Solution

For

f (x) = V_{0} x / L

, direct integration gives

Δ E_{Rad} = \frac{q ^{2} μ _{0}}{3 π m ^{2} c} m /2 \frac{V _{0}}{L} (E - E - V_{0}), E > V_{0},

(152)

and

Δ E_{Rad} = \frac{2 q ^{2} μ _{0}}{3 π m ^{2} c} m /2 \frac{V _{0}}{L} E, E < V_{0} .

(153)

The reflected expression increases up to threshold and the transmitted expression decreases above it. The supremum is therefore approached from below at

E = V_{0}

and equals

\frac{2 q ^{2} μ _{0}}{3 π m ^{2} c} m /2 \frac{V _{0}^{3/2}}{L} .

(154)

Solved by gpt-5.6-sol high.

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