A moving-coil galvanometer used historically to measure electric charge (a "ballistic…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise

Problem

A moving-coil galvanometer used historically to measure electric charge (a "ballistic galvanometer") has its coil suspended so that the angular deflection

\theta(t)

obeys

I\ddot{\theta}+b\dot{\theta}+k\theta = N\,i(t),

where

I

is the coil's moment of inertia,

b

the viscous damping constant of the suspension,

k

the torsional restoring constant, and

N

the torque produced per unit current. Take

I = 2.0\times10^{-7}\,\mathrm{kg\,m^2}

k = 5.0\times10^{-4}\,\mathrm{N\,m\,rad^{-1}}

b = 2.0\times10^{-6}\,\mathrm{N\,m\,s\,rad^{-1}}

and

N = 2.0\times10^{-2}\,\mathrm{N\,m\,A^{-1}}

. Friction other than the stated damping is ignored. The coil hangs at rest,

\theta(0)=0

\dot{\theta}(0)=0

, when a brief current pulse

i(t)=I_0

for

0 \le t \le \tau_p

(zero otherwise) passes through it, with

I_0 = 0.050\,\mathrm{A}

and

\tau_p = 2.0\,\mathrm{ms}

. ((a)) Find the damped angular frequency

\omega_d

of the coil and its natural period

T

, and confirm that

\tau_p \ll T

. ((b)) Because the pulse is far shorter than the period, model it as an instantaneous kick in angular velocity

\Delta\dot{\theta}=NI_0\tau_p/I

delivered at

t=0

, with

\theta

unchanged during the brief pulse, and use the convolution solution built from the causal impulse response to find the resulting maximum deflection

\theta_{\max}

and the time at which it occurs. ((c)) Explain in one sentence why the causal Green function used in part (b) guarantees

\theta(t)=0

for

t < 0

, before the pulse arrives.

Related theory

Ordinary differential equations

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