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?
To solve these problems, we can use the equations for resistivity and current. The resistivity of copper is known, and Ohm's law can be used to calculate the current. Let's go step by step:
1) (a) To find the length of the wire, we need to use the formula for resistance:
Resistance (R) = Resistivity (ρ) x Length (L) / Cross-sectional area (A)
The resistivity of copper (ρ) is a known value of 1.68 x 10^-8 Ω∙m.
We can rearrange the formula to solve for the length of the wire (L):
L = (Resistance x Area) / Resistivity
Given that the resistance (R) is 0.600 Ω and the mass of copper used is 1.15 g, we need to find the area of the wire.
The mass of copper can be converted to volume using the density of copper, which is 8.96 g/cm^3:
Volume = Mass / Density = 1.15 g / 8.96 g/cm^3
From here, we can calculate the dimensions of the wire, assuming it is cylindrical in shape:
Volume = πr^2h
Where r is the radius and h is the length of the wire.
We can rearrange the formula for volume to solve for the height:
h = Volume / (πr^2)
We can substitute this value for h in the formula for the length to calculate it.
(b) To find the diameter of the wire, we know the radius (r) and can convert it to diameter using the formula:
Diameter = 2 x Radius
Now let's calculate step by step:
1) (a) Calculate the Length of the wire:
1. Calculate the volume of copper:
Volume = Mass / Density = 1.15 g / 8.96 g/cm^3
2. Calculate the height of the wire:
h = Volume / (πr^2)
3. Calculate the length of the wire:
L = (Resistance x Area) / Resistivity
(b) Calculate the Diameter of the wire:
Diameter = 2 x Radius
2) (a) To find the maximum current that the copper block can carry, we can use Ohm's law:
Current (I) = Voltage (V) / Resistance (R)
Given that the voltage (V) is 4.0 V, and the resistance can be calculated using the resistivity and the dimensions of the block, we can solve for the maximum current (I).
(b) To find the minimum current, we need to determine the resistance, which is dependent on the shape of the block. To find the minimum current, we need to find the maximum resistance. We can calculate this by considering the longest diagonal of the block as the path of current flow.
Now let's solve step by step:
2) (a) Calculate the maximum current that the block can carry:
I = V / R
(b) Calculate the minimum current:
1. Calculate the maximum resistance:
R = Resistivity x (l / A)
2. Calculate the minimum current:
I = V / R
Please note that in both questions, the dimensions of the wire and block are not provided. To solve completely, we need the dimensions (e.g., radius, length, and side lengths).
To solve these questions, we will make use of certain equations and principles from physics. I'll explain the steps to find the answers to each part of the questions.
1) (a) To find the length of the wire, we can use the formula relating resistance, resistivity, length, and cross-sectional area. The formula is:
Resistance (R) = (Resistivity (ρ) * Length (L)) / Cross-sectional Area (A)
Since we are given the resistance of the wire and assuming that the copper is used completely, we can rearrange the formula to solve for length:
Length (L) = (Resistance (R) * Cross-sectional Area (A)) / Resistivity (ρ)
For this question, we are given the resistance (R = 0.600 Ω) and the mass (m = 1.15 g) of copper. We need to find the cross-sectional area (A) and resistivity (ρ) of copper.
To find the cross-sectional area (A), we need the diameter (d) of the wire. We can use the formula:
Cross-sectional Area (A) = (π/4) * d^2
To find the resistivity (ρ), we can look up the value for copper in a reference. It is approximately 1.7 x 10^-8 Ω·m.
Once we have the values for resistance (R), cross-sectional area (A), and resistivity (ρ), we can substitute them into the formula for length (L) to find the answer in meters (m).
(b) To find the diameter of the wire, we can rearrange the formula for cross-sectional area (A) as follows:
Diameter (d) = √(4 * A / π)
Substitute the value of the cross-sectional area (A) and calculate the diameter (d) in micrometers (µm).
2) To find the currents that the block of copper can carry, we need to use Ohm's Law and some geometric principles.
(a) To find the maximum current that the block can carry, we need to calculate the resistance of the block using the formula:
Resistance (R) = Resistivity (ρ) * (Length (L) / Cross-sectional Area (A))
Assuming the block is made entirely of copper, we can use the resistivity value for copper (approximately 1.7 x 10^-8 Ω·m) and substitute the given values (length, width, and height) to calculate the resistance (R). Then, using Ohm's Law (V = I * R), we can solve for the maximum current (I) in Amperes (A).
(b) To find the minimum current, we consider the case where the current flows along the longest dimension of the block. Since electrical resistance is directly proportional to the length of the conductor, we take the longest length (34 cm) as the length (L) and substitute it into the formula for resistance. Then, using Ohm's Law (V = I * R), we can solve for the minimum current (I) in Amperes (A).
By following these steps, you should be able to find the answers to the questions.