How you can prove the resistance in parqllel circuit?

Answer 1

The proof of the equivalent value of resistors in series lies in using Ohm's law along with Kirchoff's voltage and current laws.

For a circuit of two resistors in parallel, across a voltage source, start with Kirchoff's voltage law. The signed sum of voltages drops across each element in a series circuit is zero. Said another way, the voltages across two parallel elements is the same. That means the voltage across each resistor is the same, and it is the same as the voltage across the voltage source.

Since you now know the voltage across each resistor, you can calculate its current using Ohm's law.

Now look at Kirchoff's current law. The signed sum of the currents entering a node is zero. Said another way, the current at every point in a series circuit is the same. Said yet another way, the way we are going to need it, is that the sum of the currents entering a node is equal to the sum of the currents leaving a node.

You know the current in each resistor. Add them up and you get the current leaving the voltage source which is also the total current through the resistors. Knowing that current, and the voltage across them, go back to Ohm's law and plug in voltage and current, giving you net resistance.