Answer :
To solve this problem, we need to calculate the resistance of each layer and then determine the relative rate of heat dissipation, which is proportional to the resistance in each layer when layers are in series.
Here are the steps to find the solution:
1. Convert Thickness to Meters:
- Skin Thickness: [tex]\( 0.01 \, \text{mm} = 0.01 \times 10^{-3} \, \text{m} = 0.00001 \, \text{m} \)[/tex]
- Fat Thickness: [tex]\( 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} = 0.002 \, \text{m} \)[/tex]
- Muscle Thickness: [tex]\( 10 \, \text{mm} = 10 \times 10^{-3} \, \text{m} = 0.01 \, \text{m} \)[/tex]
2. Given Resistivity:
- Resistivity of Skin: [tex]\( 10^6 \, \Omega \cdot m \)[/tex]
- Resistivity of Fat: [tex]\( 15 \, \Omega \cdot m \)[/tex]
- Resistivity of Muscle: [tex]\( 2 \, \Omega \cdot m \)[/tex]
3. Calculate Resistance for Each Layer Using [tex]\( R = \rho \frac{L}{A} \)[/tex]:
- Since all layers have the same cross-sectional area, it cancels out in the ratio calculations, so we can simplify to [tex]\( R = \rho \times \text{thickness} \)[/tex].
- Resistance of Skin:
[tex]\[ R_{\text{skin}} = 10^6 \times 0.00001 = 10 \, \Omega \][/tex]
- Resistance of Fat:
[tex]\[ R_{\text{fat}} = 15 \times 0.002 = 0.03 \, \Omega \][/tex]
- Resistance of Muscle:
[tex]\[ R_{\text{muscle}} = 2 \times 0.01 = 0.02 \, \Omega \][/tex]
4. Total Resistance:
- Since the layers are in series, the total resistance [tex]\(R_{\text{total}}\)[/tex] is the sum of individual resistances:
[tex]\[ R_{\text{total}} = R_{\text{skin}} + R_{\text{fat}} + R_{\text{muscle}} \][/tex]
[tex]\[ R_{\text{total}} = 10 + 0.03 + 0.02 = 10.05 \, \Omega \][/tex]
5. Relative Rate of Heat Dissipation:
The rate of heat dissipation in each layer is proportional to the resistance of that layer. We find the relative rate by dividing the resistance of each layer by the total resistance:
- Skin:
[tex]\[ \text{Relative Rate}_{\text{skin}} = \frac{R_{\text{skin}}}{R_{\text{total}}} = \frac{10}{10.05} \approx 0.9950 \][/tex]
- Fat:
[tex]\[ \text{Relative Rate}_{\text{fat}} = \frac{R_{\text{fat}}}{R_{\text{total}}} = \frac{0.03}{10.05} \approx 0.003 \][/tex]
- Muscle:
[tex]\[ \text{Relative Rate}_{\text{muscle}} = \frac{R_{\text{muscle}}}{R_{\text{total}}} = \frac{0.02}{10.05} \approx 0.0020 \][/tex]
Summarizing, the relative rates of heat dissipation in each layer are approximately:
- Skin: [tex]\( 0.9950 \)[/tex] or [tex]\( 99.50\% \)[/tex]
- Fat: [tex]\( 0.003 \)[/tex] or [tex]\( 0.30\% \)[/tex]
- Muscle: [tex]\( 0.002 \)[/tex] or [tex]\( 0.20\% \)[/tex]
Thus, the layer with the highest relative rate of heat dissipation is the skin, followed by fat, and then muscle. This aligns with the relative resistances of each layer.
Here are the steps to find the solution:
1. Convert Thickness to Meters:
- Skin Thickness: [tex]\( 0.01 \, \text{mm} = 0.01 \times 10^{-3} \, \text{m} = 0.00001 \, \text{m} \)[/tex]
- Fat Thickness: [tex]\( 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} = 0.002 \, \text{m} \)[/tex]
- Muscle Thickness: [tex]\( 10 \, \text{mm} = 10 \times 10^{-3} \, \text{m} = 0.01 \, \text{m} \)[/tex]
2. Given Resistivity:
- Resistivity of Skin: [tex]\( 10^6 \, \Omega \cdot m \)[/tex]
- Resistivity of Fat: [tex]\( 15 \, \Omega \cdot m \)[/tex]
- Resistivity of Muscle: [tex]\( 2 \, \Omega \cdot m \)[/tex]
3. Calculate Resistance for Each Layer Using [tex]\( R = \rho \frac{L}{A} \)[/tex]:
- Since all layers have the same cross-sectional area, it cancels out in the ratio calculations, so we can simplify to [tex]\( R = \rho \times \text{thickness} \)[/tex].
- Resistance of Skin:
[tex]\[ R_{\text{skin}} = 10^6 \times 0.00001 = 10 \, \Omega \][/tex]
- Resistance of Fat:
[tex]\[ R_{\text{fat}} = 15 \times 0.002 = 0.03 \, \Omega \][/tex]
- Resistance of Muscle:
[tex]\[ R_{\text{muscle}} = 2 \times 0.01 = 0.02 \, \Omega \][/tex]
4. Total Resistance:
- Since the layers are in series, the total resistance [tex]\(R_{\text{total}}\)[/tex] is the sum of individual resistances:
[tex]\[ R_{\text{total}} = R_{\text{skin}} + R_{\text{fat}} + R_{\text{muscle}} \][/tex]
[tex]\[ R_{\text{total}} = 10 + 0.03 + 0.02 = 10.05 \, \Omega \][/tex]
5. Relative Rate of Heat Dissipation:
The rate of heat dissipation in each layer is proportional to the resistance of that layer. We find the relative rate by dividing the resistance of each layer by the total resistance:
- Skin:
[tex]\[ \text{Relative Rate}_{\text{skin}} = \frac{R_{\text{skin}}}{R_{\text{total}}} = \frac{10}{10.05} \approx 0.9950 \][/tex]
- Fat:
[tex]\[ \text{Relative Rate}_{\text{fat}} = \frac{R_{\text{fat}}}{R_{\text{total}}} = \frac{0.03}{10.05} \approx 0.003 \][/tex]
- Muscle:
[tex]\[ \text{Relative Rate}_{\text{muscle}} = \frac{R_{\text{muscle}}}{R_{\text{total}}} = \frac{0.02}{10.05} \approx 0.0020 \][/tex]
Summarizing, the relative rates of heat dissipation in each layer are approximately:
- Skin: [tex]\( 0.9950 \)[/tex] or [tex]\( 99.50\% \)[/tex]
- Fat: [tex]\( 0.003 \)[/tex] or [tex]\( 0.30\% \)[/tex]
- Muscle: [tex]\( 0.002 \)[/tex] or [tex]\( 0.20\% \)[/tex]
Thus, the layer with the highest relative rate of heat dissipation is the skin, followed by fat, and then muscle. This aligns with the relative resistances of each layer.
