A graphite rod is 1.0 m long, with a 1.0 cm by 0.5 cm rectangular cross sectional area. What is the electrical resistance between its long ends? What must be the diameter of a circular 2.0 m long copper rod if its resistance is to be the same?

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To calculate the electrical resistance of the graphite rod, we can use the formula:

R = ρ * (L / A)

Where:
R is the electrical resistance,
ρ is the resistivity of the material,
L is the length of the rod, and
A is the cross-sectional area of the rod.

First, let's calculate the cross-sectional area of the graphite rod:

A = length * width
= (1.0 cm) * (0.5 cm)
= 0.5 cm^2

Next, we need to convert the area to square meters:

A = 0.5 cm^2 * (1 m / 100 cm)^2
= 0.5 cm^2 * (0.01 m / cm)^2
= 0.5 cm^2 * 0.0001 m^2 / cm^2
= 0.00005 m^2

Now, let's determine the resistivity of graphite. The resistivity (ρ) can vary depending on the purity and temperature, but a typical value for graphite at room temperature is approximately 3.5 x 10^-5 ohm-meter (Ω⋅m).

Using the resistivity and the cross-sectional area, we can now calculate the electrical resistance of the graphite rod:

R = (3.5 x 10^-5 Ω⋅m) * (1.0 m / 0.00005 m^2)
= (3.5 x 10^-5 Ω⋅m) * (1.0 m) * (1 / 0.00005 m^2)
= 3.5 x 10^-5 Ω⋅m * 1.0 m * 20000
= 0.7 Ω

Therefore, the electrical resistance between the long ends of the graphite rod is 0.7 ohms.

Now, let's determine the diameter of the copper rod that will have the same resistance. Since the copper rod is circular, we can use the formula for the resistance of a cylindrical conductor:

R = ρ * (L / A)

In this case, we have the resistance (R) of 0.7 Ω and the length (L) of 2.0 m. We need to find the cross-sectional area (A) and the resistivity (ρ) of copper.

Rearranging the formula, we get:

A = ρ * (L / R)

The resistivity of copper at room temperature is typically around 1.7 x 10^-8 Ω⋅m.

Substituting the values into the formula:

A = (1.7 x 10^-8 Ω⋅m) * (2.0 m / 0.7 Ω)
= (1.7 x 10^-8 Ω⋅m) * (2.0 m) * (1 / 0.7 Ω)
= 4.857 x 10^-8 Ω⋅m * m * Ω / Ω
= 4.857 x 10^-8 m^2

Finally, to find the diameter of the copper rod, we can use the formula for the cross-sectional area of a circle:

A = π * (d^2 / 4)

Rearranging the formula, we get:

d^2 = (4 * A) / π

Substituting the value of A that we calculated:

d^2 = (4 * 4.857 x 10^-8 m^2) / π
≈ 6.157 x 10^-8 m^2 / π

Taking the square root of both sides, we can find the diameter:

d ≈ sqrt(6.157 x 10^-8 m^2 / π)
≈ 1.977 x 10^-4 m

Therefore, the diameter of the circular copper rod that will have the same resistance as the graphite rod is approximately 1.977 x 10^-4 meters.