Problem 70 - Physics

Question

When a light bulb is connected across the terminals of a battery, the battery delivers 24 \(\mathrm{W}\) of power to the bulb. A voltage of \(11.8 \mathrm{~V}\) exists between the terminals of the battery, which has an internal resistance of \(0.10 \Omega\). What is the emf of the battery?

Step-by-Step Solution

Answer

The emf of the battery is approximately 12.0 V.

Step 1: Identify Known Values

We are given the power delivered by the battery, \( P = 24\, \mathrm{W} \), the voltage between the terminals of the battery, \( V = 11.8\, \mathrm{V} \), and the internal resistance of the battery, \( r = 0.10\, \Omega \).

Step 2: Use Power Formula to Find Current

Use the power formula \( P = V \cdot I \) to solve for the current \( I \) flowing through the circuit.Rearrange the formula to find the current:\[ I = \frac{P}{V} = \frac{24}{11.8} \approx 2.03 \text{ A} \].

Step 3: Determine Voltage Drop Across Internal Resistance

The voltage drop across the internal resistance \( r \) is given by Ohm's Law, \( V_r = I \cdot r \).Substitute the known values to find \( V_r \):\[ V_r = 2.03 \times 0.10 = 0.203 \text{ V} \].

Step 4: Calculate EMF of the Battery

The electromotive force (emf), \( \mathcal{E} \), of the battery can be found by adding the internal voltage drop \( V_r \) to the external voltage \( V \):\[ \mathcal{E} = V + V_r = 11.8 + 0.203 \approx 12.003 \text{ V} \].

Key Concepts

Internal Resistance

Internal resistance is a subtle yet crucial aspect of electrical circuits, particularly in batteries. It refers to the inherent resistance a power source offers to the flow of current within itself. This occurs because of the composition and structure of the battery, such as the resistance of the chemicals and components inside it.
In practice, internal resistance affects the battery's efficiency. When current flows, some energy is lost as heat due to this resistance. This means not all the energy produced by the battery is used to power external devices like a light bulb.
Here's why internal resistance is important:

• Heat Loss: Internal resistance causes the battery to warm up, wasting energy as heat.
• Reduced Output: It lowers the voltage that is available at the battery's terminals.
• Measurement of Health: An increase in internal resistance often indicates that a battery is aging or deteriorating.

When considering a battery's performance, knowing the value of the internal resistance can help predict how well the battery will work under load conditions.

Ohm's Law

Ohm's Law is a foundational principle in understanding electrical circuits. It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points. The relationship is mathematically expressed as:\[ V = I imes R \] Where:

• \( V \) is the voltage across the two points,
• \( I \) is the current flowing through the conductor,
• \( R \) is the resistance of the conductor.

Ohm's Law helps in calculating parameters within electrical circuits when two of the three quantities are known.
In our problem, Ohm's Law is used to determine the voltage drop across the internal resistance of the battery. By rearranging the formula, we find this voltage drop through multiplying the current by the internal resistance.
Understanding this law allows one to predict how changes in one property of the circuit will affect the others, providing a tool to analyze and solve various electrical problems.

Voltage Drop

Voltage drop refers to the reduction in voltage in an electrical circuit as it passes through a component, such as a resistor or a battery's internal resistance. It occurs because some of the electrical energy is "used up" - converted into other forms like heat - as the current passes through a component.
In our example exercise, the voltage drop across a battery's internal resistance is calculated using Ohm's Law. This drop means that the voltage available across the battery's terminals is less than the emf (electromotive force) inside the battery.
Important aspects of voltage drop include:

• Impact on Performance: A high voltage drop can lead to insufficient power being delivered to the load.
• Efficiency: It affects how efficiently electrical energy is transmitted through the circuit.
• Design Considerations: Engineers often design circuits to minimize voltage drop to ensure efficient operation.

By understanding and calculating voltage drop, one can ensure that a circuit is operating effectively and that the devices powered by the circuit receive adequate voltage to function properly.