There is an infinite triangular lattice; the edges of the lattice are made of wire, and the resistance of each edge is R. Let us denote the corners of a triangular lattice face by A, B, and C. The wire connecting B and
C
is cut off. Determine the resistance between A and B. (a.)
To determine the resistance between A and B, we can consider the equivalent resistance between A and B in the triangular lattice with the wire connecting B and C cut off.
Without loss of generality, let's assign a resistance of R/2 to the wire connecting B and C. This is because cutting off the wire effectively results in two parallel resistors, each with a resistance of R/2.
Now, let's consider the triangular lattice with the wire connecting B and C cut off. We can see that there are two paths from A to B:
1. Path 1: A --> B directly
2. Path 2: A --> C --> B
Path 1 has a total resistance of R/2, as there is only one resistor along this path.
Path 2 has a total resistance of 2(R/2) = R, as there are two resistors in parallel along this path.
Since the two paths are in parallel, the equivalent resistance between A and B is given by the formula:
1/Req = 1/(R/2) + 1/R
Simplifying this expression gives:
1/Req = 2/R + 1/R
1/Req = 3/R
Therefore, the equivalent resistance between A and B is R/3.
Answer: (a.) R/3
To determine the resistance between points A and B after the wire connecting B and C is cut off, we can use the principle of electrical circuit analysis.
1. We will assume that the resistance along each edge is R.
2. Since the wire connecting B and C is cut off, we can consider the triangular lattice as three resistors connected in series: A-B, A-C, and C-B.
3. The resistance between A and B can be found using the formula for resistors in series:
R_total = R1 + R2 + R3
4. The resistance of A-B is simply R, the resistance of each individual edge.
5. For A-C and C-B, each resistor is traversed twice in the triangular lattice, so their resistance is twice as much. Therefore, A-C and C-B each have a resistance of 2R.
6. Substituting the values into the formula, we have:
R_total = R + 2R + 2R = 5R
Therefore, the resistance between points A and B after the wire connecting B and C is cut off is 5R.
To determine the resistance between point A and point B, we can use the concept of equivalent resistance.
Since the wire connecting points B and C is cut off, we can imagine the lattice being divided into two separate parts: one containing points A and B, and the other containing points B and C.
To find the equivalent resistance between points A and B, we need to analyze the resistances in the lattice connections. Each individual wire in an infinite triangular lattice has a resistance of R.
Let's consider the lattice segment between points A and B. We can see that there are two possible paths to reach point B from point A, as illustrated below:
A
/ \
/ R \
C ------ B
We can calculate the resistance of each path using the formula for resistors in series:
For the upper path:
- Resistance from A to C = R (as there is only one wire between A and C)
- Resistance from C to B = R (as there is only one wire between C and B)
- Total resistance for upper path = R + R = 2R
For the lower path:
- Resistance from A to B = 2R (as there are two parallel wires between A and B)
Now, we have two possible paths with different resistances. To find the equivalent resistance between points A and B, we need to determine which path has the lower resistance.
Comparing the resistances, we can see that the lower path has a lower resistance of 2R, while the upper path has a higher resistance of 2R. Therefore, the current flowing from point A to point B will choose the lower path since it offers the path of least resistance.
Hence, the resistance between points A and B in the given lattice, with the wire between points B and C cut off, is 2R.