[Physics] Change in current on adding a resistor

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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.

Best Answer

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,

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

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