Question

How can you light a \(1.0-\mathrm{W}, 1.5-\mathrm{V}\) bulb with your \(12.0-\mathrm{V}\) car battery?

Short answer

Answer: The resistance value of the required resistor is approximately 15.67 Ohms. A suitable choice would be a 15 Ohm resistor. The circuit should be connected by placing the resistor in series with the 1.0W, 1.5V bulb and the 12V car battery.

Step-by-step solution

Calculate the current required to light the bulb

Using the power and voltage rating of the bulb, we can solve for the current required to light the bulb using the formula: Power = Voltage × Current \[ I = \frac{Power}{Voltage} = \frac{1.0 W}{1.5 V} = 0.67 A \]

Calculate the resistance needed for the bulb

Now, we can find the resistance of the bulb using Ohm's Law: Voltage = Current × Resistance \[ R_{bulb} = \frac{Voltage}{Current} = \frac{1.5 V}{0.67 A} = 2.24 \Omega \]

Determine the voltage drop across the resistor required to light the bulb

We will use a resistor in series with the bulb to drop the voltage from 12V (car battery) to 1.5V (bulb voltage). So the voltage drop across the resistor should be: \[ V_{resistor} = V_{battery} - V_{bulb} = 12 V - 1.5 V = 10.5 V \]

Calculate the resistance value of the resistor

Since we know the voltage drop across the resistor (10.5V) and the current (0.67A) flowing through the circuit, we can use Ohm's Law once again to find the resistance value of the resistor required: \[ R_{resistor} = \frac{V_{resistor}}{I} = \frac{10.5 V}{0.67 A} = 15.67 \Omega \]

Choose an appropriate resistor value and connect it in series with the bulb

We need to find a resistor with a value close to 15.67 Ohms. In this case, a 15 Ohm resistor could be a suitable choice. Connect this resistor in series with the 1.0W, 1.5V bulb and the 12V car battery. The bulb should now light up, as the resistor will drop 10.5V leaving 1.5V across the bulb.

Key concepts

Electrical Resistance

Electrical resistance is a fundamental concept in the realm of electronics and physics, referring to the opposition that materials present to the flow of electric current. Imagine you are trying to push water through a pipe; the resistance would be anything that makes it harder for the water to pass through, such as the pipe's length or its diameter. Similarly, in an electrical circuit, resistance is determined by the physical properties of the conductor, such as its length, cross-sectional area, type of material, and temperature.

When we examine a bulb, for instance, its resistance can be calculated using the formula derived from Ohm's Law, which states that the resistance \(R\) is equal to the voltage \(V\) across the material divided by the current \(I\) flowing through it: \[ R = \frac{V}{I} \.
\] Therefore, by knowing two of these quantities, the third can be determined. In the textbook problem, the bulb's resistance is crucial for designing a circuit that will allow it to function correctly using a higher power source, such as a car battery.

Voltage Drop Calculation

Understanding voltage drop calculation is essential for designing circuits and ensuring the proper function of electronic components. A voltage drop occurs when an electrical current flows through a passive element, such as a resistor, and loses some of its energy. The lost energy appears as a reduced voltage across that component.

The voltage drop across an individual component in a circuit can be determined using Ohm's Law. To calculate the voltage drop \(V_{drop}\), simply multiply the component's resistance \(R\) by the current flowing through it \(I\): \[ V_{drop} = I \times R \.
\] In our textbook example, by connecting a resistor in series with the bulb, we ensure that the voltage drop across the resistor accounts for the difference between the car battery's voltage and the bulb's required voltage. The correct resistor can then be chosen to create the precise voltage drop needed, which, in this case, is 10.5V.

Series Circuit

What is a Series Circuit?

A series circuit is a type of electrical circuit where components are arranged in a linear chain, so all current flows through each component sequentially. In such a circuit, the current is the same through all components because there's only one path for the current to take. However, the voltage across each component can vary depending on the component's resistance.

In our exercise, connecting the bulb and the resistor in series is crucial because it allows for the use of a single current path to control the voltage across the bulb. As a rule, the total resistance of a series circuit is the sum of the resistances of all components, which we calculate using \[ R_{total} = R_1 + R_2 + \dots + R_n \.
\] The design of a series circuit ensures the bulb receives only the voltage it requires to light up, while the rest of the voltage is dropped across the resistor, as we see in the proposed solution. Adequate series circuit design promotes safety and efficiency in electronic systems, ensuring that each component operates within its required voltage range.