Understanding Series LEDs and Resistors

When multiple LEDs are connected in series, they share the same current. A current-limiting resistor is crucial to protect the LEDs from excessive current, which could damage them. The resistor limits the current to the desired level (specified by the LED's requirements).

In this problem, we have a stack of 4 LEDs connected in series, each requiring a specific voltage (3 V) and current (3 mA). The power source provides a DC voltage of 15 V.

Calculating the Required Resistor Value

To find the value of the current-limiting resistor, we need to follow these steps:

1. Calculate the total voltage drop across the LEDs:

   Since the LEDs are in series, the total voltage drop is the sum of the voltage drops across each individual LED.

   Total LED Voltage ($ V_{LED_{total}} $) = Number of LEDs $\times$ Voltage per LED

   Using LaTeX notation:

   $ V_{LED_{total}} = 4 \times 3 \, \text{V} = 12 \, \text{V} $

2. Determine the voltage across the resistor:

   The voltage across the resistor ($ V_R $) is the difference between the source voltage ($ V_{source} $) and the total voltage drop across the LEDs ($ V_{LED_{total}} $).

   Voltage across Resistor ($ V_R $) = Source Voltage ($ V_{source} $) - Total LED Voltage ($ V_{LED_{total}} $)

   Using LaTeX notation:

   $ V_R = 15 \, \text{V} - 12 \, \text{V} = 3 \, \text{V} $

3. Identify the required current:

   The current required for the LEDs is given as 3 mA. Since the LEDs are in series, this is the current that must flow through the resistor as well.

   Required Current ($ I_{LED} $) = 3 mA

   Convert mA to Amperes (A):

   $ I_{LED} = 3 \, \text{mA} = 3 \times 10^{-3} \, \text{A} $

4. Calculate the resistance using Ohm's Law:

   Ohm's Law states that Resistance ($ R $) = Voltage ($ V $) / Current ($ I $).

   Resistor Value ($ R $) = Voltage across Resistor ($ V_R $) / Required Current ($ I_{LED} $)

   Using LaTeX notation:

   $ R = \frac{V_R}{I_{LED}} = \frac{3 \, \text{V}}{3 \times 10^{-3} \, \text{A}} $

   $ R = \frac{3}{0.003} \, \Omega = 1000 \, \Omega $

   Since $ 1000 \, \Omega $ is equal to 1 k$ \Omega $, the required resistance is 1 k$ \Omega $.

Conclusion

Therefore, the value of the current-limiting resistor needed for this series LED circuit is 1 k$ \Omega $. This value ensures that the current flowing through the LEDs does not exceed their specified limit of 3 mA, protecting them from damage while allowing them to operate correctly with the given 15 V source.