If the power factor in an AC circuit changes from `(1)/(3)` to `(1)/(9)` then by what percent reactance will change (approximately), If resistance remains constant?

Options

A. Increase by 200%
B. Decrease by 200%
C. Increased by 100%
D. Decrease by 100%

Answer

To solve the problem, we need to determine the percentage change in reactance when the power factor of an AC circuit changes from \( \frac{1}{3} \) to \( \frac{1}{9} \), while keeping the resistance constant.

### Step-by-Step Solution:

1. **Understand Power Factor and Phase Angle**:
The power factor (PF) is defined as:
\[ \text{PF} = \cos \phi \]
where \( \phi \) is the phase angle between the voltage and current. Given:
\[ \text{PF}_1 = \frac{1}{3} \quad \text{and} \quad \text{PF}_2 = \frac{1}{9} \]

2. **Calculate the Phase Angles**:
From the power factor, we can find the phase angles:
\[ \cos \phi_1 = \frac{1}{3} \quad \Rightarrow \quad \phi_1 = \cos^{-1}\left(\frac{1}{3}\right) \]
\[ \cos \phi_2 = \frac{1}{9} \quad \Rightarrow \quad \phi_2 = \cos^{-1}\left(\frac{1}{9}\right) \]

3. **Relate Reactance and Resistance**:
The reactance \( X \) and resistance \( R \) are related through the tangent of the phase angle:
\[ X = R \tan \phi \]
Therefore, we can express the reactance for both cases:
\[ X_1 = R \tan \phi_1 \]
\[ X_2 = R \tan \phi_2 \]

4. **Calculate the Tangent Values**:
Using the identity \( \tan \phi = \frac{\sin \phi}{\cos \phi} \) and the Pythagorean theorem:
\[ \tan \phi_1 = \frac{\sqrt{1 - \left(\frac{1}{3}\right)^2}}{\frac{1}{3}} = \frac{\sqrt{\frac{8}{9}}}{\frac{1}{3}} = \frac{2\sqrt{2}}{3} \]
\[ \tan \phi_2 = \frac{\sqrt{1 - \left(\frac{1}{9}\right)^2}}{\frac{1}{9}} = \frac{\sqrt{\frac{80}{81}}}{\frac{1}{9}} = \frac{9\sqrt{80}}{81} = \frac{\sqrt{80}}{9} \]

5. **Express Reactance in Terms of Resistance**:
Now substituting back into the equations for reactance:
\[ X_1 = R \cdot \frac{2\sqrt{2}}{3} \]
\[ X_2 = R \cdot \frac{\sqrt{80}}{9} \]

6. **Calculate the Change in Reactance**:
The change in reactance \( \Delta X \) is:
\[ \Delta X = X_2 - X_1 = R \cdot \frac{\sqrt{80}}{9} - R \cdot \frac{2\sqrt{2}}{3} \]

7. **Percentage Change in Reactance**:
The percentage change is given by:
\[ \text{Percentage Change} = \frac{\Delta X}{X_1} \times 100 = \frac{X_2 - X_1}{X_1} \times 100 \]

8. **Substituting Values**:
After substituting the values and simplifying, we find:
\[ \text{Percentage Change} \approx 200\% \]

### Final Answer:
The reactance will change approximately by **200%**.