An electron starts from one plate of a charged closely spaced (vertical) parallel plate arrangement with a velocity of 1.78×104m/s to the right. Its speed on reaching the other plate, 2.25cm away, is 3.63×104m/s .
If the plates are square with an edge length of 26.0cm , determine the charge on each.
To solve this problem, we can use the concepts of electric potential and energy conservation.
Step 1: Calculate the change in electric potential energy (ΔPE) of the electron as it moves from one plate to the other. We can use the formula:
ΔPE = q * ΔV
where q is the charge of the electron and ΔV is the change in electric potential.
Step 2: Calculate the initial and final electric potentials (V_i and V_f) at the two plates. For a uniformly charged plate, the electric potential can be calculated using the formula:
V = k * q / r
where k is the Coulomb's constant (9 x 10^9 N m^2/C^2), q is the charge on the plate, and r is the distance from the plate.
Step 3: Calculate the change in electric potential (ΔV) using the formula:
ΔV = V_f - V_i
where V_i is the initial electric potential and V_f is the final electric potential.
Step 4: Calculate the charge on each plate of the parallel plate arrangement. Since the plates are closely spaced, the electric field between them is approximately uniform, and we assume the electron gains its entire kinetic energy as electric potential energy.
Let's perform the calculations:
Given values:
Initial velocity (v_i) = 1.78 x 10^4 m/s
Final velocity (v_f) = 3.63 x 10^4 m/s
Distance between the plates (d) = 2.25 cm = 0.0225 m
Edge length of the plates (a) = 26.0 cm = 0.26 m
Step 1: Calculate the change in electric potential energy (ΔPE) of the electron.
ΔPE = (1/2) * m * (v_f^2 - v_i^2)
= (1/2) * (9.11 x 10^-31 kg) * [(3.63 x 10^4 m/s)^2 - (1.78 x 10^4 m/s)^2]
= 4.39 x 10^-17 J
Step 2: Calculate the initial and final electric potentials (V_i and V_f) at the two plates.
V_i = k * q / a
V_f = k * q / (a + d)
where a is the edge length of the plates, and d is the distance between the plates.
Step 3: Calculate the change in electric potential (ΔV).
ΔV = V_f - V_i
Step 4: Calculate the charge on each plate (q).
ΔPE = q * ΔV
Now let's perform the calculations step by step:
Step 2 (continued): Calculate the initial and final electric potentials (V_i and V_f) at the two plates.
V_i = (9 x 10^9 N m^2/C^2) * q / a
V_f = (9 x 10^9 N m^2/C^2) * q / (a + d)
Step 3: Calculate the change in electric potential (ΔV).
ΔV = V_f - V_i
= (9 x 10^9 N m^2/C^2) * q / (a + d) - (9 x 10^9 N m^2/C^2) * q / a
Step 4 (continued): Calculate the charge on each plate (q).
ΔPE = q * ΔV
4.39 x 10^-17 J = q * [(9 x 10^9 N m^2/C^2) * q / (a + d) - (9 x 10^9 N m^2/C^2) * q / a]
Simplify the equation by substituting the known values:
4.39 x 10^-17 J = q * [(9 x 10^9 N m^2/C^2) * q / (0.26 m + 0.0225 m) - (9 x 10^9 N m^2/C^2) * q / 0.26 m]
Now we can solve this equation to find the charge on each plate.
To determine the charges on each plate, we can use the principle of conservation of energy.
The initial kinetic energy of the electron is given by:
KE1 = (1/2) * m * v1^2
where m is the mass of the electron and v1 is the initial velocity.
The final kinetic energy of the electron is given by:
KE2 = (1/2) * m * v2^2
where v2 is the final velocity.
Since we know the initial and final velocities, we can calculate the initial and final kinetic energies.
The potential difference (V) between the plates can be calculated using the distance between the plates (d) and the final velocity (v2) of the electron:
V = (1/2) * m * v2^2 = q * d * E
where q is the charge on the electron, E is the electric field between the plates, and d is the distance between the plates.
Since we know the distance between the plates, we can solve for the electric field (E) using the final velocity and the given distance.
E = (v2^2) / (2 * d)
Using the electric field value, we can calculate the potential difference (V) between the plates.
Since V is equal to q * d * E, we can solve for the charge on each plate using the equation:
q = V / (d * E)
Substituting the values of V, d, and E, we can calculate the charge on each plate.