an infinite ladder network of resistance is constructed with 1 ohm and 2 ohm resistances .calculate equivalent resistance between points a and b
To calculate the equivalent resistance between points A and B in an infinite ladder network, we can follow these steps:
Step 1: Identify the repeating pattern of the ladder network.
In this case, we have the repeating pattern of 1 ohm and 2 ohm resistors. The ladder network can be depicted as follows:
|-----1 ohm-----|-----2 ohm-----|
| |
| |
A ---|-----2 ohm-----|-----1 ohm-----|--- B
| |
| |
|-----1 ohm-----|-----2 ohm-----|
Step 2: Simplify the repeating pattern.
We can replace the repeating pattern with an equivalent resistor, let's say R_eq, between A and B.
Step 3: Determine the value of the equivalent resistor R_eq.
To do this, we need to consider the resistance between points A and B and divide it into two parts.
Part 1: The resistances on the left-hand side of R_eq.
Between A and the first vertical line, there's a 1 ohm resistor. Between the first and second vertical lines, there's a 2 ohm resistor. This pattern repeats indefinitely. We can sum up the resistances in that part as follows:
R_left = 1 ohm + 2 ohm + 1 ohm + 2 ohm + ...
Part 2: The resistances on the right-hand side of R_eq.
Between B and the last vertical line, there's a 2 ohm resistor. Between the last and second-to-last vertical lines, there's a 1 ohm resistor. This pattern also repeats indefinitely. We can sum up the resistances in that part as follows:
R_right = 2 ohm + 1 ohm + 2 ohm + 1 ohm + ...
Step 4: Calculate the value of R_eq.
The equivalent resistance R_eq can be calculated using the formula for infinite geometric series:
R_eq = R_left + R_right
= (1 + 2 + 1 + 2 + ...) ohm + (2 + 1 + 2 + 1 + ...) ohm
= (1 + 2 + 1 + 2 + ...) ohm + (2 + 1 + 2 + 1 + ...) ohm
To find the sum of these infinite series, we need to calculate the sum of the first two terms and then divide by 1 minus the common ratio.
Sum of first two terms = 1 + 2 = 3
Common ratio = 2
Using the formula for the sum of an infinite geometric series:
R_eq = (Sum of first two terms) / (1 - Common ratio)
= 3 / (1 - 2)
= 3 / (-1)
= -3 ohm
Therefore, the equivalent resistance between points A and B in the infinite ladder network is -3 ohms.
To calculate the equivalent resistance between points A and B in an infinite ladder network of 1 ohm and 2 ohm resistances, we can use the concept of series and parallel resistances.
Here's how you can approach the problem:
1. Imagine dividing the ladder network into segments like this:
```
____ 1Ω ____ 2Ω ____ 1Ω ____
| |
|A| | Segments | |B|
|____ 1Ω ____ 2Ω ____ 1Ω ____
```
2. Start by calculating the resistance of each segment. In this case, we have three segments: a 1Ω resistor, a 2Ω resistor, and a repeated pattern of another 1Ω resistor followed by a 2Ω resistor.
- Segment 1: 1Ω resistor
- Segment 2: 2Ω resistor
- Segment 3: 1Ω resistor followed by a 2Ω resistor (repeated pattern)
3. To calculate the equivalent resistance of Segment 1 (1Ω resistor), it is simply the resistance value of the resistor itself, which is 1Ω.
4. To calculate the equivalent resistance of Segment 2 (2Ω resistor), it is also the resistance value of the resistor itself, which is 2Ω.
5. To calculate the equivalent resistance of Segment 3 (1Ω resistor followed by a 2Ω resistor), we can find the equivalent resistance of the two resistors in parallel. The formula for calculating the equivalent resistance of two resistors in parallel is:
```
1 / Req = 1 / R1 + 1 / R2
```
In this case, the equivalent resistance (Req) is the resistance between the points where the two resistors meet. Therefore:
```
1 / Req = 1 / 1Ω + 1 / 2Ω
```
To simplify the expression, you can find the least common multiple (LCM) of the denominators, which is 2Ω in this case. Then, multiply each term by the LCM:
```
1 / Req = 2 / 2Ω + 1 / 2Ω
= 3 / 2Ω
```
Taking the reciprocal, you get:
```
Req = 2Ω / 3
```
6. Now, to find the equivalent resistance between points A and B, we can treat the ladder network as an infinite series of these three segments. Since the pattern keeps repeating, you can see that each segment is identical. Therefore, the equivalent resistance between points A and B is the sum of the equivalent resistances of all three segments.
```
Req(A to B) = Segment 1 + Segment 2 + Segment 3 + ...
= 1Ω + 2Ω + (1Ω || 2Ω) + ...
= 1Ω + 2Ω + (2Ω / 3)
```
Simplifying the expression:
```
Req(A to B) = 3Ω + 2Ω / 3
= (9Ω + 2Ω) / 3
= 11Ω / 3
```
Therefore, the equivalent resistance between points A and B in the infinite ladder network of 1Ω and 2Ω resistors is 11/3 ohms.