Problem:

A battery of 10V and negligible internal resistance is connected across the diagonally opposite corners of a cubical network consisting of 12 resistors, each of 1 Ohm resistance. Find the equivalent resistance and total current.

Solution:

To solve this problem, we need to determine the equivalent resistance and total current in the given network. We can break down the solution into the following steps:

Step 1: Identifying the Network:
The given network is a cubical network consisting of 12 resistors. Each resistor has a resistance of 1 Ohm. The battery is connected across the diagonally opposite corners of the network.

Step 2: Identifying the Diagonal Resistors:
Since the battery is connected across the diagonally opposite corners of the network, we can identify the diagonal resistors. In a cube, the diagonally opposite corners are connected by three mutually perpendicular diagonals. Therefore, each diagonal consists of 4 resistors.

Step 3: Identifying the Diagonal Resistors in the Network:
In the given network, we can identify three mutually perpendicular diagonals. Let's label them as diagonal A, diagonal B, and diagonal C. Each diagonal consists of 4 resistors. We can represent the network as follows:

```
A B C

|------| |------| |------|
| | | | | |
| 1 |---2---| 3 |---4---| 5 |
| | | | | |
|------| |------| |------|

|------| |------| |------|
| | | | | |
| 6 |---7---| 8 |---9---| 10 |
| | | | | |
|------| |------| |------|

|------| |------| |------|
| | | | | |
| 11 |--12---| 13 |--14---| 15 |
| | | | | |
|------| |------| |------|
```

Step 4: Reducing the Network:
To find the equivalent resistance, we need to reduce the given network. We can do this by replacing diagonal A, B, and C with a single equivalent resistor.

Step 5: Finding the Equivalent Resistance:
To find the equivalent resistance of the network, we need to calculate the resistance between the diagonally opposite corners of the cube.

Step 6: Applying the Formula:
In a cube, the resistance between diagonally opposite corners is given by the formula:
R_eq = 2 * (R1 + R2 + R3)

Since each resistor has a resistance of 1 Ohm, we can substitute the values in the formula:
R_eq = 2 * (1 + 1 + 1) = 2 * 3 = 6 Ohms

Therefore, the equivalent resistance of the network is 6 Ohms.