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Suppose we have a battery with terminal PD $V$ connected to an external resistor of resistance $R_1$. Now if another resistor say $R_2$ is added in parallel to $R_1$ , does the potential drop across $R_1$ change?

I know that resistances in parallel have the same potential drop across them but is this true if we add/subtract additional resistances in parallel?

My thinking is that as the total resistance of the circuit $\frac{R_1 R_2}{R_1 + R_2} < R_1$ the current increases while keeping the voltage same($V=IR$). Is this true or does the potential difference also change?

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The voltage across $R_1$ and $R_2$ will be the same, $V$.

You are right that the net resistance will decrease to $(1/R_1+1/R_2)^{-1}$ and this change will be compensated by an increase in the current $I$ (Ohm's Law). $V$ will stay constant.

Why?

Because each component in a parallel circuit has two common nodes with each of the other components in the parallel circuit. Consequently there are only two nodes in a parallel circuit that can be at a different Voltage potential.

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See this question for more technical explanations as to why it happens Why does voltage remains same over Parallel Circuit

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