Step 1: Convert the Persian wheel's discharge to a per hour rate first.
The discharge is given as 230 litres per minute, so multiplying by 60 gives the hourly rate: \[ 230 \times 60 = 13800 \text{ litres/hour} = 13.8 \text{ m}^3/\text{hour} \]
Step 2: Multiply by the total operating time to get the total volume delivered.
The wheel runs for 50 hours, so the total volume of water applied to the field is \[ V = 13.8 \text{ m}^3/\text{hr} \times 50 \text{ hr} = 690 \text{ m}^3 \]
Step 3: Convert the irrigated area to square metres and divide.
The area is 1 hectare, which is \( 10{,}000 \text{ m}^2 \), so the average depth spread over the field is \[ d = \frac{690 \text{ m}^3}{10{,}000 \text{ m}^2} = 0.069 \text{ m} \]
Step 4: Convert to centimetres to match the answer choices.
\[ 0.069 \text{ m} \times 100 = 6.9 \text{ cm} \] which lines up exactly with one of the given options.
\[ \boxed{6.9 \text{ cm}} \]