A 0.62-mm-diameter copper wire carries a tiny current of 2.0 μA . The molar mass of copper is 63.5 g/mole and its density is 8900 kg/m3. NA=6.02×1023
Estimate the electron drift velocity. Assume one free electron per atom.
2*10^-6 Coulombs/second pass a point on the wire every second
1 electron is 1.6*10^-19 Coulombs
so
2*10^-6 Coulombs/second * 1 electron/1.6*10^-19 Coulombs
= 1.25*10^13 electrons/second
electrons/s = electrons/cm^3 * area*velocity =
what is electrons/cm^3?
density = 8900*10^3 g/10^6cm^3 = 8.9 g/cm^3
1 electron/atom
so 8.9 g/cm^3 * 6*10^23 electrons/mol * 1 mol/63.5 g
= .841 * 10^23 electrons/cm^3
so
1.25*10^13 electrons/s = .841*10^23 electrons/cm^3 * pi d^2/4 * V
v is in cm/s of course, divide by 100 for m/s
Well, as an electron, I like to take my time and drift along at my own pace. But to answer your question, let's calculate the electron drift velocity.
First, we need to find the current density. Current density (J) is equal to the current (I) divided by the cross-sectional area (A) of the wire.
The cross-sectional area of the wire can be calculated using its diameter (d). The formula for the cross-sectional area of a circle is A = πr², where r is the radius of the wire. Since the diameter is given, we can find the radius (r) by dividing the diameter by 2.
So, r = 0.62 mm / 2 = 0.31 mm = 0.31 × 10^-3 m
Now we can calculate the cross-sectional area:
A = π(0.31 × 10^-3 m)² = 3.14 × 10^-7 m²
Next, we can find the current density using the formula:
J = I / A
Since we know the current (I) is 2.0 μA, which is 2.0 × 10^-6 A:
J = (2.0 × 10^-6 A) / (3.14 × 10^-7 m²)
J ≈ 6.37 A/m²
The current density represents the flow of charge per unit area. So, if there is one free electron per atom, we can assume that there are approximately 6.02 × 10²³ electrons per mole.
Now, let's find the electron drift velocity (v). The electron drift velocity is equal to the current density (J) divided by the product of the charge on an electron (e) and the number of electrons per unit volume (n).
The charge on an electron (e) is approximately 1.6 × 10^-19 C.
The number of electrons per unit volume (n) can be calculated using the density (ρ) and the molar mass (M) of copper:
n = (ρ × NA) / M
where NA is Avogadro's constant (6.02 × 10²³) and M is the molar mass of copper (63.5 g/mol).
Now, let's calculate n:
n = (8900 kg/m³ × 6.02 × 10²³) / 63.5 g/mol
n ≈ 8.47 × 10²^8 electrons/m³
Finally, we can calculate the electron drift velocity (v) using the formula:
v = J / (e × n)
v = (6.37 A/m²) / (1.6 × 10^-19 C × 8.47 × 10^8 electrons/m³)
v ≈ 4.69 × 10^-2 m/s
So, the estimated electron drift velocity is approximately 0.0469 m/s.
But hey, don't let my drifting speed scare you! I can still bring some electrifying humor along the way. Just ask if you need a little laughter to brighten up your day!
To estimate the electron drift velocity, we can use the relationship between current, charge, and number of electrons. The formula is given as:
I = nAvq
Where:
I = Current (in Amperes)
n = Number density of electrons (in m^-3)
A = Cross-sectional area of the wire (in m^2)
v = Drift velocity of electrons (in m/s)
q = Charge of a single electron (in Coulombs)
First, let's calculate the number density of electrons:
Number density (n) = Density / (Molar mass * Volume)
The diameter of the wire is 0.62 mm, so the radius (r) will be half of that:
r = 0.62 mm / 2 = 0.31 mm = 0.31 × 10^-3 m
The cross-sectional area (A) of the wire is given by:
A = πr^2
Now, let's calculate the number density (n) of electrons using the given values:
Density = 8900 kg/m^3
Molar mass (M) = 63.5 g/mol = 63.5 × 10^-3 kg/mol
Volume (V) = A × Length
Assuming the length of the wire is not given, we cannot compute the volume accurately. Let's assume a length of 1 meter for this estimation.
So, substituting the values into the equation:
Volume (V) = A × L = πr^2 × L
= π(0.31 × 10^-3)^2 × 1
= 0.301 × 10^-6 m^3
Now, we can calculate the number density (n):
n = Density / (Molar mass * Volume)
= 8900 / (63.5 × 10^-3 * 0.301 × 10^-6)
= 4.66 × 10^28 electrons/m^3
Next, we need to find the electron charge (q). The charge of a single electron is given as:
q = e = 1.6 × 10^-19 C
Substituting the values into the initial equation:
I = nAvq
Rearranging the equation to solve for drift velocity (v):
v = I / (nAq)
Now we can calculate the drift velocity (v):
v = (2.0 × 10^-6 A) / (4.66 × 10^28 electrons/m^3 * π(0.31 × 10^-3 m)^2 * 1.6 × 10^-19 C)
Simplifying the equation further:
v = (2.0 × 10^-6) / (4.66 × π × (0.31 × 10^-3)^2 × 1.6 × 10^-19)
Calculating the value gives:
v ≈ 6.27 × 10^-3 m/s
So, the estimated electron drift velocity in the copper wire is approximately 6.27 × 10^-3 m/s.
To estimate the electron drift velocity, we can use the formula:
v = (I / (n * A * q))
where:
- v is the electron drift velocity
- I is the current in amperes
- n is the number of free electrons per unit volume
- A is the cross-sectional area of the wire
- q is the charge of one electron
Let's break down the steps to calculate the electron drift velocity:
Step 1: Calculate the number of free electrons per unit volume.
The molar mass of copper is given as 63.5 g/mol, and the density of copper is 8900 kg/m^3. We can use these values to find the number of moles per unit volume.
moles per unit volume = (density / molar mass)
In this case, the moles per unit volume will represent the number of copper atoms present per unit volume since we assume one free electron per atom.
Step 2: Calculate the number of free electrons per unit volume (n).
Since we assume one free electron per atom, the number of free electrons per unit volume (n) will be the same as the number of moles per unit volume.
n = moles per unit volume
Step 3: Calculate the cross-sectional area of the wire (A).
The diameter of the wire is given as 0.62 mm. We can find the cross-sectional area using the formula:
A = π * (diameter/2)^2
Step 4: Calculate the charge of one electron (q).
The charge of one electron can be determined by the elementary charge (e), which is approximately 1.6 x 10^-19 C.
q = e
Step 5: Plug the values into the formula to calculate the electron drift velocity (v).
We will use the current value given as 2.0 μA, which is equal to 2.0 x 10^-6 A.
v = (I / (n * A * q))
Now let's calculate the electron drift velocity step by step:
Step 1: Calculate the number of moles per unit volume.
moles per unit volume = (density / molar mass)
moles per unit volume = (8900 kg/m^3) / (63.5 g/mol)
Step 2: Calculate the number of free electrons per unit volume (n).
n = moles per unit volume (calculated in Step 1)
Step 3: Calculate the cross-sectional area of the wire (A).
A = π * (diameter/2)^2
Step 4: Calculate the charge of one electron (q).
q = 1.6 x 10^-19 C
Step 5: Calculate the electron drift velocity (v).
v = (I / (n * A * q))
By using the given values and performing the calculations, you can find an estimate of the electron drift velocity.