A photovoltaic plate with a power output of 30 mW cm⁻² and an efficiency of 40%. i) Determine the amount of solar energy falling on it per unit time. ii) How many such plates can supply energy directl

    $$
    I_{out} = 30 \text{ mW cm}^{-2} = 30 \times 10^{-3} \text{ W} \times \left(\frac{1 \text{ m}}{100 \text{ cm}}\right)^2
    $$
    $$
    I_{out} = 30 \times 10^{-3} \text{ W} \times \frac{10^4}{\text{ m}^2} = 300 \text{ Wm}^{-2}
    $$

    The efficiency ($\eta$) is given by:
    $$
    \eta = \frac{\text{Power output per unit area} (I_{out})}{\text{Power input per unit area} (I_{in})}
    $$
    Rearranging for $I_{in}$:
    $$
    I_{in} = \frac{I_{out}}{\eta}
    $$

    Given $\eta = 40\% = 0.40$.
    $$
    I_{in} = \frac{300 \text{ Wm}^{-2}}{0.40}
    $$
    $$
    I_{in} = 750 \text{ Wm}^{-2}
    $$
    The amount of solar energy falling on it per unit time (per unit area) is $\boxed{\text{750 Wm}^{-2}}$.

*   ii) How many such plates can supply energy directly to a 3 horse power electric pump switched on for 20 minutes if the intensity of solar radiation is assumed constant within this time and $1 \text{ hp} = 800 \text{ W}$?

    Given pump power = 3 hp.
    $$
    P_{pump} = 3 \text{ hp} \times 800 \text{ W/hp}
    $$
    $$
    P_{pump} = 2400 \text{ W}
    $$

    $$
    A_{plate} = 1.5 \text{ m} \times 1.0 \text{ m}
    $$
    $$
    A_{plate} = 1.5 \text{ m}^2
    $$

    Using the power output per unit area $I_{out} = 300 \text{ Wm}^{-2}$ from part (i).
    $$
    P_{out,plate} = I_{out} \times A_{plate}
    $$
    $$
    P_{out,plate} = 300 \text{ Wm}^{-2} \times 1.5 \text{ m}^2
    $$
    $$
    P_{out,plate} = 450 \text{ W}
    $$

    $$
    N = \frac{\text{Total power required by pump}}{\text{Power output of one plate}}
    $$
    $$
    N = \frac{2400 \text{ W}}{450 \text{ W}}
    $$
    $$
    N = 5.333...
    $$
    Since you cannot use a fraction of a plate, you must round up to ensure the pump receives sufficient power.  
    The number of plates required is $\boxed{\text{6 plates}}$.