A pair of parallel plates are charged with uniform charge densities of –32pC/m2 and +32pC/m2 . The distance between the plates is 2.5mm .
If a free electron is released at rest from the negative plate, find its speed when it reaches the positive plate.
Find the time it takes the electron to travel across the 2.5mm gap.
To find the speed of the electron when it reaches the positive plate, we can use the principle of conservation of energy.
The electric potential difference between the plates can be calculated using the formula:
V = Ed
Where V is the potential difference, E is the electric field, and d is the distance between the plates.
Given that the distance between the plates is 2.5mm and the electric field is uniform between the plates, we can find the electric field by dividing the potential difference by the distance:
E = V/d
Now, the electric field between the plates is produced by the charges on the plates. The electric field due to a uniformly charged plate is given by:
E = σ / (2ε₀)
Where σ is the charge density on the plates and ε₀ is the permittivity of free space.
In this case, the negative plate has a charge density of -32pC/m² and the positive plate has a charge density of +32pC/m². Substituting these values into the formula, we get:
E = (-32pC/m²) / (2ε₀) for the negatively charged plate
E = (32pC/m²) / (2ε₀) for the positively charged plate
Since the charges on the plates are equal in magnitude, the electric field between the plates is the sum of the fields produced by each plate:
E_total = E_negative + E_positive
Now we can substitute the values for the electric field and distance into the formula for potential difference:
V = E_total * d
Substituting the expressions for E_total and d, we get:
V = (E_negative + E_positive) * d
To find the potential difference, we need to know the value of ε₀. The value of ε₀ is approximately 8.85 x 10⁻¹² C²/Nm².
Once we have the potential difference, we can find the change in electric potential energy of the electron as it moves from the negative plate to the positive plate. This change in potential energy is equal to the kinetic energy gained by the electron.
The change in electric potential energy can be calculated using the formula:
ΔPE = qΔV
Where ΔPE is the change in electric potential energy, q is the charge of the electron, and ΔV is the potential difference between the plates.
The charge of an electron is approximately -1.6 x 10⁻¹⁹ C.
Substituting the values into the formula, we get:
ΔPE = (-1.6 x 10⁻¹⁹ C) * ΔV
Now, the change in electric potential energy is equal to the kinetic energy gained by the electron:
ΔPE = (1/2)mv²
Where m is the mass of the electron and v is its speed.
Substituting the values into the formula, we get:
(1/2)mv² = (-1.6 x 10⁻¹⁹ C) * ΔV
We can solve this equation for v:
v = √[(2 * (-1.6 x 10⁻¹⁹ C) * ΔV) / m]
To find the time it takes for the electron to travel across the 2.5mm gap, we can use the equation for velocity:
v = d/t
Where v is the speed of the electron and t is the time taken to travel the distance d.
Rearrange the equation to solve for t:
t = d/v
Substituting the values for d and v, we get:
t = (2.5 x 10⁻³ m) / v
Now you can substitute the values, calculate the speed, and then use it to find the time taken for the electron to cross the gap.