This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Here are the solutions to the problems: Problem 1: a) A metal wire of length 79 cm and diameter 0.31 mm has a resistance of 5.0 . Find the resistivity of the metal wire. Step 1: Convert given values to SI units. Length L = 79 cm = 0.79 m Diameter d = 0.31 mm = 0.31 × 10^-3 m Resistance R = 5.0 Step 2: Calculate the cross-sectional area A of the wire. The formula for the area of a circle is A = ( d^2)/(4). A = (0.31 × 10^-3 m)^24 A = (9.61 × 10^-8 m^2)4 A ≈ 7.5498 × 10^-8 m^2 Step 3: Use the resistance formula R = (L)/(A) to find resistivity . Rearranging the formula for : = (R · A)/(L) Substitute the values: = (5.0 ) × (7.5498 × 10^-8 m^2)0.79 m = 3.7749 × 10^-7 · m0.79 ≈ 4.778 × 10^-7 · m Rounding to two significant figures: Resistivity = 4.8 × 10^-7 · m Problem 2: b) Determine the resistance of an aluminium cable of length 2.0 × 10^4 m and diameter 4.0 mm. The resistivity of aluminium is 3.0 × 10^-8 · m. Step 1: Convert given values to SI units. Length L = 2.0 × 10^4 m Diameter d = 4.0 mm = 4.0 × 10^-3 m Resistivity = 3.0 × 10^-8 · m Step 2: Calculate the cross-sectional area A of the cable. A = ( d^2)/(4) A = (4.0 × 10^-3 m)^24 A = (16.0 × 10^-6 m^2)4 A = × 4.0 × 10^-6 m^2 A ≈ 1.2566 × 10^-5 m^2 Step 3: Use the resistance formula R = (L)/(A). Substitute the values: R = (3.0 × 10^-8 · m) × 2.0 × 10^4 m1.2566 × 10^-5 m^2 R = 6.0 × 10^-4 · m^21.2566 × 10^-5 m^2 R ≈ 47.74 Rounding to two significant figures: Resistance R = 48 Problem 3: c) A manganin wire of cross-sectional area 4.0 × 10^-7 m^2 and resistivity 4.3 × 10^-7 · m has a resistance of 0.05 . Calculate the length of the wire. Step 1: Identify the given values. Cross-sectional area A = 4.0 × 10^-7 m^2 Resistivity = 4.3 × 10^-7 · m Resistance R = 0.05 Step 2: Use the resistance formula R = (L)/(A) to find length L. Rearranging the formula for L: L = (R · A)/() Substitute the values: L = (0.05 ) × (4.0 × 10^-7 m^2)4.3 × 10^-7 · m L = 0.2 × 10^-7 · m^24.3 × 10^-7 · m L = (0.2)/(4.3) m L ≈ 0.04651 m Rounding to two significant figures: Length L = 0.047 m