Proof of Relationship in Parallel Circuit Using Kirchoff's Law

[soul5]

Homework Statement

How do I prove the relationship between the equivalent resistance of a parallel circuit and the values of the constituent resistors exist?

Homework Equations

V=IR

The Attempt at a Solution

Is it because:

I total= I1+I2+I3

and V is the same in parallel so.

V/R total= V/R1 + V/R2 + V/R3

and divide by V so

1/R= 1/R1 + 1/R2 + 1/R3.

is that why?

 

[Dick]

That's exactly why. Well done.

 

[Moetasim]

Your understanding is partially correct. The relationship between the equivalent resistance of a parallel circuit and the values of the constituent resistors exists due to Kirchoff's Current Law (KCL) and Kirchoff's Voltage Law (KVL).

KCL states that the sum of currents entering a node (or junction) in a circuit must equal the sum of currents leaving the node. In a parallel circuit, the current entering the node is equal to the total current (Itotal), while the currents leaving the node are equal to the individual currents through each resistor (I1, I2, I3). Therefore, Itotal = I1 + I2 + I3.

KVL states that the sum of voltage drops around a closed loop in a circuit must equal the sum of voltage sources in the loop. In a parallel circuit, the voltage across each resistor is the same, so the voltage drop across the equivalent resistance (Rtotal) is equal to the voltage drop across each individual resistor (R1, R2, R3). Therefore, V = V1 = V2 = V3.

Using Ohm's Law (V=IR), we can rewrite these equations as:

Itotal = V/Rtotal and I1 = V/R1, I2 = V/R2, I3 = V/R3.

Substituting these into the KCL equation, we get:

V/Rtotal = V/R1 + V/R2 + V/R3.

Dividing both sides by V, we get the relationship:

1/Rtotal = 1/R1 + 1/R2 + 1/R3.

This proves that the equivalent resistance of a parallel circuit is equal to the reciprocal of the sum of the reciprocals of the individual resistances. Therefore, the relationship between the equivalent resistance and the values of the constituent resistors exists.