Question

In an emergency, you need to run a radio that uses \(30.0 \mathrm{~W}\) of power when attached to a \(10.0-\mathrm{V}\) power supply. The only power supply you have access to provides \(25.0 \mathrm{kV}\) but you do have a large number of \(25.0-\Omega\) resistors. If you want the power to the radio to be as close as possible to \(30.0 \mathrm{~W}\), how many resistors should you use, and how should they be connected (in series or in parallel)?

Short answer

Answer: Use 8 resistors connected in parallel.

Step-by-step solution

Calculate the total resistance required

In order to calculate the required total resistance, we have to use the formula for power: $$ P = \frac{V^2}{R_{total}} $$ Here, \(P\) represents the power, \(V\) represents the voltage, and \(R_{total}\) is the total resistance. Since we want the power to be as close as possible to 30.0 W, we can plug in the values and solve for the total resistance: $$ R_{total} = \frac{V^2}{P}$$ $$ R_{total} = \frac{(10.0 \mathrm{V})^2}{30.0 \mathrm{W}}$$ $$ R_{total} = 3.33 \Omega $$

Calculate number of resistors

Now we have an idea of the total resistance needed, which is \(3.33 \Omega\). Given that we only have access to \(25.0-\Omega\) resistors, we need to find out the number of resistors required and their connection to get as close as possible to the desired total resistance. Let's consider the two configurations: series and parallel. 1. Series: If we connect resistors in series, their resistances simply add up: $$ R_{series} = R_1 + R_2 + \cdots + R_n $$ To get close to the required resistance of \(3.33 \Omega\), we would need fewer resistors. However, connecting \(25.0-\Omega\) resistors in series will result in a higher total resistance than the desired value. Therefore, this configuration will not work. 2. Parallel: If we connect resistors in parallel, the reciprocal of the total resistance is equal to the sum of the reciprocals of their individual resistances: $$ \frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} $$ Now, let's plug in the numbers and solve for the number of resistors \(n\) required: $$ \frac{1}{R_{total}} = n \cdot \frac{1}{25.0 \Omega} $$ $$ n = \frac{1}{R_{total}} \cdot 25.0 \Omega $$ Using the required total resistance of \(3.33 \Omega\): $$ n = \frac{1}{3.33 \Omega} \cdot 25.0 \Omega $$ $$ n \approx 7.50 $$ Since we need a whole number of resistors, we can round this value up to 8. Thus, we need 8 resistors connected in parallel.

Conclusion

To achieve a power dissipation as close as possible to 30.0 W using a 10.0-V power supply and 25.0-\(\Omega\) resistors, we need to use 8 resistors connected in parallel.

Key concepts

Understanding Ohm's Law

Ohm's Law is a fundamental principle in the field of electronics that relates the voltage across a conductor to the current through it, and the resistance of the conductor. It's given by the formula:

\[ V = I \times R \]
Where \(V\) is the voltage in volts (V), \(I\) is the current in amperes (A), and \(R\) is the resistance in ohms (Ω). If you know any two of these quantities, you can calculate the third. This relationship is crucial when solving circuit-related problems, such as calculating the number of resistors needed to achieve a certain power output with a given voltage. In our example, we needed to calculate the total resistance required for a radio to safely operate at a specific power level, which we did by rearranging Ohm's Law in the context of power:

\[ R_{total} = \frac{V^2}{P} \]

Series and Parallel Circuits Explained

In the realm of electrical circuits, there are two basic ways to connect components: series and parallel. Understanding the difference is vital for any circuit analysis.

Series Circuits

In series circuits, components are connected end-to-end, so the same current flows through each component. The total resistance is the sum of all individual resistances, as shown by the formula:
\[ R_{series} = R_1 + R_2 + \. + R_n \]
However, adding resistors in series always increases total resistance, making it unsuitable if a lower resistance is needed.

Parallel Circuits

Conversely, in parallel circuits, the ends of all components are connected together, and each component has the same voltage across it. The formula for total resistance in a parallel circuit is:
\[ \frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \. + \frac{1}{R_n} \]
The result is that the total resistance decreases as more components are added in parallel, which is useful in cases like our problem where a much lower resistance is needed than what individual resistors can provide.

The Concept of Power Dissipation

Power dissipation in electrical circuits is the process by which an electrical device converts electric energy into heat or another form of energy, like light or mechanical energy. It's typically an indication of how much work or energy is being used by a component when current passes through it. The power dissipated by a resistor, for example, can be calculated by:
\[ P = I^2 \times R \]
or in terms of voltage and resistance by:
\[ P = \frac{V^2}{R} \]
This is why in the problem at hand, the calculation of the total resistance necessary to achieve a desired power of 30.0 watts from a 10-volt supply was crucial. By adjusting the resistance through configuring multiple resistors in parallel, the power dissipation could be controlled to prevent damage to the radio and provide the necessary power for operation. The choice of parallel over series was key to achieving the correct resistance value and, consequently, the correct power dissipation for the emergency radio.