how does a parallel circuit change when a branch is added?\nthe total resistance increases, so the current…

Explanation:

Step1: Analyze the formula for total resistance in parallel circuits

The formula for total resistance (R_{total}) in a parallel circuit with resistors (R_1,R_2,\cdots,R_n) is (\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots+\frac{1}{R_n}). When a new branch (a new resistor (R_{n + 1})) is added, (\frac{1}{R_{total}}) increases (because (\frac{1}{R_{total}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots+\frac{1}{R_n}+\frac{1}{R_{n+1}})). Since (y = \frac{1}{x}) is a decreasing function ((x>0)), when (\frac{1}{R_{total}}) increases, (R_{total}) decreases.

Step2: Use Ohm's law (I=\frac{V}{R})

Assume the voltage (V) of the power - supply is constant. According to Ohm's law (I=\frac{V}{R}), when (R = R_{total}) decreases (and (V) is constant), the total current (I) in the circuit increases. Also, in a parallel circuit, the voltage across each branch is the same as the source voltage. The brightness of a bulb (assuming it is a resistive load) depends on the power (P=\frac{V^{2}}{R}) (since (V) across each bulb remains the same as the source voltage in a parallel circuit, the brightness of each individual bulb does not change when a new branch is added)

Answer:

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