1) Suppose that you wish to fabricate a uniform wire out of 1.15 g of copper. Assume the wire has a resistance R = 0.600 , and all of the copper is used.
(a) What will be the length of the wire?
m?
(b) What will be the diameter of the wire?
µm?
2) A rectangular block of copper has sides of length 5 cm, 18 cm, and 34 cm. If the block is connected to a 4.0 V source across two of its opposite faces of the rectangular block, what are the currents that the block can carry?
(a) the maximum current
A?
(b) the minimum current
A?
1) To find the length of the wire, we can use Ohm's Law, which states that resistance (R) is equal to the resistivity (ρ) multiplied by the length (L) divided by the cross-sectional area (A).
(a) We are given the resistance (R) as 0.600 Ω and the mass of copper as 1.15 g. To find the length (L), we need to calculate the resistivity (ρ) first.
The resistivity of copper can be found in a reference table or by doing a quick search online. Let's assume the resistivity of copper is 1.7 x 10^-8 Ωm.
The resistivity (ρ) can be calculated using the formula:
ρ = (m / A) * L
Rearranging the formula to solve for length (L), we have:
L = (R * A) / ρ
To calculate the cross-sectional area (A), we need to determine the diameter of the wire and then use it to find the area of a circle.
(b) The diameter of the wire can be calculated from its mass. We will assume that the wire is uniform, so the mass will be evenly distributed along its length. The density of copper can be found in a reference table or by doing a quick search online. Let's assume the density of copper is 8.92 g/cm^3.
The volume (V) of the wire can be calculated as:
V = mass / density
From the volume (V), we can find the diameter (D) of the wire using the formula for the volume of a cylinder:
V = π * (D^2) * L / 4
Rearranging the formula to solve for diameter (D), we have:
D = sqrt((4 * V) / (π * L))
Now let's substitute the given values into the formulas:
(a) Length of the wire (L):
L = (R * A) / ρ
(b) Diameter of the wire (D):
D = sqrt((4 * V) / (π * L))
2) To find the currents that the block can carry, we can use Ohm's Law.
(a) The maximum current is achieved when the resistance is at its lowest value. In this case, the resistance will be determined by the cross-sectional area that is in the path of the current flow. Considering the sides given, we can determine the cross-sectional area (A) of the block.
(b) The minimum current is achieved when the resistance is at its highest value. In this case, the resistance will be determined by the longest path of the current through the block. Considering the sides given, we can determine the resistance (R) of the block.
Then, using Ohm's Law (I = V / R), we can calculate the currents:
(a) Maximum current (I_max) = V / R_max
(b) Minimum current (I_min) = V / R_min
Substitute the given values into the formulas to find the currents.