If the current through a resistor in a circuit increases by 3%, the power dissipated by the resistor

Options

A. increases approximately by 3%
B. increases approximately by 6%
C. increases approximately by 9%
D. decreases approximately by 3%

Explanation

To solve the problem of how the power dissipated by a resistor changes when the current through it increases by 3%, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Power Formula**:
The power (P) dissipated by a resistor is given by the formula:
\[ P = I^2 R \]
where \(I\) is the current through the resistor and \(R\) is the resistance.

2. **Identifying the Change in Current**:
We are given that the current increases by 3%. This can be expressed as:
\[ \Delta I = 0.03 I \]
where \(\Delta I\) is the change in current.

3. **Calculating the New Current**:
The new current \(I'\) after the increase can be written as:
\[ I' = I + \Delta I = I + 0.03 I = 1.03 I \]

4. **Calculating the New Power**:
The new power \(P'\) after the increase in current can be calculated using the new current:
\[ P' = (I')^2 R = (1.03 I)^2 R = 1.0609 I^2 R \]

5. **Finding the Percentage Change in Power**:
The original power \(P\) is:
\[ P = I^2 R \]
The change in power \(\Delta P\) is:
\[ \Delta P = P' - P = (1.0609 I^2 R) - (I^2 R) = 0.0609 I^2 R \]
The percentage change in power can be calculated as:
\[ \text{Percentage Change in Power} = \left(\frac{\Delta P}{P}\right) \times 100 = \left(\frac{0.0609 I^2 R}{I^2 R}\right) \times 100 = 6.09\% \]

6. **Conclusion**:
Therefore, the power dissipated by the resistor increases by approximately 6.09% when the current increases by 3%.