A carbon resistor is made of a cylinder of carbon, of diameter $7.4 \mathrm{~mm}$. If the resistivity of carbon is $3.5 \times 10$ $-5 \mathrm{~W} \cdot \mathrm{m}$, and a potential difference of $0.15 \mathrm{~V}$ causes a current of $12 \mathrm{~A}$ to flow through the resistor, what is the length of the cylinder?
A) $1.5 \mathrm{~cm}$
B) $0.5 \mathrm{~cm}$
C) $6.1 \mathrm{~cm}$
D) $0.15 \mathrm{~cm}$

Question

Quizplus · Accepted Answer

The resistance of the resistor is given by:$$R = \rho \frac{L}{A}$$where:* $R$ is the resistance in ohms* $\rho$ is the resistivity in ohm-meters* $L$ is the length of the cylinder in meters* $A$ is the cross-sectional area of the cylinder in square metersThe cross-sectional area of the cylinder is given by:$$A = \pi r^2$$where:* $r$ is the radius of the cylinder in metersSubstituting the given values into the equations above, we get:$$R = (3.5 \times 10^{-5} \Omega \cdot m) \frac{L}{\pi (3.7 \times 10^{-3} m)^2}$$$$R = 1.5 \times 10^{-2} \Omega \frac{L}{4.24 \times 10^{-5} m^2}$$$$R = 3.54 \times 10^3 \Omega \frac{L}{m}$$The potential difference across the resistor is given by:$$V = IR$$where:* $V$ is the potential difference in volts* $I$ is the current in amperesSubstituting the given values into the equation above, we get:$$0.15 V = 12 A \cdot R$$$$R = 0.0125 \Omega$$Equating the two expressions for $R$, we get:$$3.54 \times 10^3 \Omega \frac{L}{m} = 0.0125 \Omega$$$$L = 3.54 \times 10^{-6} m = 3.54 cm$$Therefore, the length of the cylinder is $3.54 cm$.

A Carbon Resistor Is Made of a Cylinder of Carbon 7.4 mm7.4 \mathrm{~mm}7.4 mm

A Carbon Resistor Is Made of a Cylinder of Carbon $7.4 \mathrm{~mm}$