To solve the problem regarding the unknown resistance in the post office box experiment, we will follow these steps:
### Step-by-Step Solution:
1. **Understand the Given Information:**
- We have the ratio \( \frac{Q}{P} = \frac{1}{10} \).
- When \( R = 142 \, \Omega \), the deflection is towards the right.
- When \( R = 143 \, \Omega \), the deflection is towards the left.
2. **Set Up the Wheatstone Bridge Equation:**
- In a Wheatstone bridge, at null deflection, the relationship is given by:
\[
\frac{Q}{P} = \frac{x}{R}
\]
- Here, \( x \) is the unknown resistance, and \( R \) is the known resistance.
3. **Calculate Unknown Resistance for \( R = 142 \, \Omega \):**
- Substitute the values into the equation:
\[
\frac{1}{10} = \frac{x}{142}
\]
- Rearranging gives:
\[
x = \frac{142}{10} = 14.2 \, \Omega
\]
4. **Calculate Unknown Resistance for \( R = 143 \, \Omega \):**
- Substitute the new value of \( R \):
\[
\frac{1}{10} = \frac{x}{143}
\]
- Rearranging gives:
\[
x = \frac{143}{10} = 14.3 \, \Omega
\]
5. **Determine the Range of Unknown Resistance:**
- From the calculations, we find that when \( R = 142 \, \Omega \), \( x = 14.2 \, \Omega \) and when \( R = 143 \, \Omega \), \( x = 14.3 \, \Omega \).
- Therefore, the range of the unknown resistance \( x \) is:
\[
14.2 \, \Omega \leq x < 14.3 \, \Omega
\]
### Final Answer:
The range of the unknown resistance \( x \) is from \( 14.2 \, \Omega \) to \( 14.3 \, \Omega \).
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