Problem:
A wire of resistance 0.5 ohm is bent into a circle of radius 1 m. The same wire is connected across a diameter AB as shown in the figure. Find the equivalent resistance.

Solution:

Given:
Resistance of the wire, R = 0.5 ohm
Radius of the circle, r = 1 m

Approach:
To find the equivalent resistance, we need to analyze the circuit and determine the effective resistance. We can break down the problem into two parts:

1. Resistance of the circular part.
2. Resistance across the diameter.

1. Resistance of the circular part:
The given wire is bent into a circle. The resistance of the circle can be calculated using the formula: R = ρ × (L/A), where ρ is the resistivity of the wire, L is the length of the wire, and A is the cross-sectional area of the wire. Since the wire is the same throughout, ρ and A will remain constant.

In this case, the wire is bent into a circle of radius 1 m. The circumference of the circle will be 2πr. Therefore, the length of the wire (L) will be 2πr.

Substituting the values into the formula, we have:
R_circle = ρ × (L/A) = ρ × (2πr/A)

2. Resistance across the diameter:
The wire is connected across the diameter AB. In this case, the length of the wire (L) will be equal to the diameter of the circle, which is 2r.

Using the resistance formula, we have:
R_diameter = ρ × (L/A) = ρ × (2r/A)

Combining the resistances:
To find the equivalent resistance, we need to add the resistances of the circular part and the diameter:
R_equivalent = R_circle + R_diameter

Substituting the values, we have:
R_equivalent = ρ × (2πr/A) + ρ × (2r/A)

Simplifying the expression:
R_equivalent = ρ × (2πr + 2r) / A

Conclusion:
The equivalent resistance of the given circuit can be calculated using the formula R_equivalent = ρ × (2πr + 2r) / A, where ρ is the resistivity of the wire, r is the radius of the circle, and A is the cross-sectional area of the wire.