A very brief current pulse in a wire is modelled as

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Fourier analysis and integral transforms

Mathematical Methods for Physics, Ch Delta functions, distributions and generalised functions

Problem

A very brief current pulse in a wire is modelled as I(t)=I0 δ(t−t0)I(t) = I_0\,\delta(t-t_0), with I0=2.5×10−3 CI_0 = 2.5\times10^{-3}\,\mathrm{C} (a coefficient carrying units of charge, since δ(t)\delta(t) has units s−1\mathrm{s}^{-1}) and t0=1.0×10−3 st_0 = 1.0\times10^{-3}\,\mathrm{s}. ((a)) Find the total charge delivered, Q=∫−∞∞I(t) dtQ = \int_{-\infty}^{\infty} I(t)\,\mathrm{d}t. ((b)) The voltage across the wire is V(t)=V0 e−t/τV(t) = V_0\,e^{-t/\tau} with V0=12 VV_0 = 12\,\mathrm{V} and τ=5.0×10−3 s\tau = 5.0\times10^{-3}\,\mathrm{s}. Find the energy delivered, W=∫−∞∞V(t)I(t) dtW = \int_{-\infty}^{\infty} V(t)I(t)\,\mathrm{d}t. ((c)) Using the transform of a shifted spike, show that the magnitude ∣I~(ω)∣|\tilde I(\omega)| of the pulse's Fourier transform is the same at every angular frequency, and give its value.

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