Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 36C / c

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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},

(1)

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} .

(2)

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} .

(3)

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36C
Paper 3
ii
2021
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 36C / c

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