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Two rooms, each a cube \SI{4.0}{m} per side, share a \SI{12}{cm} thick brick wall (\SI{0.75}{\watt\per\meter\per\kelvin}). Because of a number of \SI{100}{W} lightbulbs in one room, the air is at \SI{30}{\degreeCelsius}, while in the other room it is at \SI{10}{\degreeCelsius}. How many lightbulbs are needed to maintain the temperature difference across the wall?
\Geg{ \ell &= \SI{4.0}{m}\\ d &= \SI{12}{cm} =\SI{0.12}{m}\\ \lambda &= \SI{0.75}{\watt\per\meter\per\kelvin}\\ P_1 &= \SI{100}{W}\\ T_2&= \SI{30}{\degreeCelsius} =\SI{303}{K}\\ T_1&= \SI{10}{\degreeCelsius} =\SI{283}{K} } \Ges{Number (of lightbulbs)}{[N]=\si{}} Tbe heat transferred through the brick wall is: \begin{align} \Phi &= \lambda \frac{T_2T_1}{d}A\\ &= \SI{2.0e3}{W} \end{align} Hence, a number of \begin{align} N &= \frac{\Phi}{P_1} = \frac{\lambda \frac{T_2T_1}{d}A}{P_1} = \frac{\lambda (T_2T_1)A}{dP_1}\\ &= 20 \end{align} light bulbs would be sufficient to maintain the temperature. \Lsg{ N &= \frac{\lambda (T_2T_1)A}{dP_1}\\ &= 20 }
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