Physics: Electricity and Magnetism - Series and Parallel Circuits

Calculate the total resistance and the current in a circuit where two resistors, $R_1 = 4\ \Omega$ and $R_2 = 6\ \Omega$, are connected in series to a $20V$ battery.

1. Total Resistance ($R_t$): $R_t = R_1 + R_2 = 4\ \Omega + 6\ \Omega = 10\ \Omega$
2. Current ($I$): $I = \frac{V}{R_t} = \frac{20V}{10\ \Omega} = 2A$

In a series circuit, resistances are simply added together. Once the total resistance is found, Ohm's law ($I = V/R$) is used to find the current flowing from the source.

Two resistors of $12\ \Omega$ and $6\ \Omega$ are connected in parallel across a $12V$ power supply. Calculate the total resistance ($R_p$) and the total current ($I_{total}$).

1. Total Resistance ($R_p$): $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{12} + \frac{1}{6}$ $\frac{1}{R_p} = \frac{1}{12} + \frac{2}{12} = \frac{3}{12}$ $R_p = \frac{12}{3} = 4\ \Omega$
2. Total Current ($I_{total}$): $I_{total} = \frac{V}{R_p} = \frac{12V}{4\ \Omega} = 3A$

In a parallel circuit, the reciprocal of the total resistance is the sum of the reciprocals of individual resistances. Note that the total resistance ($4\ \Omega$) is less than the smallest individual resistor ($6\ \Omega$).

Three resistors with values $R_1 = 2\ \Omega$, $R_2 = 3\ \Omega$, and $R_3 = 5\ \Omega$ are connected in series to a $20V$ DC supply. Determine the total resistance and the voltage drop across $R_2$.

1. Calculate Total Resistance ($R_s$): $R_s = R_1 + R_2 + R_3$ $R_s = 2 + 3 + 5 = 10\ \Omega$

2. Calculate Total Current ($I$): $I = \frac{V}{R_s} = \frac{20}{10} = 2A$

3. Calculate Voltage Drop across $R_2$ ($V_2$): $V_2 = I \times R_2$ $V_2 = 2 \times 3 = 6V$

In a series circuit, we find the total resistance by summing all individual resistors. Since the current is the same everywhere, we use the total current and the specific resistance of $R_2$ to find its individual voltage drop using Ohm's Law.

Two identical lamps, each with a resistance of $10\ \Omega$, are connected in parallel to a $5V$ battery. Calculate the total current flowing out of the battery.

1. Calculate Total Resistance ($R_p$): $\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2}$ $\frac{1}{R_p} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10}$ $R_p = \frac{10}{2} = 5\ \Omega$

2. Calculate Total Current ($I_{total}$): $I_{total} = \frac{V}{R_p}$ $I_{total} = \frac{5}{5} = 1A$

For parallel circuits, we use the reciprocal formula to find the equivalent resistance. Because the lamps are identical and in parallel, the total resistance is halved. We then apply Ohm's law using the source voltage and this equivalent resistance.