Question

how does a parallel circuit change when a branch is added?
the total resistance increases, so the current in the circuit decreases.
the total resistance decreases, so the current in the circuit increases.
the total resistance increases, so bulbs shine at a lower brightness.
the total resistance decreases, so bulbs shine at a higher brightness.

Explanation:

1. Recall the formula for total resistance in a parallel circuit: For resistors \( R_1, R_2, \dots, R_n \) in parallel, the total resistance \( R_{total} \) is given by \( \frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}+\dots+\frac{1}{R_n} \). When a new branch (resistor) is added, we are adding another positive term to the right - hand side of the equation. This means that \( \frac{1}{R_{total}} \) increases, so \( R_{total} \) (the total resistance) decreases.
2. Recall Ohm's Law: \( I = \frac{V}{R} \), where \( V \) is the voltage of the power source (constant in a parallel circuit, as each branch has the same voltage as the source), \( I \) is the total current, and \( R \) is the total resistance. Since \( V \) is constant and \( R_{total} \) decreases, the total current \( I \) in the circuit increases.
3. Analyze the brightness of bulbs: In a parallel circuit, each branch has the same voltage as the source. The brightness of a bulb (which is a resistor) in a branch depends on the power \( P=\frac{V^{2}}{R_{bulb}} \) (since \( V \) across the bulb and \( R_{bulb} \) are constant for a given bulb). So adding a new branch does not change the voltage across the existing bulbs, and thus their brightness remains the same. So we can eliminate the options related to the change in bulb brightness.
4. Evaluate the options:

• Option 1: Says total resistance increases (wrong, as we saw total resistance decreases) and current decreases (wrong, current should increase). Eliminate.
• Option 2: Says total resistance decreases (correct) and current in the circuit increases (correct, from Ohm's Law).
• Option 3: Says total resistance increases (wrong) and bulbs shine at lower brightness (wrong, brightness of existing bulbs doesn't change). Eliminate.
• Option 4: Says total resistance decreases (correct), but says bulbs shine at higher brightness (wrong, brightness of existing bulbs doesn't change as their voltage and resistance are constant). Eliminate.

Answer:

B. The total resistance decreases, so the current in the circuit increases.