Worksheet: Resistance and Resitivity of Conductors

In this worksheet, we will practice relating the dimensions of and the motion of free electrons through an object to its resistance.

Q1:

A wire made of an unknown substance has a resistance of 125 mΩ. The wire has a length of 1.8 m and a cross-sectional area of 2.35×10 m2. What is the resistivity of the substance from which the wire is made?

  • A9.6×10 Ω⋅m
  • B5.3×10 Ω⋅m
  • C9.6×10 Ω⋅m
  • D1.6×10 Ω⋅m
  • E1.6×10 Ω⋅m

Q2:

A copper wire with a resistance of 12.8 mΩ has a cross-sectional area of 1.15×10 m. Find the length of the wire. Use 1.7×10 Ω⋅m for the resistivity of copper.

Q3:

The diagram shows two identical circuits. Each circuit’s conducting wire has had a section of it greatly magnified to show the ions that the conducting wire is comprised of and the free electrons that move between these ions. The conducting wires are both made of the same substance.

Which of the following statements correctly states how the resistivities of the sections of conducting wire in diagram (a) and diagram (b) compare to each other?

  • AThe resistivity of the section in diagram (a) is greater than the resistivity of the section in diagram (b).
  • BThe resistivity of the section in diagram (b) is greater than the resistivity of the section in diagram (a).
  • CThe resistivities are the same for both sections.

Which of the following statements correctly states how the cross-sectional areas of the conducting wires compare to each other?

  • AThe cross-sectional area of the wire in diagram (b) is greater than the cross-sectional area of the wire in diagram (a).
  • BThe cross-sectional areas of the wires are the same.
  • CThe cross-sectional area of the wire in diagram (a) is greater than the cross-sectional area of the wire in diagram (b).

Which of the following statements correctly states how the average time taken for a free electron to cross from one side of the section to the other side in diagram (a) compares to the average time taken for a free electron to cross from one side of the section to the other side in diagram (b)?

  • AThe time taken for the section in diagram (b) is greater.
  • BThe time taken for the section in diagram (a) is greater.
  • CThe time taken is the same for both sections.

Which of the following statements correctly states how the resistance of the section of conducting wire in diagram (a) compares to the resistance of the section of conducting wire in diagram (b)?

  • AThe resistance of the section in diagram (b) is greater.
  • BThe resistance of the section in diagram (a) is greater.
  • CThe resistance is the same for both sections.

Q4:

The wires that carry current from a power station to a substation are 7.25 km long. They are made of copper with a resistivity of 1.7×10 Ω⋅m. The current through the wires is 450 mA. The power dissipated by the wires is required to be no more than 15 W. What is the minimum cross-sectional area required for the wires that transmit the current?

  • A3.7×10 m2
  • B1.7×10 m2
  • C1.7×10 m2
  • D8.2×10 m2
  • E3.7×10 m2

Q5:

Which of the following formulas correctly relates the resistivity, 𝜌, of a substance to the resistance of an object with a length, 𝑙, that is made of the substance if the object has a cross-sectional area 𝐴 and a resistance 𝑅?

  • A𝑅=𝜌𝐴𝑙
  • B𝜌=𝑅𝑙𝐴
  • C𝑅=𝜌𝐴𝑙
  • D𝑅=𝜌𝑙𝐴

Q6:

A copper wire with a resistance of 22 mΩ has a length of 6.2 m. Find the cross-sectional area. Use 1.7×10 Ω⋅m for the resistivity of copper.

  • A4.8×10 m2
  • B7.5×10 m2
  • C8.0×10 m2
  • D2.3×10 m2
  • E2.1×10 m2

Q7:

A current of 1.4 A in a copper wire is carried by free electrons. The cross-sectional area of the wire is 2.5×10 m2. Find the average speed at which free electrons pass through the wire. Use a value of 1.6×10 C for electron charge and a value of 8.46×10 m−3 for the density of free electrons in copper.

  • A4.7×10 m/s
  • B5.3×10 m/s
  • C2.1×10 m/s
  • D2.4×10 m/s
  • E4.1×10 m/s

Q8:

Diagrams (a) and (b) show sections of identical conducting objects consisting of three equally separated ions. Free electrons that have a velocity greater than a velocity 𝑣escape move away from the nearest ion to a neighboring ion. Free electrons with velocities lower than 𝑣escape do not move to a neighboring ion. Diagrams (c) and (d) show sections of conducting objects in which the spacing between ions is greater than it is for the objects shown in diagrams (a) and (b). The density of free electrons in the object in diagram (d) is the same as the free electron density in the objects shown in diagrams (a) and (b). The density of free electrons in the object in diagram (c) is greater than the free electron density in the objects shown in diagrams (a) and (b). The average speed at which a single typical electron moves in the objects in diagrams (c) and (d) is 𝑣escape.

