PRELIMINARY STAGE 2024
Contest 7
Tepa SHS: 40 points (Winner)
Bright SHS: 39 points
Sekondi College: 27 points
Contest 8
Queen of Peace SHS: 34 points (Winner)
Sonrise Christian High School: 31 points
Presby SHS, Bompata: 30 points
Contest 9
Navrongo SHS: 39 points (Winner)
St. Mary’s Seminary SHS, Lolobi: 36 points
Bolgatanga Girls’ SHS: 29 points
FIRST SET OF QUESTIONS
PREAMBLE
The electrical resistivity of an alloy is $4.0\times10^{-8}\ \Omega\cdot\text{m}$.
Find the resistance of alloy wire of the given dimensions.
VARIABLES DEFINED:
* $R$ = Resistance (ohms, $\Omega$)
* $\rho$ = Electrical resistivity ($4.0\times10^{-8}\ \Omega\cdot\text{m}$)
* $L$ = Length of wire (meters, m)
* $A$ = Cross-sectional area (meters squared, $\text{m}^2$)
1. Length equals 5.0 meters and cross-sectional area equals $2.0\times10^{-6}\text{ m}^2$.
ANSWER: 0.10 $\Omega$
SOLUTION:
$R = \dfrac{\rho \times L}{A}$
$R = \dfrac{(4.0\times10^{-8}) \times 5.0}{2.0\times10^{-6}}$
$R = \dfrac{4\times10^{-8} \times 5}{2\times10^{-6}}$
$R = 2 \times 5 \times10^{-2}$
$R = 1.0\times10^{-1}\ \Omega$
$R = 0.10\ \Omega$
2. Length equals 2.0 meters and cross-sectional area equals $5.0\times10^{-7}\text{ m}^2$.
ANSWER: 0.16 $\Omega$
SOLUTION:
$R = \dfrac{\rho \times L}{A}$
$R = \dfrac{(4.0\times10^{-8}) \times 2.0}{5.0\times10^{-7}}$
$R = \dfrac{4\times10^{-8} \times 2}{5\times10^{-7}}$
$R = \dfrac{4 \times 2}{5} \times10^{-1}$
$R = 1.6\times10^{-1}\ \Omega$
$R = 0.16\ \Omega$
3. Length equals 0.40 meters and cross-sectional area equals $8.0\times10^{-8}\text{ m}^2$.
ANSWER: 0.20 $\Omega$
SOLUTION:
$R = \dfrac{\rho \times L}{A}$
$R = \dfrac{(4.0\times10^{-8}) \times 0.40}{8.0\times10^{-8}}$
$R = \dfrac{4\times10^{-8} \times 4\times10^{-1}}{8\times10^{-8}}$
$R = \dfrac{4}{2} \times10^{-1}$
$R = 0.20\ \Omega$
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PRACTICE QUESTIONS
1. QUESTION: Find the resistance of a wire of length 1.2 m and cross-sectional area $1.0\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $1.5\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.018 Ω or $1.8\times10^{-2}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(1.5\times10^{-8}) \times 1.2}{1.0\times10^{-6}}$
$R = \dfrac{15\times10^{-9} \times 12\times10^{-1}}{10^{-6}}$
$R = 15 \times 12 \times10^{-4}$
$R = 0.018\ \Omega$
2. QUESTION: Find the resistance of a wire of length 3.0 m and cross-sectional area $3.0\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $8.0\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.080 Ω or $8.0\times10^{-2}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(8.0\times10^{-8}) \times 3.0}{3.0\times10^{-6}}$
$R = \dfrac{8\times10^{-8} \times 3}{3\times10^{-6}}$
$R = 0.080\ \Omega$
3. QUESTION: Find the resistance of a wire of length 3.5 m and cross-sectional area $2.0\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $2.8\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.049 Ω or $4.9\times10^{-2}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(2.8\times10^{-8}) \times 3.5}{2.0\times10^{-6}}$
$R = \dfrac{9.80\times10^{-8}}{2.0\times10^{-6}}$
$R = \dfrac{98\times10^{-9}}{2\times10^{-6}}$
$R = 0.049\ \Omega$
4. QUESTION: Find the resistance of a wire of length 2.0 m and cross-sectional area $2.0\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $4.0\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.040 Ω or $4.0\times10^{-2}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(4.0\times10^{-8}) \times 2.0}{2.0\times10^{-6}}$
$R = \dfrac{4\times10^{-8} \times 2}{2\times10^{-6}}$
$R = 2 \times 2 \times10^{-2}$
$R = 0.040\ \Omega$
