# Change in current on adding a resistor

Is it correct to say that the current flowing through a given resistor remains constant even if another resistor is connected in parallel with it, assuming a source of constant EMF and no other components in the circuit?

My attempt: Consider two resistor of resistances $$R_1$$ and $$R_2$$. Initially, $$I = \frac V{R_1}$$

After connecting, let $$I$$ be the total current and $$I_1$$ and $$I_2$$ the currents through the individual resistances.

$$I = \frac {V(R_1 + R_2)}{R_1R_2}$$

$$I_1 = \frac{IR_2}{R_1 + R_2} = \frac{V}{R_1}$$

Somehow this dosen't seem correct to me.

• Think of the voltage $V$ over the resistor. Does that change when another one is added? – Steeven Jun 18 '17 at 12:31

Is it correct to say that the current flowing through a given resistor remains constant even if another resistor is connected in parallel with it, assuming a source of constant EMF and no other components in the circuit?

Yes and you've showed that using KCL and then current division. But, as a comment points out, the result follows simply from your problem statement.

(1) the resistors are connected in parallel which means they have the same voltage across

(2) there is a voltage source across the parallel resistors and no other components in the circuit so the voltage across either resistor is $V$, the terminal voltage of the voltage source

(3) the voltage produced by the voltage source is constant

That's really all there is to it. Since the terminal voltage of the voltage source is constant, adding yet another resistor in parallel does not change $V$; $V$ is independent of the total current.

By Ohm's law, the current through the resistor is the voltage across divided by the resistance.

Thus,

$$I_n = \frac{V}{R_n},\qquad V\; \mathrm{constant}$$

Yes you are correct in saying that "In the given situation, current flowing through a given resistor remains constant even if another resistor is connected in parallel with it". Because Ohm's Law says that current through a resistor depends only on the voltage across itself and its own resistance. When you have connected two resistors in parallel with a constant EMF source of potential difference V, then voltage across each resistor is V and so by Ohm's Law individual currents will be V/R1 and V/R2. When you remove R2 from the circuit, still the potential difference across R1 is V, hence by Ohm's Law current through it will be V/R1. But notice that total current changes when you remove or add a parallel resistor. Emf source has to provide the current to all the resistors. The power delivered by the EMF source also changes when you add or remove a parallel resistor.