Final answer:
To calculate the power delivered to an electron in a television tube at a certain displacement, one must find the electron's velocity at that point and then use the kinetic energy and the distance to estimate the power. This involves using principles of work, energy, and kinematics.
Step-by-step explanation:
To find the power delivered to an electron at a given instant in a television CRT, we can use the work-energy principle and the definition of power.
The work done on the electron to bring it to speed v from rest over a distance d is equal to the kinetic energy (KE) the electron has at the speed v. So, KE = (1/2)mv²where m is the mass of the electron and v is its speed.
Power P, at an instant, is given by the derivative of work with respect to time. However, since we don't have the function of work with respect to time, we look at small changes such that P ≈ ΔW/Δt. Since ΔW can be approximated by the kinetic energy at that instant and Δt can be replaced by Δd/v (where Δd is the displacement and v is the velocity at that displacement), we get P ≈ (1/2)mv²/(Δd/v).
To get the velocity at 1.0 cm displacement, we'll assume that the acceleration is uniform and use the relation v²= u² + 2as, where u is the initial velocity (which is 0 in this case), a is the acceleration, and s is the displacement. With the final velocity v_f given to be 8.4 × 10⁷ m/s at a total displacement d_total of 2.5 cm, we can find a and then v at 1.0 cm:
a = v_f² / (2 × d_total)
Now using this acceleration, we find v for a displacement of 1.0 cm:
v = √(0 + 2 × a × 1.0 cm) = √(2 × a × 0.01 m)
With this velocity, we can now calculate power at 1.0 cm displacement:
P = (1/2)m × (v²) / (0.01 m / v) = (1/2)m × v³¹/ 0.01 m
Finally, we plug in the mass of the electron (m = 9.11 × 10⁻³¹ kg) and our found velocity v to calculate the power P.
Use this approach to calculate the power, and compare the result with the given options to find the correct answer.