5. QUESTION: Find the resistance of a wire of length 4.5 m and cross-sectional area $1.2\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $5.6\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.21 Ω or $2.1\times10^{-1}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(5.6\times10^{-8}) \times 4.5}{1.2\times10^{-6}}$
$R = \dfrac{56\times10^{-9} \times 45\times10^{-1}}{12\times10^{-7}}$
$R = \dfrac{14 \times 45}{3} \times10^{-3}$
$R = 14 \times 15 \times10^{-3}$
$R = 0.21\ \Omega$
6. QUESTION: Find the resistance of a wire of length 1.5 m and cross-sectional area $2.5\times10^{-7}\text{ m}^2$ made of an alloy of resistivity $9.8\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.59 Ω or $5.9\times10^{-1}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(9.8\times10^{-8}) \times 1.5}{2.5\times10^{-7}}$
$R = \dfrac{1.47\times10^{-7}}{2.5\times10^{-7}}$
$R = \dfrac{147\times10^{-9}}{25\times10^{-8}}$
$R = \dfrac{147}{25} \times10^{-1}$
$R \approx 0.59\ \Omega$
7. QUESTION: Find the resistance of a wire of length 7.5 m and cross-sectional area $2.5\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $5.0\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.15 Ω or $1.5\times10^{-1}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(5.0\times10^{-8}) \times 7.5}{2.5\times10^{-6}}$
$R = \dfrac{5\times10^{-8} \times 75\times10^{-1}}{25\times10^{-7}}$
$R = 5 \times 3 \times10^{-2}$
$R = 0.15\ \Omega$
8. QUESTION: Find the resistance of a wire of length 8.5 m and cross-sectional area $3.4\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $2.4\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.060 Ω or $6.0\times10^{-2}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(2.4\times10^{-8}) \times 8.5}{3.4\times10^{-6}}$
$R = \dfrac{2.04\times10^{-7}}{3.4\times10^{-6}}$
$R = \dfrac{204\times10^{-9}}{34\times10^{-7}}$
$R = 0.060\ \Omega$
9. QUESTION: Find the resistance of a wire of length 6.0 m and cross-sectional area $6.0\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $6.0\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.060 Ω or $6.0\times10^{-2}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(6.0\times10^{-8}) \times 6.0}{6.0\times10^{-6}}$
$R = \dfrac{6\times10^{-8} \times 6}{6\times10^{-6}}$
$R = 0.060\ \Omega$
10. QUESTION: Find the resistance of a wire of length 1.5 m and cross-sectional area $1.5\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $2.5\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.025 Ω or $2.5\times10^{-2}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(2.5\times10^{-8}) \times 1.5}{1.5\times10^{-6}}$
$R = \dfrac{25\times10^{-9} \times 15\times10^{-1}}{15\times10^{-7}}$
$R = 0.025\ \Omega$
11. QUESTION: Find the resistance of a wire of length 4.0 m and cross-sectional area $5.0\times10^{-6}\text{ m}^2$ made of an alloy of resistivity $2.0\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.016 Ω or $1.6\times10^{-2}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(2.0\times10^{-8}) \times 4.0}{5.0\times10^{-6}}$
$R = \dfrac{2\times10^{-8} \times 4}{5\times10^{-6}}$
$R = \dfrac{2 \times 4}{5} \times10^{-2}$
$R = 0.016\ \Omega$
12. QUESTION: Find the resistance of a wire of length 6.5 m and cross-sectional area $9.0\times10^{-7}\text{ m}^2$ made of an alloy of resistivity $7.2\times10^{-8}\ \Omega\cdot\text{m}$.
ANSWER: 0.52 Ω or $5.2\times10^{-1}$ Ω
SOLUTION:
$R = \dfrac{\rho L}{A}$
$R = \dfrac{(7.2\times10^{-8}) \times 6.5}{9.0\times10^{-7}}$
$R = \dfrac{72\times10^{-9} \times 65\times10^{-1}}{9\times10^{-7}}$
$R = 8 \times 65 \times10^{-3}$
$R = 0.52\ \Omega$