The potential energy barrier in field emission is not rectangular, but resembles a ramp, a...

The potential energy barrier in field emission is not rectangular, but resembles a ramp, as shown in Figure 6.16. Here we compare tunneling probability calculated by the crudest approximation to that calculated by a better one. In method 1, calculate T by treating the barrier as an actual ramp in which U-E is initially φ, but falls off with a slope of M. Use the formula given in Exercise 37. In method 2, the cruder one, assume a barrier whose height excceds E by a constant φ/2 (the same as the average excess for the ramp) and whose width is the same as the distance the particle tunnels through in the ramp

(a) Show that the ratio, is

(b) Do the methods differ more when tunneling probability is relatively high or relatively low?