Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In a series connection with n elements it is true that (voltage):

$$V = V_1 + V_2 + ... +V_n$$

and (resistance):

$$R = R_1 + R_2 + ... +R_n$$

If I know one of these I can infer the other. But is it possible to prove any of them without the other?

share|cite|improve this question
Do you know the definition of $U$ in terms of the more elementary electric field? – Fabian Aug 7 '12 at 8:04
I know the definition as energy per charge. – Lucy Brennan Aug 7 '12 at 8:28
If you now imagine having two islands in series. You need the energy (per charge) $U_1$ to bring a unit charge in the first island and the energy $U_2$ to bring it from the first to the second. Then because energy itself is additive you need the energy $U_1+U_2$ to bring it directly to the second island (so the first equation is more elementary and derives from the additivity of energy). – Fabian Aug 7 '12 at 10:13

But is it possible to prove any of them without the other?

(1) By KVL, the voltage across the N resistors is:

$V = V_{R_1} + V_{R_2} + ... + V_{R_N}$

(2) For a series connection, by definition, there is only one current, $I$.

By Ohm's Law, the voltage across any of the series resistors is:

$V_{R_n} = I \cdot R_n$


$V = I \cdot R_1 + I \cdot R_2 + ... + I \cdot R_N = I \cdot (R_1 + R_2 + ... R_N) = I \cdot R$

$R = R_1 + R_2 + ... + R_N$

share|cite|improve this answer

Here is a very slow derivation of $U = U_1 + ... + U_n$

The energy transformed in both of them must equal the sum of the energy transformed in each of them:

$P = P_1 + P_2 + ... + P_n$

According to the definition of electric power:

$P = IV$

By combining the two:

$I*V = I_1*V_1 + I_2*V_2 + ... + I_n*V_n$

But since the current is the same at any point in a series connection, $I$ can be crossed out on both sides.

$V = V_1 + V_2 + ... + V_n$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.