Series Resistors

• Two resistors connected in series must have the same current.
  • All electrons must pass through one resistor before reaching the other; hence, they have the same current.
• Total resistance in series is given by: R{total} = R1 + R_2
  • For example, if you have resistors of 15 ohms and 30 ohms:
    R_{total} = 15 + 30 = 45 ext{ ohms}

Parallel Resistors

• Resistors connected in parallel provide alternative paths for current, leading to a lower total resistance compared to any individual resistor.
• Total resistance in parallel is given by: \frac{1}{R{total}} = \frac{1}{R1} + \frac{1}{R_2}
  • Example:
  • For a 15 ohm and a 30 ohm resistor:
    \frac{1}{R_{total}} = \frac{1}{15} + \frac{1}{30}

Ohm’s Law and Voltage

• Ohm's Law states that V = I \cdot R
  • Where:
  • V = voltage (in volts)
  • I = current (in amperes)
  • R = resistance (in ohms)
• For a circuit with a 10 volts power supply and resistors connected, if the total resistance is given, you can find the current. Example:
  If resistance across the light bulb is known, the circuit current calculation gives $I = 0.5 ext{ A}$.

Determining Voltage Across Resistors

• The voltage drop across each resistor in a series circuit can be found using Vi = I \cdot Ri
  • Example: For a circuit with a total voltage of 100 volts and a resistor of 5 ohms,
    V_1 = 6.67 A * 5 \approx 33.3 ext{ volts}
  • The remaining voltage, thus being 66.7 volts across another resistor.

Setups for Circuit Analysis

• When analyzing circuits, always draw the circuit diagram clearly.
• Use consistent variable definitions for clarity in calculations while working on combinations of resistors in series and parallel.

Effective Resistance in Mixed Circuits

• When resistors are in both series and parallel:
  1. Compute effective resistance for parallel resistors first, substituting their combined value back into the series resistance calculation.
  2. Example: If you have a resistance of 4k ohms and 2k ohms in parallel, find their effective resistance and then combine it with another resistor in series.

Current Flow in Parallel

• Current through parallel resistors may differ, but the voltage across each is the same.
• The total current entering splits between multiple paths based on their resistances:
  I = I1 + I2

Power in Resistors

• The power dissipated in resistors can be expressed as: P = V \cdot I or via Ohm's Law, substituting for voltage: P = I^2 \cdot R
  • For a given current, power increases as resistance increases.

Practical Applications

• In light bulbs, the brightness can decrease when more bulbs are added in series due to the divided voltage.
• Additionally, multiple resistors in series consume more voltage, while in parallel they each maintain the same voltage.

Summary of Circuit Concepts

• Understanding series and parallel configurations is crucial in circuit design.
• Always consider the effects of topology on total resistance and current flow to calculate performance under different conditions using Ohm's Law.