Question

The tungsten filament of bulb has resistance equal to \(18 \Omega\) at \(27^{\circ} \mathrm{C}\) temperature \(0.25 \mathrm{~A}\) of current flows, when \(45 \mathrm{~V}\) is connected to it If \(\alpha=4.5 \times 10^{-3} \mathrm{~K}^{-1}\) for a tungsten then find the temperature of the filament. (A) \(2160 \mathrm{~K}\) (B) \(1800 \mathrm{~K}\) (C) \(2070 \mathrm{~K}\) (D) \(2300 \mathrm{~K}\)

Short answer

The temperature of the tungsten filament is \(2300 \mathrm{K}\), which corresponds to answer choice \((D)\).

Step-by-step solution

Write the given information

We are given the following information: - The resistance of the tungsten filament at 27℃ (initial temperature) is 18 ohms - The current flowing through the filament is 0.25 A - The voltage connected to it is 45 V - The temperature coefficient of resistance for tungsten is \(4.5 \times 10^{-3} \mathrm{K}^{-1}\)

Calculate the power dissipated in the filament

Using Ohm's Law, the power dissipation in the filament can be calculated as: \[P = IV\] Where \(P\) is the power, \(I\) is the current, and \(V\) is the voltage. Substitute the given values: \[P = (0.25 \mathrm{~A})(45 \mathrm{~V})\] \[P = 11.25 \mathrm{~W}\]

Determine temperature change

Now, we need to find the temperature change from the initial temperature 27℃. Divide the formula for resistance by \(R_0\): \[\frac{R_t}{R_0} = 1 + \alpha \Delta T\] Since we are looking for the final temperature, we can use Ohm's Law to find \(R_t\) by substituting \(V = 45 \mathrm{V}\) and \(I = 0.25 \mathrm{A}\): \[R_t = \frac{V}{I} = \frac{45 \mathrm{~V}}{0.25 \mathrm{~A}} = 180 \Omega\] Now substitute the given values: \[\frac{180 \Omega}{18 \Omega} = 1 + (4.5 \times 10^{-3} \mathrm{K}^{-1}) \Delta T\]

Solve for the temperature change

Now, solve for \(\Delta T\): \[10 = 1 + (4.5 \times 10^{-3} \mathrm{K}^{-1}) \Delta T\] \[9 = (4.5 \times 10^{-3} \mathrm{K}^{-1}) \Delta T\] \[\Delta T = \frac{9}{(4.5 \times 10^{-3} \mathrm{K}^{-1})} = 2000 \mathrm{K}\]

Calculate the final temperature of the filament

Now we can calculate the final temperature by adding the initial temperature to the temperature change: \[T = T_0 + \Delta T\] \[T = 27 ℃ + 2000 \mathrm{K}\] Convert the initial temperature to Kelvin by adding 273 to 27℃: \[T = (273 + 27) \mathrm{K} + 2000 \mathrm{K} = 2300 \mathrm{K}\] Therefore, the temperature of the tungsten filament is 2300 K, which corresponds to answer choice (D).

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle used to understand the relationship between voltage, current, and resistance in an electrical circuit. The formula for Ohm's Law is \( V = IR \), where \( V \) represents voltage, \( I \) is the current, and \( R \) denotes resistance. This law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points.

In the context of our tungsten filament bulb, Ohm's Law helps us establish the initial condition by calculating the resistance of the filament at a given voltage and current. Knowing these variables is crucial to further analysis of power dissipation and changes in resistance due to temperature variations.

• Voltage (\( V \)): The electrical potential difference, measured in volts.
• Current (\( I \)): The flow of electric charge, measured in amperes.
• Resistance (\( R \)): The hindrance to current flow, measured in ohms.

Recognizing the interdependence of these elements is essential for solving most electrical problems.

Power Dissipation

Power dissipation in an electrical circuit is the process through which electrical energy is converted into heat. This is a common occurrence in resistors and often needs to be evaluated to ensure that circuit components do not overheat.

Power dissipation can be calculated using the formula \( P = IV \), where \( P \) is the power in watts, \( I \) is the current in amperes, and \( V \) is the voltage in volts. Alternatively, it can be expressed as \( P = I^2R \) or \( P = \frac{V^2}{R} \). These derivations come from combining the basic definition of power with Ohm's Law.

For our tungsten filament, we calculate power dissipation to be \( 11.25 \) watts, which emphasizes the amount of heat energy being produced and dispelled. Understanding power dissipation is critical to ensure that the light bulb operates safely and effectively.

• Ensures thermal management for components.
• Helps in selecting appropriate components for the circuit.
• Influences efficiency and durability of devices.

Tungsten Filament Resistance

Tungsten is frequently used as a filament material in light bulbs due to its high melting point and excellent electrical resistivity. The resistance of the filament is crucial as it determines how much of the electrical energy is used as light and how much as heat.

Resistance in metallic wires like tungsten increases with temperature. For example, at room temperature (27°C), our tungsten filament has a resistance of \( 18 \Omega \). As the filament heats up to higher temperatures due to electrical power, its resistance increases significantly.

The resistance change due to temperature is determined using the temperature coefficient of resistance (\( \alpha \)), specific to the material being used. For tungsten, \( \alpha = 4.5 \times 10^{-3} \text{ K}^{-1} \), which provides a measure of how much the resistance will change with each degree of temperature increase.

• High-temperature resilience makes tungsten ideal for bulb filaments.
• Resistance increase converts electrical energy to heat efficiently.
• Supports the functioning of incandescent bulbs by providing the necessary brightness.

Calculating Temperature Change

The temperature change of a material like a tungsten filament in a bulb can be determined by measuring how its resistance varies with temperature. This involves using the formula \( \frac{R_t}{R_0} = 1 + \alpha \Delta T \), where:

• \( R_t \): Resistance at the final temperature.
• \( R_0 \): Resistance at the initial temperature.
• \( \alpha \): Temperature coefficient of resistance.
• \( \Delta T \): Change in temperature.

In our example, by calculating the new resistance (\( R_t = 180 \Omega \)) using Ohm's Law, and using the initial resistance (\( 18 \Omega \)), we derive the equation to solve for \( \Delta T \). We find that the temperature change is \( 2000 \text{ K} \).

Finally, adding the initial temperature to this change gives the final temperature in Kelvin. This calculation is crucial in ensuring the filament withstands the significant temperature changes when in operation, pointing to proper design and material choices for electrical components.