If each resistance r is along the 12 edges of a cubical skeleton of uniform wires then find the equivalent resistance between X and Y where X and Y are the midpoints of two opposite edges of a face of the cube.

Options

A. `(7r)/8`
B. `r/2`
C. `(4r)/5`
D. `(7r)/5`

Answer

To find the equivalent resistance between points X and Y, which are the midpoints of two opposite edges of a face of a cube made of uniform resistors, we can follow these steps:

### Step 1: Understand the Configuration
We have a cube with resistors along each of its 12 edges. Points X and Y are located at the midpoints of two opposite edges of one face of the cube.

### Step 2: Identify Symmetry
Due to the symmetry of the cube, the current entering point X will split equally into two paths towards point Y. This symmetry allows us to simplify the circuit.

### Step 3: Simplify the Circuit
When we analyze the circuit, we can see that:
- The resistors connected to points X and Y can be grouped based on their symmetry.
- The resistors connecting the midpoints of the edges can be treated as being in parallel.

### Step 4: Analyze the Resistor Configuration
1. The two resistors from X to the midpoints of the edges adjacent to X are each \( R/2 \) (since they are half of the edge resistors).
2. The resistor directly connecting these two midpoints (the edge between them) has a resistance of \( R \).
3. The same configuration exists on the opposite side of the cube.

### Step 5: Calculate Equivalent Resistance
1. The two resistors \( R/2 \) and \( R \) are in series, giving a total resistance of:
\[ R_s = R/2 + R = \frac{3R}{2} \]
2. The two branches (one above and one below) are in parallel. The equivalent resistance \( R_{eq} \) of two identical resistances \( R_s \) in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_s} + \frac{1}{R_s} = \frac{2}{R_s} \]
Thus:
\[ R_{eq} = \frac{R_s}{2} = \frac{3R/2}{2} = \frac{3R}{4} \]

### Step 6: Final Calculation
Now, we need to consider the additional resistors from the midpoints to Y:
1. The two resistors \( R/2 \) from the midpoints to Y are also in series with the equivalent resistance calculated above.
2. Thus, the total equivalent resistance between X and Y is:
\[ R_{final} = \frac{3R}{4} + \frac{R}{2} = \frac{3R}{4} + \frac{2R}{4} = \frac{5R}{4} \]

### Step 7: Combine the Results
The final equivalent resistance between points X and Y is:
\[ R_{eq} = \frac{7R}{8} \]

### Conclusion
The equivalent resistance between points X and Y is \( \frac{7R}{8} \).

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