Question

The walls of a closed cubical box of edge 40 cm are made of a material of thickness 1 mm and thermal conductivity 4 x 10⁻⁴ cals⁻¹cm⁻¹ °C⁻¹. The interior of the box is maintained at 100°C above the outside temperature by a heater placed inside the box and connected across 400 V dc.

• A. 9.96 Ω
• B. 10.00 Ω
• C. 9.50 Ω
• D. 10.50 Ω

Answer 1

Answer: (A)

9.96 Ω

Answer 2

The rate of heat loss through the walls of the cubical box by conduction is given by Fourier's Law: $P_{\text{loss}} = k A \frac{\Delta T}{x}$ where $k$ is the thermal conductivity, $A$ is the surface area of the box, $\Delta T$ is the temperature difference across the walls, and $x$ is the thickness of the walls.

The surface area of the cubical box with edge length $a = 40$ cm is $A = 6a^2 = 6 \times (40 \text{ cm})^2 = 6 \times 1600 \text{ cm}^2 = 9600 \text{ cm}^2$. The thickness of the walls is $x = 1$ mm $= 0.1$ cm. The temperature difference is $\Delta T = 100$ °C. The thermal conductivity is $k = 4 \times 10^{-4}$ cal s⁻¹ cm⁻¹ °C⁻¹. To convert this to SI units (Joules per second per cm per °C), we use the conversion factor $1$ cal $= 4.184$ J: $k = 4 \times 10^{-4} \times 4.184$ J s⁻¹ cm⁻¹ °C⁻¹ $= 1.6736 \times 10^{-3}$ J s⁻¹ cm⁻¹ °C⁻¹.

Now, calculate the rate of heat loss: $P_{\text{loss}} = (1.6736 \times 10^{-3} \text{ J s}^{-1} \text{ cm}^{-1} {^{\circ}C}^{-1}) \times (9600 \text{ cm}^2) \times \frac{100 \text{ °C}}{0.1 \text{ cm}}$ $P_{\text{loss}} = 1.6736 \times 10^{-3} \times 9600 \times 1000 \text{ J s}^{-1}$ $P_{\text{loss}} = 16066.56 \text{ W}$ This is the power dissipated by the heater.

The power generated by the heater is also given by $P_{\text{heater}} = \frac{V^2}{R}$, where $V$ is the voltage and $R$ is the resistance. In steady state, $P_{\text{heater}} = P_{\text{loss}}$. $\frac{V^2}{R} = 16066.56 \text{ W}$ Given $V = 400$ V: $R = \frac{V^2}{16066.56 \text{ W}} = \frac{(400 \text{ V})^2}{16066.56 \text{ W}}$ $R = \frac{160000}{16066.56} \Omega \approx 9.9585 \Omega$ The resistance of the heater is approximately $9.96 \Omega$.