Identical-sized wires made of the materials that the objects in diagrams (a) and (d) are made of are connected across the same potential difference. Which of the following statements most correctly describes how the current through wire I, which is made of the material from diagram (a), compares to the current through wire II, which is made of the material from diagram (d)?

  • AThe current in wire II is greater than the current in wire I.
  • BThe current in wire I is greater than the current in wire II.
  • CThe current is the same in both wires.

Identical-sized wires made of the materials that the objects in diagrams (c) and (d) are made of are connected across the same potential difference. Which of the following statements most correctly describes how the current through wire I, which is made of the material from diagram (c), compares to the current through wire II, which is made of the material from diagram (d)? Take into account the mutual repulsion of electrons.

  • AThe current in wire I is greater than the current in wire II.
  • BThe current is the same in both wires.
  • CThe current in wire II is greater than the current in wire I.

Q9:

The diagram shows two very similar circuits. Each circuit’s conducting wire has had a section of it greatly magnified to show the ions that the conducting wire is comprised of and the free electrons that move between these ions.

Which of the following statements correctly states how the resistivities of the sections of the conducting wire in diagram (a) and diagram (b) compare to each other?

  • AThe resistivity of the section in diagram (a) is greater.
  • BThe resistivity of both sections is the same.
  • CThe resistivity of the section in diagram (b) is greater.

Which of the following statements correctly states how the cross-sectional areas of the conducting wires compare to each other?

  • AThe cross-sectional area of the wire in diagram (b) is greater than the cross-sectional area of the wire in diagram (a).
  • BThe cross-sectional areas of the wires are the same.
  • CThe cross-sectional area of the wire in diagram (a) is greater than the cross-sectional area of the wire in diagram (b).

Which of the following statements correctly states how the number of free electrons per meter of length of the conducting wire in diagram (a) compares to the number of free electrons per meter of length of the conducting wire in diagram (b)?

  • AThe number of free electrons per meter of length of the conducting wire in diagram (b) is greater than the number of free electrons per meter of length of the conducting wire in diagram (a).
  • BThe number of free electrons per meter of length of the conducting wire in diagram (a) is greater than the number of free electrons per meter of length of the conducting wire in diagram (b).
  • CThe number of free electrons per meter of length is the same for both wires.

Which of the following statements correctly states how the average time taken for a free electron to cross from one side of the section to the other side in diagram (a) compares to the average time taken for a free electron to cross from one side of the section to the other side in diagram (b)?

  • AThe average time taken for a free electron to cross from one side of the section to the other side in diagram (b) is greater than the average time taken for a free electron to cross from one side of the section to the other side in diagram (a).
  • BThe average time taken for a free electron to cross from one side of the section to the other side in diagram (a) is greater than the average time taken for a free electron to cross from one side of the section to the other side in diagram (b).
  • CThe average time taken for a free electron to cross from one side of the section to the other side is the same for both sections.

Which of the following statements correctly states how the resistance of the section of conducting wire in diagram (a) compares to the resistance of the section of conducting wire in diagram (b)?

  • AThe resistance of the section in diagram (b) is greater.
  • BThe resistance of the section in diagram (a) is greater.
  • CThe resistance of both sections is the same.

Q10:

A current of 77 mA in a conducting wire of an unknown material is carried by free electrons. The cross-sectional area of the conductor is 1.5×10 m2. Find the density of free electrons in the material if the average speed of the free electrons along the wire is 0.18 mm/s. Use a value of 1.6×10 C for the electron charge.

  • A1.8×10 m−3
  • B3.0×10 m3
  • C1.8×10 m−3
  • D1.8×10 m−3
  • E3.0×10 m−3

Q11:

The diagram shows two very similar circuits. Each circuit’s conducting wire has had a section of it greatly magnified to show the ions that the conducting wire is comprised of and the free electrons that move between these ions. The conducting wires are both made of the same substance, but the conducting wires in the circuit in diagram (b) are of greater thickness than the conducting wires in the circuit in diagram (a).

Which of the following statements correctly states how the resistivities of the sections of the conducting wires in diagram (a) and diagram (b) compare to each other?

  • AThe resistivity of the section in diagram (b) is greater than the resistivity of the section in diagram (a).
  • BThe resistivity of the section in diagram (a) is greater than the resistivity of the section in diagram (b).
  • CThe resistivities are the same for both sections.

Which of the following statements correctly states how the cross-sectional areas of the conducting wires compare to each other?

  • AThe cross-sectional areas of the wires are the same.
  • BThe cross-sectional area of the wire in diagram (a) is greater than the cross-sectional area of the wire in diagram (b).
  • CThe cross-sectional area of the wire in diagram (b) is greater than the cross-sectional area of the wire in diagram (a).

