The electrical resistance R of a wire varies directly as its length L and inversely as the… — JAMB UTME

The electrical resistance $R$ of a wire varies directly as its length $L$ and inversely as the square of its diameter $d$ . A wire of length $50\text{ m}$ and diameter $2\text{ mm}$ has a resistance of $5\text{ ohms}$ . Calculate the resistance (in ohms) of a wire made of the same material with a length of $80\text{ m}$ and a diameter of $4\text{ mm}$ .

Answer: 2 ohms

Answer

The resistance of the wire is

2\text{ ohms}

The variation model is

R = \frac{k L}{d^2}

. Substituting

R = 5\text{ ohms}

L = 50\text{ m}

, and

d = 2\text{ mm}

gives

5 = \frac{50k}{4}

, so

k = 0.4

. Substituting

k = 0.4

L = 80\text{ m}

, and

d = 4\text{ mm}

yields

R = \frac{0.4 \times 80}{4^2} = \frac{32}{16} = 2\text{ ohms}

Step-by-Step Solution

Set up the variation formula

R = \frac{k L}{d^2}

Resistance varies directly as length

L

and inversely as the square of diameter

d

Calculate the constant of variation

k

k = 0.4

Substitute the initial values

R = 5

L = 50

, and

d = 2

into the variation equation.

Compute the new resistance

R

R = 2\text{ ohms}

Substitute

k = 0.4

L = 80

, and

d = 4

into the formula.

Key Concept

Direct and Inverse Joint Variation

Estimated Time:1m 30s

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