Question

Two parallel plates are \(4.0 \mathrm{cm}\) apart. The bottom plate is charged positively and the top plate is charged negatively, producing a uniform electric field of \(5.0 \times 10^{4} \mathrm{N} / \mathrm{C}\) in the region between the plates. What is the time required for an electron, which starts at rest at the upper plate, to reach the lower plate? (Assume a vacuum exists between the plates.)

Short answer

Answer: It takes approximately \(9.56 \times 10^{-9}\text{ s}\) for an electron to travel from the upper plate to the lower plate.

Step-by-step solution

Identify the Known Parameters

We're given: 1. Distance between plates (d): \(4.0\text{ cm}\) (converting to meters, \(d = 0.04\text{ m}\)) 2. Electric field (E): \(5.0 \times 10^{4} \text{ N/C}\) 3. Electron's initial speed (v_0): \(0\text{ m/s}\) (starts at rest) 4. Charge of the electron (e): \(-1.6 \times 10^{-19}\text{ C}\) 5. Mass of electron (m): \(9.11 \times 10^{-31}\text{ kg}\) We need to find the time (t) it takes for the electron to reach the lower plate.

Calculate Acceleration Due to Electric Field

The force acting on an electron due to an electric field is given by: $$F = e \times E$$ Where F is the force, e is the electron's charge, and E is the electric field strength. Since F = m × a, where m is the mass of the electron and a is the acceleration, we can solve for acceleration: $$a = \frac{e \times E}{m}$$ Substitute the given values: $$a = \frac{-1.6 \times 10^{-19}\text{ C} \times 5.0 \times 10^{4} \text{ N/C}}{9.11 \times 10^{-31}\text{ kg}}$$ Calculate the value of a: $$a = -8.77 \times 10^{13} \text{ m/s}^2$$ The negative sign indicates the acceleration is towards the positively charged bottom plate.

Use Kinematic Equations to Find Time

Using the equation \(d = v_0 t + \frac{1}{2}at^2\), where d is the distance, v_0 is the initial velocity, a is the acceleration, and t is the time, we can solve for the time t: $$0.04\text{ m} = 0\text{ m/s} \times t -\frac{1}{2}(8.77 \times 10^{13}\text{ m/s}^2) \times t^2$$ Simplify the equation: $$0.04\text{ m} = -4.385 \times 10^{13}\text{ m/s}^2 \times t^2$$ Solve for t: $$t^2 = \frac{0.04\text{ m}}{-4.385 \times 10^{13}\text{ m/s}^2}$$ $$t^2 = 9.124 \times 10^{-16}\text{ s}^2$$ $$t = \sqrt{9.124 \times 10^{-16}\text{ s}^2}$$ $$t = 9.56 \times 10^{-9}\text{ s}$$ The time required for the electron to reach the lower plate is approximately \(9.56 \times 10^{-9}\text{ s}\).