, you'll have twelve formulas; three each of the 
forms:

     E = ...     I = ...     R = ...   and   P = ...

Then, given any two quantities, you can solve for the desired one.

For more on the substitution method, check out the following entries 
from our Ask Dr. Math archives:

   Solving Systems of Equations by Substitution
   http://mathforum.org/dr.math/problems/lynn8.24.98.html   

   System of Equations and the Substitution Method
   http://mathforum.org/dr.math/problems/poindexter6.7.98.html   

   Solving Pairs of Equations by Substitution
   http://mathforum.org/dr.math/problems/lana11.26.97.html   

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/