Problem 42 - Physics

Question

The current in a \(47-\Omega\) resistor is 0.12 A. This resistor is in series with a \(28-\Omega\) resistor, and the series combination is connected across a battery. What is the battery voltage?

Step-by-Step Solution

Answer

The battery voltage is 9 V.

Step 1: Understanding the Series Circuit

In a series circuit, the total resistance is the sum of the resistances of each component. The total resistance \(R_t\) can be calculated by adding the resistance of the \(47-\Omega\) resistor and the \(28-\Omega\) resistor: \(R_t = 47 + 28\).

Step 2: Calculate Total Resistance

Calculate the total resistance: \(R_t = 47\, \Omega + 28\, \Omega = 75\, \Omega\).

Step 3: Ohm's Law to Find Voltage

To find the battery voltage, apply Ohm's Law: \(V = IR\), where \(I\) is the current and \(R\) is the resistance. Use the current provided (0.12 A) and the total resistance (75 Ω):\[ V = 0.12 \, A \times 75 \, \Omega \]

Step 4: Calculate Battery Voltage

Substitute the values into Ohm's Law to find the battery voltage: \[ V = 0.12 \, A \times 75 \, \Omega = 9 \, V \]

Key Concepts

Series Circuit

A series circuit is an electrical circuit in which components are connected end-to-end so that the same current flows through each component. This means:

• The current is constant along all the components in a series.
• The total voltage across the series circuit is the sum of the voltages across each component.
• The total resistance is the sum of the resistances of all components.

For example, when you connect a 47-ohm resistor and a 28-ohm resistor in series, as given in the problem, they share the same current. The understanding of how components interact in a series path is crucial, particularly when calculating total resistance.

Total Resistance

Total resistance in a series circuit is an essential concept. It helps us determine how much the circuit will oppose the flow of electric current. Since a series circuit presents a single path for current, the total resistance is simply the sum of each resistor's resistance in the circuit. For instance:

• With a 47-ohm resistor and a 28-ohm resistor in series, the total resistance is:

\[ R_t = 47 \, \Omega + 28 \, \Omega = 75 \, \Omega \]This formula is straightforward but powerful. By understanding how to effectively calculate total resistance, you set the groundwork for using other principles, such as Ohm's Law, to determine additional parameters like the total current or voltage.

Battery Voltage

Battery voltage in a series circuit can be calculated using Ohm's Law, which states: \[ V = I \times R \]This formula shows the relationship between voltage (V), current (I), and resistance (R). To find the voltage of the battery in this circuit, you multiply the total current by the total resistance:

• Use the current provided: 0.12 A.
• Use the total resistance: 75 Ω (as calculated from the series circuit).
• Substitute these values into Ohm's Law:

\[ V = 0.12 \, A \times 75 \, \Omega = 9 \, V \]This calculation tells you that the battery must provide 9 volts to maintain a current of 0.12 A through the entire circuit. Understanding battery voltage helps in ensuring the proper functioning of the circuit and in selecting the right battery for your needs.