Question

In the figure, an electron with an initial kinetic energy of 4.00 keV enters region 1 at time t = 0. That region contains a uniform magnetic field directed into the page, with magnitude 0.00510 T. The electron goes through a half-circle and then exits region 1, headed toward region 2 across a gap of 25.0 cm. There is an electric potential difference ΔV = 2000 V across the gap, with a polarity such that the electron's speed increases uniformly as it traverses the gap. Region 2 contains a uniform magnetic field directed out of the page, with magnitude 0.0184 T. The electron goes through a half-circle and then leaves region 2. At what time t does it leave?

Answer 1

As the electron has 4keV, it will enter the region-1 with 3.75x107 m/s. As soon as it enters the magnetic field the magnetic force tries to rotate in circular path but the electron completes half revolution and comes out of region-1. The magnetic force provides centripetal force. Let t1 be the time taken by the electron to complete half revolution. t1 = 3.498 ns

After coming from region-1, it travels 25cm to enter region-2. During this journey the electron is accelerated by 2kV potential difference. Hence its energy becomes 6keV. Hence it enters the region-2 with 4.6x107 m/s. As soon as it enters the magnetic field the magnetic force tries to rotate in circular path but the electron completes half revolution and comes out of region-2. The magnetic force provides centripetal force. Let t3 be the time taken by the electron to complete half revolution. t3 = 0.9696 ns

From region-1 to region-2 the electron travels 25cm with uniform acceleration. Its initial velocity is 3.75x107 m/s and final velocity is 4.6x107 m/s. The distance travelled is the product of average velocity and time t2. t2 = 5.988 ns

The total time is 10.4556 ns.