The voltage division rule (voltage divider) is a simple rule which can be used in solving circuits to simplify the solution. Applying the voltage division rule can also solve simple circuits thoroughly. The statement of the rule is simple:

Voltage Division Rule: The voltage is divided between two series resistors in direct proportion to their resistance.


It is easy to prove this. In the following circuit

Voltage Divider
Voltage Divider

the Ohm’s law implies that
v_1(t)=R_1 i(t) (I)
v_2(t)=R_2 i(t) (II)

Applying KVL
-v(t)+v_1(t)+v_2(t)=0 \rightarrow v(t)=v_1(t)+v_2(t).

Therefore
v(t) = R_1 i(t)+R_2 i(t)= (R_1 +R_2) i(t).

Hence
i(t) = \frac{v(t)}{R_1 +R_2}.

Substituting in I and II
v_1(t)=R_1 \frac{v(t)}{R_1 +R_2},
v_2(t)=R_2 \frac{v(t)}{R_1 +R_2}.

Consequently

v_1(t)= \frac{R_1}{R_1 +R_2} v(t),
v_2(t)=\frac{R_2}{R_1 +R_2} v(t).


which shows that the voltage is divided between two series resistors in direct proportion to their resistance. The rule can be easily extended to circuits with more than two resistors. For example,

Voltage division among four resistors
Voltage Division among four resistors

v_1(t)= \frac{R_1}{R_1 +R_2+R_3+R_4} v(t),
v_2(t)=\frac{R_2}{R_1 +R_2+R_3+R_4} v(t),
v_3(t)=\frac{R_3}{R_1 +R_2+R_3+R_4} v(t),
v_4(t)=\frac{R_4}{R_1 +R_2+R_3+R_4} v(t).

The voltage division rule can be used solve simple circuits or to simplify solving complicated circuits.
For example, check out this problem.

One of the common mistakes in using the voltage division rule is to use the formula for resistors which are in parallel with other elements. For example, the voltage division rule cannot be used in the following circuit directly.
voltage divider - misleading case - a

It will be incorrect if one tries to find V_x using voltage divider by neglecting the other 6 \Omega resistor as
voltage divider - misleading case - b

So, V_x \neq \frac{6 \Omega}{2 \Omega + 6 \Omega} 15 V. However, if solving other parts of a circuits confirms that the current of the other element/branch is zero, the voltage division rule can be still applied. For example, suppose that the following network is a piece of a larger circuit.
voltage divider - piece of a larger circuit
Let’s assume that the analysis of the circuit shows that I_x=0. In this case, V_x= \frac{6 \Omega}{2 \Omega + 6 \Omega} 12 V = 9 V regardless of where A and B are connected.

The voltage devision rule can be used to ease solving problems. For example, the voltage division rule is used in the following problem to find the Thévenin voltage:
Thévenin’s Theorem – Circuit with An Independent Source
.

Published by Yaz

Hi! Yaz is here. I am passionate about learning and teaching. I try to explain every detail simultaneously with examples to ensure that students will remember them later too.

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17 Comments

  1. a parallel plate capacitor has area=2 mm and plate separation distance, d=1 mm. find how much charge is stored when the capacitor si connected to a 10 volts battery.

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  2. Really usefull.Especially the voltage divider thing, I struggle with it in Polytechnic and Iam still struggling with it but thanks, I have cleared the confusion after going through your information.

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  3. really sooorry for last one. That is my younger brother’s deeds.valuable information. but you didnt mention any thing about parallel circuits.

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  5. The voltage divider formulas and the solved problems are very useful to me. I corrected several circuits in our company. Thanks a lot for the support.

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  8. the questions are very challenging and concept oriented…hence keep it up!!!

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  9. Confused about the line ‘now substitute in I and II’ what exactly is substituted? I cannot see where this step came from.

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  11. Hi Yaz,

    You seem to have a good page, but the graphics have either fallen away over time or otherwise are not present.

    Thanks for the math, though — it was useful review and simplified expression!

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  12. Pingback: A copper wire and an iron wire - Potential difference | Physics Forums
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  14. Hi
    Good and simple to understand. hope to see simple calculation to solve complicated circuit which comprises voltage and current sources

    Greg

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  15. hey,
    really confused about the part where parallel circuits are given and voltage division rule cannot be applied directly. Isn’t voltage supposed to be equal across parallel resistances?

    Reply

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