Problem 2 - Physics for Scientists and Engineers with Modern Physics

Question

(I) Four \(1.50-\mathrm{V}\) cells are connected in series to a \(12-\Omega\) lightbulb. If the resulting current is \(0.45 \mathrm{~A},\) what is the internal resistance of each cell, assuming they are identical and neglecting the resistance of the wires?

Step-by-Step Solution

Answer

The internal resistance of each cell is approximately 0.3325 Ω.

Step 1: Calculate Total Voltage

Each cell has a voltage of 1.50 V, and there are 4 cells connected in series. Therefore, the total voltage supplied by the cells is given by: \(4 \times 1.50 = 6.00 \, \text{V}\).

Step 2: Use Ohm's Law to Determine Total Resistance

Using Ohm's Law, \(V = IR\), where \(V = 6.00 \, \text{V}\) (total voltage) and \(I = 0.45 \, \text{A}\) (current). Substitute these values to find the total resistance: \[R_{\text{total}} = \frac{V}{I} = \frac{6.00}{0.45} = 13.33 \, \Omega\].

Step 3: Calculate the Total Internal Resistance

The total resistance is the sum of the lightbulb resistance and the internal resistances of the cells. Let the internal resistance of each cell be \(r\). Since there are 4 cells, the total internal resistance is \(4r\). The equation is thus: \[12 \, \Omega + 4r = 13.33 \, \Omega\].

Step 4: Solve for the Internal Resistance of Each Cell

Rearrange the equation from the previous step to solve for \(r\): \[4r = 13.33 - 12 = 1.33 \, \Omega\]. Thus, \[r = \frac{1.33}{4} \approx 0.3325 \, \Omega\].

Key Concepts

Internal Resistance

Internal resistance is an important concept when discussing electrical components like batteries. It refers to the resistance within the battery itself, which opposes the flow of electrical current. This resistance is critical because it affects the overall performance and efficiency of the battery.
In practical terms, internal resistance can cause a drop in the voltage available from a battery. This is why the voltage measured directly across the battery terminals is often slightly less than the rated voltage of the battery. With higher internal resistance, less voltage is available to the connected circuit.

• Can lead to reduced efficiency of the electrical device.
• Causes power loss in the form of heat within the battery.
• Affects how long a device can operate on a single charge.

In scenarios like the original exercise, internal resistance becomes a key factor in calculations when determining the effective resistance and current in a circuit. Understanding how to calculate it ensures a better estimation of how a circuit will perform in real-world conditions.

Series Circuits

Series circuits are one of the basic ways to connect electrical components. In a series circuit, the components are connected end-to-end in a single pathway for current flow. This arrangement means that the same current flows through all components, but the voltage is divided among them.
Key characteristics of series circuits include:

• The total resistance is the sum of the individual resistances. So, when multiple resistors (or cells) are in series, their resistances add up.
• A common current runs through all components, making the circuit simple to analyze.
• If one component fails, the entire circuit is broken, much like old-fashioned Christmas lights.

Series circuits are commonly used in situations where matched electrical parameters are necessary, like the series connection of cells in the original exercise. Here, the total voltage provided by the cells is the sum of each individual cell's voltage.

Electrical Current

Electrical current is the flow of electric charge through a conductor, such as a wire. In most cases, this flow is due to negatively charged electrons moving through a metal conductor.
The unit of current is the ampere (A), which measures the rate at which charge is flowing. Current can either be direct (DC), flowing in one direction, or alternating (AC), which changes direction periodically.
Understanding current is fundamental because it's a key factor in how circuits operate. Ohm's Law, represented by the equation \(V = IR\), shows the relationship between voltage (V), current (I), and resistance (R). The current flowing through a circuit will affect how other components in the circuit perform.

• Directly affects how much a device can perform work.
• Essential for designing circuits with specific electrical properties.
• Too much current can damage components, highlighting the importance of correct calculations and design.