Finding Current in a Parallel Circuit with Multiple Resistors

[Ry122]

Homework Statement

Im aware that you can find the current passing through each resistor in a parallel circuit by using the equation I=v/r but for homework i was asked how to find the current by using the ratios between each resistor.
eg.
---5ohms---- Resistor 1
- 20ohm - Resistor 2
- 10ohms - Resistor 3
----| - |-----
I=20AMPS
If only resistor 1 and 2 exists i can do it by dividing 5 by 5+20 and dividing
20 by 2+20. Inverting the results so 5/25 is for resistor 2 and vice versa then multiplying the results by 20amps. This gives the amps passing through each resistor.
I've no idea how to do it if there is 3 or more resistors.

Homework Equations

The Attempt at a Solution

 

[denverdoc]

On the right track--we know that current is inversely proportional to resistance, so let's set up the eqn with a little i which is the current flowing thru the 20 ohm resistor: In fact, this makes use of the V=iR eqn at least implicitly, assuming 20 amps total passes thru the branch.

i+20/10*i+20/5*i=20A=7i. i=20/7, etc.

Now if 20 amps passes thru say the small resistor, and total current is unknown,

then (5/10)*20 will pass thru the 10 ohm and (5/20)*20 thru the 20 ohm.

 

[m2003]

In a parallel circuit, the current is divided between the different branches based on the resistance of each branch. To find the current through each resistor, you can use the following formula:

I1 = (V/R1) * (R2/(R1+R2)) * (R3/(R1+R2+R3)) * ... * (Rn/(R1+R2+...+Rn))

Where:
- I1 is the current through resistor 1
- V is the voltage across the parallel circuit
- R1, R2, R3, ... Rn are the resistances of each branch

This formula takes into account the ratios between each resistor, as you mentioned in your question. It works by first finding the current through the first resistor (V/R1), then multiplying it by the ratio of the resistance of the second resistor to the total resistance (R2/(R1+R2)). This process is repeated for each resistor, taking into account the ratios of all the resistors in the circuit.

For example, in your given circuit with 3 resistors, the formula would look like this:

I1 = (V/5) * (20/(5+20)) * (10/(5+20+10))

Simplifying this, we get:

I1 = (V/5) * (20/25) * (10/35)

I1 = (V/5) * (4/5) * (2/7)

I1 = (V/5) * (8/35)

So to find the current through resistor 1, we would multiply the voltage by 8/35. Similarly, we can find the current through the other resistors using the same formula. This method can be applied to circuits with any number of resistors.