Question

How do resistors in parallel affect the total resistance?

Answer 1

Answer:

They're going to increase the total resistance as $R_{T} = \sum\limits_{i=1}^N \left(\frac{1}{R_i} \right)^{-1}$

Explanation:

If the resistors are in parallel, the potential difference is the same for each resistor. But the total current is the sum of the currents that pass through each of the resistors. Then

$I = I_1 + I_2 + ... + I_N$

where

$I_i = \frac{V_i}{R_i}$

but

$V_i = V_j = V$ for $i,j= 1, 2,..., N$

so

$I = \frac{V}{R_1}+ \frac{V}{R_2} + ... + \frac{V}{R_N} = \left(\frac{1}{R_1} +\frac{1}{R_2} + ... + \frac{1}{R_N}\right)V = \frac{V}{R_T}$

where

$R_T = \left(\frac{1}{R_1} +\frac{1}{R_2} + ... + \frac{1}{R_N}\right)^{-1} =\sum\limits_{i=1}^N \left(\frac{1}{R_i} \right)^{-1}$