When a conducting wire is connected in the left gap and known resistance in the right gap, the balance length is 75cm. If the wire is cut into 3 equal parts and one part isconnected in the left gap, the balance length

Options

A. shifts left by 25cms
B. shifts right by 25cms
C. shifts left by 50cms
D. shifts right by 50cms

Answer

To solve the problem step by step, we will analyze the situation using the principles of a meter bridge and the relationship between resistance and balance length.

### Step 1: Understanding the Initial Setup
When a conducting wire is connected in the left gap and a known resistance (let's call it R) is in the right gap, the balance length (L) is given as 75 cm.

**Hint:** Remember that in a meter bridge, the balance condition states that the ratio of the resistances is equal to the ratio of the lengths.

### Step 2: Setting Up the Equation for Initial Balance
Using the balance condition:
\[ \frac{R}{y} = \frac{75}{100 - 75} = \frac{75}{25} = 3 \]
From this, we can deduce:
\[ R = 3y \]

**Hint:** The lengths on the meter bridge correspond to the resistances in the circuit.

### Step 3: Cutting the Wire into Equal Parts
The wire is cut into three equal parts. Therefore, the length of one part (let's call it L') is:
\[ L' = \frac{L}{3} = \frac{100 \text{ cm}}{3} \approx 33.33 \text{ cm} \]
The resistance of one part of the wire (let's call it R') can be expressed as:
\[ R' = \frac{R}{3} = \frac{3y}{3} = y \]

**Hint:** Cutting the wire into equal parts divides its resistance by the same factor.

### Step 4: Setting Up the Equation for New Balance
Now, we connect one part of the wire (R') in the left gap and the known resistance (R = 3y) remains in the right gap. The new balance length (let's call it L'') can be found using the balance condition again:
\[ \frac{R'}{R} = \frac{L''}{100 - L''} \]
Substituting the values we have:
\[ \frac{y}{3y} = \frac{L''}{100 - L''} \]
This simplifies to:
\[ \frac{1}{3} = \frac{L''}{100 - L''} \]

**Hint:** Use cross-multiplication to solve the equation.

### Step 5: Solving for the New Balance Length
Cross-multiplying gives us:
\[ L'' = \frac{1}{3}(100 - L'') \]
Multiplying through by 3 to eliminate the fraction:
\[ 3L'' = 100 - L'' \]
Combining like terms:
\[ 4L'' = 100 \]
Thus, we find:
\[ L'' = 25 \text{ cm} \]

**Hint:** Make sure to isolate L'' to find its value.

### Step 6: Conclusion
The balance length when one part of the wire is connected in the left gap is 25 cm.

**Final Answer:** The new balance length is 25 cm.