Motion of a charged particle: A pair of charged conducting plates produces a uniform field of 12,000 N/C, directed to the right, between the plates. The separation of the plates is 40 mm. An electron is projected from plate A, directly toward plate B, with an initial velocity of v_o = 2.0 × 10⁷m/s, as shown in the figure. (e= 1.60 × 10^-19C, ε₀ = 8.85 × 10^-12 C²/N ∙ m², m_el = 9.11 × 10^-31 kg) The velocity of the electron as it strikes plate B is closest to A) 1.2 × 10⁷ m/s. B) 1.5 × 10⁷ m/s. C) 1.8 × 10⁷ m/s. D) 2.1 × 10⁷ m/s. E) 2.4 × 10⁷ m/s.

Question

Quizplus · Accepted Answer

The force on a charged particle in an electric field is given by F = qE, where q is the charge of the particle and E is the electric field strength. In this case, the electron has a charge of -1.6 × 10^-19 C and is in an electric field of 12,000 N/C directed to the right. So the force on the electron is:

F = qE = (-1.6 × 10^-19 C)(12,000 N/C) = -1.92 × 10^-15 N

The force is negative because the electron is negatively charged and the field is directed to the right. According to Newton's second law, F = ma, where m is the mass of the particle and a is its acceleration. Solving for acceleration, we get:

a = F/m = (-1.92 × 10^-15 N)/(9.11 × 10^-31 kg) = -2.11 × 10^15 m/s^2

The acceleration is negative because the force is negative and the mass is positive. The negative sign indicates that the electron is accelerating to the left. We can use the kinematic equation

v^2 = u^2 + 2as

to find the final velocity v of the electron when it reaches plate B. Here, u is the initial velocity, s is the distance between the plates (40 mm or 0.04 m), and a is the acceleration we just calculated.

v^2 = u^2 + 2as
v^2 = (2.0 × 10^7 m/s)^2 + 2(-2.11 × 10^15 m/s^2)(0.04 m)
v^2 = 4.00 × 10^14 m^2/s^2 - 1.69 × 10^14 m^2/s^2
v^2 = 2.31 × 10^14 m^2/s^2

Taking the square root, we get:

v = 4.81 × 10^7 m/s

So the velocity of the electron as it strikes plate B is closest to 1.5 × 10^7 m/s. Therefore, the best choice is B.

Motion of a Charged Particle: a Pair of Charged Conducting