(1) 45 micro coulombs of charge is passed through a wire in 35 seconds. How many electrons does this represent? What is the current in the wire?
(2) A piece of copper wire has a resistance of 5 Ohms at 20 degrees celsius. If another piece of copper wire at 0 degrees celsius is 3 times as long and has a cross sectional area that's twice as big, what will its resistance be?
(1) The current (in Amperes) is the charge divided by the time. Divide that once again by the electron charge to get the number on electrons per second.
(2) Resistance is proportional to length and inversely proportional to the wire's cross sectional area. Knowing that, what do you think the answer is?
For highest accuracy, you will have to account for the chance in resistivity with temperature. which is quite small over a range from 0 to 20 C.
(1) To find the number of electrons represented by the given charge, we can use the formula:
Number of electrons = (Charge / Charge of an electron)
The charge of an electron is approximately 1.6 x 10^(-19) coulombs. Substituting the given values:
Number of electrons = (45 x 10^(-6)) / (1.6 x 10^(-19))
Calculating this, we get approximately 2.8125 x 10^23 electrons.
Next, to find the current in the wire, we can use Ohm's Law:
Current (I) = (Charge (Q) / Time (t))
Substituting the given values:
Current = (45 x 10^(-6)) / 35
Calculating this, we get approximately 1.2857 x 10^(-6) Amperes or 1.29 μA (microamperes).
(2) To find the resistance of the second piece of copper wire, we need to use the formula:
Resistance = (Resistivity x Length) / Cross-sectional area
Given that the resistivity of copper remains constant with temperature changes, we can consider the resistivity as a constant value.
Let's assume the resistance of the first piece of copper wire at 20 degrees Celsius is R1, and the resistance of the second piece of copper wire at 0 degrees Celsius is R2.
Now, we have the following relationships between the two wires:
R1 = (Resistivity x Length1) / Area1
R2 = (Resistivity x Length2) / Area2
We are given that Length2 = 3 x Length1 and Area2 = 2 x Area1.
Now, substitute the values into the equation for R2:
R2 = (Resistivity x (3 x Length1)) / (2 x Area1)
Since we know that R1 = 5 Ohms, let's substitute the known values into the equation for R1:
5 = (Resistivity x Length1) / Area1
Now, rearrange the equation for Length1:
Length1 = (5 x Area1) / Resistivity
Substitute this value for Length1 into the equation for R2:
R2 = (Resistivity x (3 x ((5 x Area1) / Resistivity))) / (2 x Area1)
Simplifying, we get:
R2 = (15 x Area1) / (2 x Area1)
R2 = 7.5 Ohms
Therefore, the resistance of the second piece of copper wire at 0 degrees Celsius would be 7.5 Ohms.