Calculating Equivalent Resistance for Parallel Resistors

To find the equivalent resistance ($R_{eq}$) when resistors are connected in parallel, we use a specific formula. The formula states that the reciprocal of the equivalent resistance is the sum of the reciprocals of all the individual resistances connected in the circuit.

The general formula for parallel resistors is:

$$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n} $$

In this problem, we have 10 resistors (so $n=10$). Each resistor has a resistance value of 10 Ω (meaning $R_1 = R_2 = ... = R_{10} = 10 \, \Omega$).

Since all the individual resistances are the same, we can simplify the formula. We are adding $\frac{1}{10 \, \Omega}$ ten times:

$$ \frac{1}{R_{eq}} = \frac{1}{10 \, \Omega} + \frac{1}{10 \, \Omega} + ... \text{ (10 times)} $$

This can be written as:

$$ \frac{1}{R_{eq}} = 10 \times \frac{1}{10 \, \Omega} $$

$$ \frac{1}{R_{eq}} = \frac{10}{10 \, \Omega} $$

$$ \frac{1}{R_{eq}} = \frac{1}{\Omega} $$

To find the equivalent resistance $R_{eq}$, we take the reciprocal of both sides:

$$ R_{eq} = \frac{1 \, \Omega}{1} $$

$$ R_{eq} = 1 \, \Omega $$

Thus, the equivalent resistance when ten 10 Ω resistors are connected in parallel is 1 Ω.