Using the same projection model as before: lead reads a signal proportional to

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Bioelectricity, the senses and the physics of the body · Bioelectricity, the senses and the physics of the body

Forces, Machines and Signals, Ch Muscle, heart and the fields outside the body

Problem

Using the same projection model as before: lead kk reads a signal proportional to cos⁡(θ−ϕk)\cos(\theta-\phi_k), with ϕI=0∘\phi_{\mathrm I}=0^\circ, ϕII=60∘\phi_{\mathrm{II}}=60^\circ, ϕIII=120∘\phi_{\mathrm{III}}=120^\circ, where θ\theta is the heart's electrical axis. This time work only with leads II and III. ((a)) Show that II+III\mathrm{II}+\mathrm{III} is proportional to 3sin⁡θ\sqrt3\sin\theta and that II−III\mathrm{II}-\mathrm{III} is proportional to cos⁡θ\cos\theta (and so equals lead I, recovering Einthoven's law). Use the sum-to-product identities cos⁡A+cos⁡B=2cos⁡ ⁣(A+B2)cos⁡ ⁣(A−B2)\cos A+\cos B = 2\cos\!\big(\tfrac{A+B}2\big)\cos\!\big(\tfrac{A-B}2\big) and cos⁡A−cos⁡B=−2sin⁡ ⁣(A+B2)sin⁡ ⁣(A−B2)\cos A - \cos B = -2\sin\!\big(\tfrac{A+B}2\big)\sin\!\big(\tfrac{A-B}2\big). ((b)) Combine the two results to show tan⁡θ=II+III3 I\tan\theta = \dfrac{\mathrm{II}+\mathrm{III}}{\sqrt3\,\mathrm{I}}. ((c)) A trace reads I=0.50 mV\mathrm{I} = 0.50\,\mathrm{mV}, II=−0.10 mV\mathrm{II} = -0.10\,\mathrm{mV}, III=−0.60 mV\mathrm{III} = -0.60\,\mathrm{mV}. Confirm these obey Einthoven's law, then find the electrical axis and say whether it falls inside the normal range of about −30∘-30^\circ to +90∘+90^\circ.

Browse the problem library