Bent Wire and Effective Resistance in a Circle

To find the effective resistance between the two points at the end of any diameter of a closed circle formed by a wire with a resistance of 20 ohms, we need to consider the concept of series and parallel resistances.

Understanding the Wire's Configuration
When the wire is bent in the form of a closed circle, it essentially forms a loop. This loop can be thought of as a combination of resistors in series and parallel.

Resistors in Series
When resistors are connected in series, their resistances add up. In the case of the wire loop, each segment of the wire contributes 20 ohms to the total resistance because the wire has a resistance of 20 ohms.

Resistors in Parallel
When resistors are connected in parallel, their effective resistance can be calculated using the formula:

1/Reff = 1/R1 + 1/R2 + 1/R3 + ...

Since the wire loop is closed, any diameter of the circle divides the loop into two halves. The two halves can be thought of as resistors in parallel.

Calculating the Effective Resistance
To find the effective resistance between the two points at the end of any diameter, we can follow these steps:

1. Divide the loop into two halves using a diameter.
2. Treat each half as a resistor in parallel.
3. Calculate the effective resistance of each half using the formula mentioned earlier.
4. Add the effective resistances of both halves to find the total effective resistance.

Since each half is a resistor in parallel, the effective resistance of each half can be calculated using the formula:

1/Reff = 1/20 + 1/20

Simplifying the equation:

1/Reff = 2/20

1/Reff = 1/10

Reff = 10 ohms

Therefore, the effective resistance between the two points at the end of any diameter of the circle is 10 ohms.

Conclusion
When a wire with a resistance of 20 ohms is bent in the form of a closed circle, the effective resistance between the two points at the end of any diameter of the circle is 10 ohms. This is because the loop can be divided into two halves, each acting as a resistor in parallel. By calculating the effective resistance of each half and adding them together, we find the total effective resistance.