The voltage across a bulb is decreased by 2%. Assuming that the resistance of the filament remains unchanged, the power of the bulb will

Options

A. decrease by 2%
B. increase by 2%
C. decrease by 4%
D. increase by 4 %

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between voltage, resistance, and power
The power \( P \) consumed by an electrical device (like a bulb) is given by the formula:
\[ P = \frac{V^2}{R} \]
where \( V \) is the voltage across the device and \( R \) is the resistance.

### Step 2: Identify the change in voltage
According to the problem, the voltage across the bulb is decreased by 2%. This can be expressed mathematically as:
\[ \Delta V = -0.02V \]
where \( V \) is the original voltage.

### Step 3: Calculate the percentage change in power
We need to find the percentage change in power \( \Delta P \) when the voltage changes. The formula for the percentage change in power is:
\[ \frac{\Delta P}{P} \times 100 \]
Using the power formula, we can express the change in power in terms of the change in voltage:
\[ P = \frac{V^2}{R} \implies \Delta P = \frac{(V + \Delta V)^2}{R} - \frac{V^2}{R} \]

### Step 4: Expand the expression for power
Substituting \( \Delta V = -0.02V \):
\[ \Delta P = \frac{(V - 0.02V)^2}{R} - \frac{V^2}{R} \]
\[ = \frac{(0.98V)^2}{R} - \frac{V^2}{R} \]
\[ = \frac{0.9604V^2 - V^2}{R} \]
\[ = \frac{-0.0396V^2}{R} \]

### Step 5: Find the percentage change in power
Now we can find the percentage change in power:
\[ \frac{\Delta P}{P} = \frac{-0.0396V^2/R}{V^2/R} = -0.0396 \]
To express this as a percentage:
\[ \frac{\Delta P}{P} \times 100 = -0.0396 \times 100 = -3.96\% \]

### Step 6: Conclusion
Since we are looking for the percentage change in power, we round it to approximately -4%. The negative sign indicates that the power has decreased. Thus, the final answer is:
\[ \text{The power of the bulb will decrease by approximately } 4\%. \]

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