Question

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, \(100 \mathrm{~W}, 60 \mathrm{~W}\) and \(40 \mathrm{~W}\) bulbs have filament resistances \(\mathrm{R}_{100}, \mathrm{R}_{60}\) and \(\mathrm{R}_{40}\). respectively, the relation between these resistances is A) \(\frac{1}{\mathrm{R}_{100}}=\frac{1}{\mathrm{R}_{40}}+\frac{1}{\mathrm{R}_{60}}\) B) \(R_{100}=R_{40}+R_{60}\) C) \(\mathrm{R}_{100}>\mathrm{R}_{60}>\mathrm{R}_{40}\) D) \(\frac{1}{\mathrm{R}_{100}}>\frac{1}{\mathrm{R}_{60}}>\frac{1}{\mathrm{R}_{40}}\)

Short answer

C) \text{R}_{100}>\text{R}_{60}>\text{R}_{40}

Step-by-step solution

- Understanding the Power and Resistance Relationship

The power rating of a bulb is given by the formula P = V^2 / R, where P is the power, V is the voltage and R is the resistance. Therefore, for the same voltage, a bulb with higher power will have lower resistance.

- Comparing Power Ratings

Assuming all bulbs are designed for the same voltage, a 100 W bulb draws more power than a 60 W bulb, and a 60 W bulb draws more power than a 40 W bulb. This implies R_100 < R_60 < R_40, because power is inversely proportional to resistance.

- Assessing Answer Options

Option A is incorrect because it suggests R_100 is the result of parallel resistances, not individual bulb resistances. Option B is incorrect because R_100 is less than either R_40 or R_60. Option D suggests an inverse relationship, which is also incorrect because it would imply R_100 has a higher resistance. Therefore, Option C must be correct as it shows R_100 as the smallest resistance and R_40 as the largest.

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle in the field of electronics and electrical engineering that reveals a direct proportionality between the voltage applied across a conductor and the resultant electric current through it, provided the temperature remains constant. It can be mathematically expressed as the formula:
\( V = IR \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance of the conductor. This law is essential in understanding the behavior of electric circuits and is the cornerstone of circuit analysis.
For instance, if a higher voltage is applied to a circuit, the current increases accordingly, assuming resistance remains unchanged. Conversely, for a higher resistance with a constant voltage, the current in the circuit would decrease. Ohm's Law not only helps predict how circuits behave but also allows the design of circuits to achieve desired electrical outputs, such as in the varying brightness of incandescent bulbs.

Electrical Resistance

Understanding Resistance

Electrical resistance is the opposition to the flow of electric current through a material. It is a property intrinsic to the material and its physical condition, often influenced by factors such as temperature, material type, length, and cross-sectional area. The unit of resistance is the ohm (Ω).

Factors Affecting Resistance

• Material: Conductors have low resistance, facilitating current flow, while insulators resist current flow significantly.
• Temperature: For most materials, as the temperature increases, resistance increases due to the more vigorous atomic vibrations that impede the flow of electrons.
• Geometry: Wider or shorter conductors have lower resistance than narrow or long ones.

In relation to an incandescent bulb, filament resistance increases with temperature, which is crucial in designing bulbs with specific power ratings.

Incandescent Bulb Physics

The physics behind an incandescent bulb is based on the conversion of electrical energy into light and heat. The filament inside the bulb, usually made of tungsten due to its high melting point, heats up when electric current flows through it, eventually glowing and emitting light.

The Role of Resistance in Bulbs

The filament's resistance is a critical aspect of bulb design. At room temperature, a bulb with a higher wattage rating will have a lower resistance, enabling it to draw more current and produce more light compared to lower wattage bulbs when connected to the same voltage source. As illustrated through the problem, a 100 watt bulb has a lower resistance than a 40 watt bulb. This knowledge is essential when comparing bulb brightness, energy consumption, and understanding the effect of resistance on the power output.