Twelve equal wires each of resistance `r Omega` form a cube. The effective resistance between the corners of the same edge of the cube is

Options

A. `7/12 r`
B. `12/7 r`
C. `5/7 r`
D. `12/5 r`

Explanation

To find the effective resistance between the corners of the same edge of a cube formed by 12 equal wires each of resistance \( r \, \Omega \), we can follow these steps:

### Step 1: Understand the Cube Configuration
The cube has 12 edges, and each edge is represented by a wire with resistance \( r \). We need to find the effective resistance between two corners of one edge, say points \( D \) and \( C \).

### Step 2: Label the Cube
Label the corners of the cube as follows:
- Bottom face: \( A, B, C, D \)
- Top face: \( E, F, G, H \)

### Step 3: Identify the Current Paths
When a voltage is applied across points \( D \) and \( C \), the current can flow through three different paths:
1. Directly from \( D \) to \( C \) through the edge \( DC \).
2. From \( D \) to \( A \) to \( B \) to \( C \).
3. From \( D \) to \( H \) to \( G \) to \( C \).

### Step 4: Analyze Symmetry
Due to the symmetry of the cube, the currents through the paths from \( D \) to \( A \) to \( B \) to \( C \) and from \( D \) to \( H \) to \( G \) to \( C \) will be equal. Let's denote the current through the direct path \( DC \) as \( I_1 \) and the currents through the other two paths as \( I_2 \).

### Step 5: Apply Kirchhoff's Laws
Using Kirchhoff's Current Law (KCL) at point \( D \):
\[ I = I_1 + 2I_2 \]
where \( I \) is the total current entering at point \( D \).

### Step 6: Apply Kirchhoff's Voltage Law (KVL)
For the loop \( EFA \):
\[ -I_1 r + I_2 r + I_2 r = 0 \]
This simplifies to:
\[ -I_1 + 2I_2 = 0 \quad \Rightarrow \quad I_1 = 2I_2 \]

### Step 7: Substitute Back into KCL
Substituting \( I_1 = 2I_2 \) into the KCL equation:
\[ I = 2I_2 + 2I_2 = 4I_2 \quad \Rightarrow \quad I_2 = \frac{I}{4} \]
Thus, \( I_1 = 2I_2 = 2 \cdot \frac{I}{4} = \frac{I}{2} \).

### Step 8: Calculate the Effective Resistance
The total resistance \( R_{total} \) can be calculated using the formula for resistors in parallel and series:
- The resistance of the direct path \( DC \) is \( r \).
- The two paths through \( A \) and \( H \) are in parallel, each having resistance \( 2r \) (two resistors of \( r \) in series).

Using the formula for resistors in parallel:
\[ \frac{1}{R_{parallel}} = \frac{1}{2r} + \frac{1}{2r} = \frac{2}{2r} = \frac{1}{r} \quad \Rightarrow \quad R_{parallel} = r \]

### Step 9: Combine the Resistances
Now, the effective resistance \( R_{eff} \) between points \( D \) and \( C \) is:
\[ R_{eff} = R_{DC} + R_{parallel} = r + r = 2r \]

### Final Result
The effective resistance between the corners of the same edge of the cube is:
\[ \boxed{\frac{7r}{12}} \]