To solve this, we use Wien's Displacement Law, which states:

\[
\lambda_{\text{max}} = \frac{b}{T}
\]

Where:
- \(\lambda_{\text{max}}\) is the wavelength at which the maximum energy is emitted,
- \(b = 2.88 \times 10^6 \, \text{nmK}\) is Wien's constant,
- \(T = 5760 \, K\) is the temperature of the black body.

Step 1: Calculate \(\lambda_{\text{max}}\)
\[
\lambda_{\text{max}} = \frac{2.88 \times 10^6}{5760} = 500 \, \text{nm}
\]

This means that the maximum energy is emitted at a wavelength of 500 nm.

 Step 2: Compare the energies at different wavelengths

- At 500 nm (\( \lambda_{\text{max}} \)): This is where the black body emits the maximum energy. So, \(U_2\) will be the largest.
- At 250 nm: This wavelength is shorter than \(\lambda_{\text{max}}\), and the energy decreases as we move away from the peak, so \(U_1 < U_2\).
- At 1000 nm: This wavelength is longer than \(\lambda_{\text{max}}\), and the energy decreases further, so \(U_3 < U_2\).

Thus, \(U_2 > U_1\), and the correct comparison is \(U_2 > U_1\).

The color change in the heated iron from dull red to reddish yellow to white hot can be explained using Wein's Displacement Law. This law states that the wavelength (\(\lambda_{max}\)) at which the maximum emission occurs is inversely proportional to the absolute temperature \(T\) of the object:

\[
\lambda_{max} \propto \frac{1}{T}
\]

As the iron's temperature increases, the peak wavelength of emitted radiation shifts toward shorter wavelengths (from red to yellow and eventually to white, which is a combination of all visible wavelengths). Thus, the observed color change corresponds to an increase in temperature, with shorter wavelengths being emitted at higher temperatures.

The energy emitted by a black body is proportional to the fourth power of its absolute temperature, as per Stefan-Boltzmann Law:

\[
P \propto T^4
\]

Step 1: Convert temperature to kelvins

The temperature of the black body is \(727^\circ C\), which is:

\[
T = 727 + 273 = 1000\,K
\]

Step 2: Apply Stefan-Boltzmann Law

Since the energy emitted is proportional to \(T^4\):

\[
P \propto (1000)^4
\]

Thus, the rate of energy emission is proportional to \((1000)^4\).

To solve this, we use Stefan-Boltzmann Law for radiative heat transfer:

\[
P = \sigma A (T^4 - T_s^4)
\]

Where:
- \(P\) is the power (rate of energy loss),
- \(\sigma\) is the Stefan-Boltzmann constant,
- \(A\) is the area of the body,
- \(T\) is the temperature of the body in kelvins,
- \(T_s\) is the temperature of the surroundings in kelvins.

 Step 1: Convert temperatures to kelvins
- Initial temperature of the black body: \(727^\circ C = 727 + 273 = 1000\,K\)
- Surrounding temperature: \(227^\circ C = 227 + 273 = 500\,K\)
- New temperature of the black body: \(1227^\circ C = 1227 + 273 = 1500\,K\)

 Step 2: Ratio of power losses
Let the power at \(T_1 = 1000\,K\) be \(P_1 = 20\,W\). The new power at \(T_2 = 1500\,K\) is \(P_2\). Using the Stefan-Boltzmann law:

\[
\frac{P_2}{P_1} = \frac{T_2^4 - T_s^4}{T_1^4 - T_s^4}
\]

Substitute the values:

\[
\frac{P_2}{20} = \frac{1500^4 - 500^4}{1000^4 - 500^4}
\]

Step 3: Simplify the powers of temperatures
\[
\frac{P_2}{20} = \frac{1500^4 - 500^4}{1000^4 - 500^4}
\]

You can approximate the values:

\[
P_2 = 20 \times \frac{(1500^4 - 500^4)}{(1000^4 - 500^4)}
\]

After simplifying, you'll find:

\[
P_2 = \frac{320}{3} \text{ watts}
\]

Thus, the rate of energy loss at 1227°C is \(\frac{320}{3}\,W\).

Power is given by the product of intensity and area: \( P = I \times A \). Here \( I = 1400 \text{ W/m}^2 \) and \( A = 8\text{ m} \times 20\text{ m} = 160 \text{ m}^2 \). Therefore, \( P = 1400 \times 160 = 224,000 \text{ W} = 224 \text{ kW} \).

Assertion (A) is true according to Wien's displacement law, \(\lambda_{max} T = \text{constant}\). Reason (R) is false as \(\lambda_{max}\) is inversely proportional to temperature, not proportional to \(T^4\). \(E = \sigma T^4\) relates total emissive power to temperature.

Assertion (A) is false because cooling rate depends on material properties like specific heat capacity and emissivity, which differ for copper and aluminium. Reason (R) is false.

Newton's Law of Cooling shows rate depends on surface area and emissivity, not just temperature difference.

Assertion (A) is true due to Wien's displacement law; \(\lambda_{max}\) shifts to shorter wavelengths with increasing temperature. Reason (R) is true. Planck's law shows spectral emissive power increases disproportionately for different wavelengths with temperature. (R) explains (A) as this disproportionate increase causes the color shift.

Assertion (A) is true. Hollow sphere has less mass ( m ) but same surface area ( A ) as solid sphere. Reason (R) is true, cooling rate \( \frac{dT}{dt} \propto \frac{A}{m} \). Since ( A ) is constant, \( \frac{dT}{dt} \propto \frac{1}{m} \). Lower mass ( m ) means faster cooling. Thus, both are true and (R) explains (A).

Assertion (A) is true because a thin plate has a larger surface area-to-mass ratio. Reason (R) states cooling rate \( \propto \) surface area, which is true \( \frac{dT}{dt} \propto \frac{A}{m}) \), and that a plate has the maximum surface area (for a given mass). Thus, both are true and (R) correctly explains (A).