if the electrc current in a lamp decreases by `5%` then the power output decreases by

Options

A. `20%`
B. `10%`
C. `5%`
D. `2.5%`

Explanation

To solve the problem of how much the power output decreases when the electric current in a lamp decreases by 5%, we can follow these steps:

### Step 1: Define the Initial Current
Let the initial current flowing through the lamp be denoted as \( I \).

### Step 2: Calculate the New Current
Since the current decreases by 5%, the new current \( I' \) can be expressed as:
\[ I' = I - 0.05I = 0.95I \]

### Step 3: Understand the Power Formula
The power \( P \) consumed by the lamp is given by the formula:
\[ P = I^2 R \]
where \( R \) is the resistance of the lamp.

### Step 4: Calculate the New Power
Using the new current \( I' \), the new power \( P' \) can be calculated as:
\[ P' = (I')^2 R = (0.95I)^2 R \]
Expanding this, we get:
\[ P' = 0.9025 I^2 R \]

### Step 5: Relate New Power to Initial Power
Now, we can express the new power \( P' \) in terms of the initial power \( P \):
\[ P' = 0.9025 P \]

### Step 6: Calculate the Decrease in Power
To find the decrease in power, we can calculate:
\[ \text{Decrease in Power} = P - P' = P - 0.9025 P = (1 - 0.9025) P = 0.0975 P \]

### Step 7: Calculate the Percentage Decrease
To find the percentage decrease in power, we use the formula:
\[ \text{Percentage Decrease} = \left( \frac{\text{Decrease in Power}}{P} \right) \times 100 \]
Substituting the decrease we found:
\[ \text{Percentage Decrease} = \left( \frac{0.0975 P}{P} \right) \times 100 = 9.75\% \]

### Conclusion
Thus, the power output decreases by approximately **9.75%** when the electric current in the lamp decreases by 5%.

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