Voltage division rule is one of the basic rules of circuit analysis. It is applicable to all series circuits and combination circuits. Series circuit always acts as a Voltage divider. In a series circuit, the same current flows through each resistance. Hence the voltage drops across each resistor are proportional to their ohmic value. In a series circuit, current flowing will be proportional to the total resistance offered by the circuit.

The voltage across each resistor, by ohm’s law, is the current flowing through the circuit multiplied by the value of resistance. Consider a circuit with ‘n’ number of resistors connected in series and with a voltage V applied across it.

The total resistance offered by this series connection of resistors to the flow of current shall be:

Req = R1 + R2 + R3 + ………… + Rn

Total current flowing through the resistor shall be:

I = V / Req ————————————— (i)

Voltage across each resistor R1 shall be

V1 = I.R1 —————————- (ii)

Voltage across each resistor R1 shall be

V2 = I.R2 —————————- (iii)

Voltage across each resistor R1 shall be

V3 = I.R3 —————————- (iv)

Voltage across each resistor R1 shall be

Vn = I.Rn —————————- (v)

Comparing (i) and (ii),

V1 = V.R1/Req

Comparing (i) and (iii),

V2 = V.R2/Req

Comparing (i) and (iv),

V3 = V.R3/Req

Comparing (i) and (v),

Vn = V.Rn/Req

Where, Vn is the voltage across the nth resistor.In general,

The voltage across any resistor in a series connection of resistors shall be equal to the ratio of the value of the resistor divided by the equivalent resistance of the circuit. This is called Voltage division rule.

For a better idea, let us consider a circuit with two resistors R1 and R2 connected in series with voltage V applied across it.

Total resistance = R1+ R2

Let the total current be I.

Voltage V = I.(R1 +R2)

Current I = V/(R1 + R2)

Voltage across R1, V1 = I.R1 = V.R1 / (R1+R2)

Voltage across R2, V2 = I.R2 = V.R2 / (R1+R2)

### Examples

**1. Find the voltage across the resistor R3 in the circuit.**

Total resistance Rt = 5 + 5 + 5 = 15 ohm

Voltage across resistance, R3 = V.R3/Rt = 5 . 5/15 = 1.67 V

**2. Find the voltage across the resistor R2 in the circuit.**

Total resistance Rt = 5 + 4= 9 ohm

Voltage across resistance R2 = V.R2/Rt = 5 . 4/9 = 2.22 V

**3. Find the voltage across the resistor R3 in the circuit.**

Total resistance Rt = 5 + 5||5 = 5 + 5/2 = 7.5 ohm

Voltage across resistance R2 = V.R2/Rt = 5 . 2.5/7.5 = 1.667 V

### Potential dividers

A simple circuit of passive components used to get a voltage that is a fraction of the input voltage is called potential divider.

Vout = (Vin.R2)/(R1+R2)

For example 3 where did you get the 2.5 from? Isn’t R2 5 ohms?

We have R2 and R3 5ohms each connected in parallel. so the effective resistance of the parallel combination becomes 2.5ohms.

To find the equivalent resistance of two resistors in parallel, you use 1/Rtotal = (1/R1 + 1/R2)

So for two 5 ohm resistors in parallel, 1/Rtotal = 1/5 + 1/5; 1/Rtotal = 2/5, hence Rtotal = 5/2 = 2.5.

Then you add that to the other 5 ohm resistor as you usually would for resistors in series