Calculating Equivalent Resistance in Parallel

When resistors are connected in parallel, the total or equivalent resistance decreases. The formula for calculating the equivalent resistance ($\text{R}_{\text{eq}}$) of resistors connected in parallel is given by the reciprocal of the sum of the reciprocals of the individual resistances. For 'n' resistors $\text{R}_1, \text{R}_2, ..., \text{R}_{\text{n}}$ connected in parallel, the formula is:

$$ \frac{1}{\text{R}_{\text{eq}}} = \frac{1}{\text{R}_1} + \frac{1}{\text{R}_2} + ... + \frac{1}{\text{R}_{\text{n}}} $$

In this problem, we have four resistors, and each resistor has a resistance of 100 $\Omega$. Let's denote them as $\text{R}_1, \text{R}_2, \text{R}_3,$ and $\text{R}_4$.

• $\text{R}_1 = 100 \, \Omega$
• $\text{R}_2 = 100 \, \Omega$
• $\text{R}_3 = 100 \, \Omega$
• $\text{R}_4 = 100 \, \Omega$

Since all four resistors have the same resistance value, we can substitute these values into the parallel resistance formula:

$$ \frac{1}{\text{R}_{\text{eq}}} = \frac{1}{100 \, \Omega} + \frac{1}{100 \, \Omega} + \frac{1}{100 \, \Omega} + \frac{1}{100 \, \Omega} $$

Adding the fractions on the right side:

$$ \frac{1}{\text{R}_{\text{eq}}} = \frac{1 + 1 + 1 + 1}{100 \, \Omega} $$

$$ \frac{1}{\text{R}_{\text{eq}}} = \frac{4}{100 \, \Omega} $$

To find the equivalent resistance ($\text{R}_{\text{eq}}$), we take the reciprocal of both sides:

$$ \text{R}_{\text{eq}} = \frac{100 \, \Omega}{4} $$

Now, perform the division:

$$ \text{R}_{\text{eq}} = 25 \, \Omega $$

So, the equivalent resistance of four 100 $\Omega$ resistors connected in parallel is 25 $\Omega$. This is significantly less than the resistance of a single resistor, which is a characteristic effect of parallel connections.

Another way to calculate the equivalent resistance when 'n' identical resistors (R) are connected in parallel is to use the simplified formula:

$$ \text{R}_{\text{eq}} = \frac{R}{n} $$

In this case, $R = 100 \, \Omega$ and $n = 4$.

$$ \text{R}_{\text{eq}} = \frac{100 \, \Omega}{4} $$

$$ \text{R}_{\text{eq}} = 25 \, \Omega $$

Both methods yield the same result.