In a cathode ray tube, a beam of electrons is injected horizontally into the exact center of the parallel-plate region. Here L = 3.0 cm and d = 0.20 cm. The top plate has a surface charge density of σ0 = +0.15 μC / m2, and the bottom plate has a surface charge density of −σ0. The electrons are injected with a speed of v = 5 × 107 m/s. After a single electron traverses a horizontal distance L, through what change in electric potential (in Volts) does it move?
I am completely lost here. Please help!
To find the change in electric potential as an electron moves through the parallel-plate region, we need to use the formula for the electric potential difference (ΔV) between two points:
ΔV = Ed
Where:
ΔV is the change in electric potential (in Volts),
E is the electric field strength (in N/C),
d is the distance between the points (in meters).
Now, let's break down the problem step by step:
1. Find the electric field strength (E) between the plates:
The electric field strength between two parallel plates is given by the formula:
E = σ / (2ε₀)
Where:
E is the electric field strength (in N/C),
σ is the surface charge density (in C/m²),
ε₀ is the permittivity of free space (8.85 × 10^-12 C²/N m²).
The top plate has a positive surface charge density, so σ = +0.15 μC/m² = +0.15 × 10^-6 C/m².
The bottom plate has a negative surface charge density, so σ = -0.15 μC/m² = -0.15 × 10^-6 C/m².
Plug in these values into the formula to find the electric field strength (E).
2. Calculate the electric potential difference (ΔV):
Now that we have the electric field strength (E) and the distance (d = L), we can find the change in the electric potential (ΔV) using the formula mentioned earlier: ΔV = Ed.
Plug in the values of E and d to calculate ΔV.
3. Calculate the change in electric potential (ΔV) for a single electron:
The electric potential change (ΔV) is measured in volts, but we also need to know it in terms of electron volts (eV). Remember that 1 eV is equal to 1.6 × 10^-19 J.
To find the change in electric potential in eV, simply divide the value of ΔV by 1.6 × 10^-19.
Follow these steps, and you should be able to calculate the change in electric potential (in Volts) through which a single electron moves.