A nichrome wire uniform cross-sectional area is bent to form a rectangular loop ABCD. Another nichrome wire of the same cross-section is connected to form the diagonal AC. Find out the ratio of the resistances across BD and AC if AB = 0.4 m and BC = 0.3 m.

Answer

To find the ratio of the resistances across BD and AC in the given rectangular loop ABCD with a diagonal AC, we will follow these steps:

### Step 1: Determine the lengths of the sides of the rectangle
Given:
- Length AB = 0.4 m
- Length BC = 0.3 m

Using the Pythagorean theorem, we can find the length of the diagonal AC:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{(0.4)^2 + (0.3)^2} = \sqrt{0.16 + 0.09} = \sqrt{0.25} = 0.5 \, \text{m} \]

### Step 2: Calculate the resistances of each segment
The resistance \( R \) of a wire is given by the formula:
\[ R = \frac{\rho L}{A} \]
where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area. Since both wires are made of nichrome and have the same cross-sectional area, the resistance is directly proportional to the length.

- Resistance of AB:
\[ R_{AB} = k \times 0.4 \, \text{m} \quad (k \text{ is a constant depending on } \rho \text{ and } A) \]

- Resistance of BC:
\[ R_{BC} = k \times 0.3 \, \text{m} \]

- Resistance of AC:
\[ R_{AC} = k \times 0.5 \, \text{m} \]

### Step 3: Calculate the equivalent resistance across AC
The total resistance across AC can be calculated as:
\[ R_{AC} = R_{AB} + R_{BC} = k \times 0.4 + k \times 0.3 = k \times (0.4 + 0.3) = k \times 0.7 \]

### Step 4: Calculate the equivalent resistance across BD
For the path BD, we have:
- Resistance from B to A (AB): \( R_{AB} = k \times 0.4 \)
- Resistance from A to D (AD): \( R_{AD} = k \times 0.3 \)
- Resistance across AC: \( R_{AC} = k \times 0.5 \)

The resistances in series for BD are:
\[ R_{BD} = R_{AB} + R_{AD} + R_{AC} = k \times 0.4 + k \times 0.3 + k \times 0.5 = k \times (0.4 + 0.3 + 0.5) = k \times 1.2 \]

### Step 5: Find the ratio of resistances \( R_{BD} \) to \( R_{AC} \)
Now, we can find the ratio of the resistances:
\[ \text{Ratio} = \frac{R_{BD}}{R_{AC}} = \frac{k \times 1.2}{k \times 0.7} = \frac{1.2}{0.7} = \frac{12}{7} \]

### Final Result
Thus, the ratio of the resistances across BD and AC is:
\[ \frac{R_{BD}}{R_{AC}} = \frac{12}{7} \]