The measurement of an unknown resistance R is to be carried out using Wheatstone bridge (see Fig. 2(EP).3). Two students perform an experiment in two way. The first student takes `R_(2)=10 Omega` and `R_(1)=5Omega`. The other student takes `R_(2)=1000Omega` and `R_(1)=500Omega`. In the standard arm, both take `R_(3)=5Omega`. Both find `R=(R_(2))/(R_(1))R_(3)=10Omega` within errors.

Options

A. The errors of measurement of the two students are the same
B. Errors of measurement do depends on the accuracy with which `R_2 and R_1` can be measured
C. If the student uses large values of `R_2 and R_1` The currents through the arms will be feeble. This will make determination of null point accurately more difficult
D. Wheatstone bridge is a very accurate instrument and has no errors of measurement

Answer

To solve the problem of measuring an unknown resistance \( R \) using a Wheatstone bridge, we can follow these steps:

### Step 1: Understand the Wheatstone Bridge Configuration
The Wheatstone bridge consists of four resistors arranged in a diamond shape. The resistors are labeled as \( R_1 \), \( R_2 \), \( R_3 \), and \( R \) (the unknown resistance). The bridge is balanced when the ratio of the resistances is equal, allowing us to calculate \( R \).

### Step 2: Write the Formula for the Wheatstone Bridge
The formula for the unknown resistance \( R \) when the bridge is balanced is given by:
\[ R = \frac{R_2}{R_1} R_3 \]

### Step 3: Calculate \( R \) for the First Student
For the first student:
- \( R_1 = 5 \, \Omega \)
- \( R_2 = 10 \, \Omega \)
- \( R_3 = 5 \, \Omega \)

Substituting these values into the formula:
\[ R = \frac{10}{5} \times 5 = 2 \times 5 = 10 \, \Omega \]

### Step 4: Calculate \( R \) for the Second Student
For the second student:
- \( R_1 = 500 \, \Omega \)
- \( R_2 = 1000 \, \Omega \)
- \( R_3 = 5 \, \Omega \)

Substituting these values into the formula:
\[ R = \frac{1000}{500} \times 5 = 2 \times 5 = 10 \, \Omega \]

### Step 5: Conclusion
Both students calculated the unknown resistance \( R \) to be \( 10 \, \Omega \). This shows that the Wheatstone bridge can yield the same result even with different resistor values, as long as the ratios are maintained.

### Step 6: Discuss the Errors
The errors in measurement can arise from inaccuracies in the resistors used. The first student's resistors are smaller, while the second student's resistors are larger. However, both students arrived at the same value of \( R \) within their respective measurement errors.

### Final Answer
Thus, the unknown resistance \( R \) measured by both students is \( 10 \, \Omega \).

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