Which of the following statements correctly states how the number of free electrons per meter of length of the conducting wire in diagram (a) compares to the number of free electrons per meter of length of the conducting wire in diagram (b)?

  • AThe number of free electrons per meter of length of the conducting wire in diagram (a) is greater than the number of free electrons per meter of length of the conducting wire in diagram (b).
  • BThe number of free electrons per meter of length of the conducting wire in diagram (b) is greater than the number of free electrons per meter of length of the conducting wire in diagram (a).
  • CThe number of free electrons per meter of length is the same for both wires.

Which of the following statements correctly states how the average time taken for a free electron to cross from one side of the section to the other side in diagram (a) compares to the average time taken for a free electron to cross from one side of the section to the other side in diagram (b)?

  • AThe average time taken for a free electron to cross from one side of the section to the other side in diagram (b) is greater than the average time taken for a free electron to cross from one side of the section to the other side in diagram (a).
  • BThe average time taken for a free electron to cross from one side of the section to the other side is the same for both sections.
  • CThe average time taken for a free electron to cross from one side of the section to the other side in diagram (a) is greater than the average time taken for a free electron to cross from one side of the section to the other side in diagram (b).

Which of the following statements correctly states how the resistance of the section of the conducting wire in diagram (a) compares to the resistance of the section of the conducting wire in diagram (b)?

  • AThe resistance of the section in diagram (a) is greater.
  • BThe resistance of the section in diagram (b) is greater.
  • CThe resistance of both sections is the same.

Q12:

Diagrams (a) and (b) show sections of identical conducting materials consisting of three equally separated ions. Free electrons that have a velocity greater than a velocity 𝑣escape move away from the nearest ion to a neighboring ion. Free electrons with velocities lower than 𝑣escape do not move to a neighboring ion. Diagram (c) shows a section of a conducting material consisting of three equally separated ions. The charge of each of these ions is the same as the charge of each of the ions in diagrams (a) and (b).

Which of the following statements correctly describes how the velocity of electrons required for them to move between ions in the material in diagram (c) compares to the velocity of electrons required for them to move between ions in the material in diagrams (a) and (b)?

  • AThe velocity of electrons required in diagram (c) is greater than the velocity of electrons required in diagrams (a) and (b).
  • BThe velocity of electrons required in diagram (c) is lower than the velocity of electrons required in diagrams (a) and (b).
  • CThe velocity of electrons required in diagram (c) is equal to the velocity of electrons required in diagrams (a) and (b).

Which of the following statements correctly describes how the resistivity of the material in diagram (c) compares to the resistivity of the material in diagrams (a) and (b)?

  • AThe resistivity of the material in diagram (c) is greater than the resistivity of the material in diagrams (a) and (b).
  • BThe resistivity of the material in diagram (c) is lower than the resistivity of the material in diagrams (a) and (b).
  • CThe resistivity of the material in diagram (c) is equal to the resistivity of the material in diagrams (a) and (b).

Q13:

A copper wire is 2.5 m long and has a cross-sectional area of 1.25×10 m2. Find the resistance of the wire. Use 1.7×10 Ω⋅m for the resistivity of copper.

Q14:

Diagrams (a) and (b) show sections of identical conducting objects consisting of three equally separated ions. Free electrons that have a velocity greater than a velocity 𝑣escape move away from the nearest ion to a neighboring ion. Free electrons with velocities lower than 𝑣escape do not move to neighboring ions. Diagrams (c) and (d) show sections of conducting objects in which the ions oscillate because the objects have significant internal energy. The spacing of ions in the object in diagram (d) is the same as it is in the objects in diagrams (a) and (b). The spacing of ions in the object in diagram (c) is greater than the spacing of ions in the objects in diagrams (a) and (b). Which of the following statements most correctly describes how the oscillation of the ions in the objects in diagrams (c) and (d) affects the motion of free electrons across the objects?

  • AThe oscillation of ions has no effect on the motion of free electrons across the objects.
  • BIn both diagrams, the oscillation of ions only reduces the net motion of free electrons across the objects.
  • CThe oscillation of ions in diagram (c) helps free electrons to move between ions more than it does in diagram (d).
  • DThe oscillation of ions in diagram (d) helps free electrons to move between ions more than it does in diagram (c).
  • EIn both diagrams, the oscillation of ions only increases the net motion of free electrons across the objects.

Q15:

A 10 V battery supplies a circuit with a 3.2 A current for 30 s. The conductor in the circuit is made of a substance with a free electron density of 1.0×10 m−3 and has a cross-sectional area of 2.0×10 m2. When analyzing the motion of free electrons in the conductor, use a value of 1.6×10 C for electron charge and a value of 9.1×10 kg for electron mass.

Find the average speed of free electrons along the conductor.

Find the instantaneous speed of a free electron in the conductor, assuming that each free electron has an equal share of the total energy of the current.

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