An electric heater with the total surface area of \(0.25 \mathrm{~m}^{2}\) and emissivity \(0.75\) is in a room where the air has a temperature of \(20^{\circ} \mathrm{C}\) and the walls are at \(10^{\circ} \mathrm{C}\). When the heater consumes \(500 \mathrm{~W}\) of electric power, its surface has a steady temperature of \(120^{\circ} \mathrm{C}\). Determine the temperature of the heater surface when it consumes \(700 \mathrm{~W}\). Solve the problem (a) assuming negligible radiation and (b) taking radiation into consideration. Based on your results, comment on the assumption made in part ( \(a\) ).

Let's start by analyzing the situation when we assume negligible radiation. In this case, 100% of the delivered power (P) is dedicated to heating the room's air. From the steady-state situation, we can establish a relationship between temperature and power consumption: \(P_1 = K (T_1 - T_\text{air})\) Here, K is a constant representing the heat transfer from the heater's surface to the air. Since we don't take into account the radiation, the value of K should remain constant when the power consumption increases. Now, we can use the same relationship for the problem scenario: \(P_2 = K (T_2 - T_\text{air})\) From these two relationships, we can find T2: \(\frac{P_1}{P_2} = \frac{T_1 - T_\text{air}}{T_2 - T_\text{air}}\) Solve for T2: \(T_2 = T_\text{air} + \frac{P_2}{P_1} (T_1 - T_\text{air})\) Finally, plug in the given values to determine T2.

Now we will take radiation into account. The Stefan-Boltzmann law helps us determine the power radiated from the heater's surface. The Stefan-Boltzmann law states that the radiated power is given by: \(P_\text{rad} = \epsilon \sigma A (T^4 - T_\text{wall}^4)\) where \(\sigma\) is the Stefan-Boltzmann constant: \(5.67 \times 10^{-8} \mathrm{W m^{-2} K^{-4}}\). As the heat given to the room's air is now decreased, we have: \(P_\text{air} = P - P_\text{rad}\) The relationships between power, temperature, and radiation can be established for the steady-state case and the problem scenario: For the initial steady-state case: \(P_1 = K (T_1 - T_\text{air}) + \epsilon \sigma A (T_1^4 - T_\text{wall}^4)\) For the problem scenario: \(P_2 = K (T_2 - T_\text{air}) + \epsilon \sigma A (T_2^4 - T_\text{wall}^4)\) There is no closed-form analytical solution for T2 in this case, so we need to solve the above equation numerically using a numerical method such as the Newton-Raphson method or a root-finding algorithm. Then, compare the results obtained in parts (a) and (b) and comment based on the differences in surface temperatures.