Physics

Physics

Problem

A circular loop expands such that its radius varies with time according to R(t)=0.050 m+2.0×10−3tR(t) = 0.050\,\text{m} + 2.0\times10^{-3}t, where tt is in seconds. The loop is placed coaxially in a non-uniform magnetic field described by B(r)=0.40 r T m−1B(r) = 0.40\,r\,\text{T\,m}^{-1}, where rr is the distance from the centre in metres. Find the magnitude of the emf induced in the loop at t=10 st = 10\,\text{s}. a) 3.7×10−5 V3.7\times10^{-5}\,\text{V} b) 7.9×10−6 V7.9\times10^{-6}\,\text{V} c) 2.5×10−5 V2.5\times10^{-5}\,\text{V} d) 1.2×10−5 V1.2\times10^{-5}\,\text{V}
Intuitive explanation

This problem combines Faraday's law with a spatially varying magnetic field—the key is that the field strength itself depends on position, not just time. You'll need to find the magnetic flux through the expanding loop by integrating the non-uniform field over the area, then differentiate that flux with respect to time to get the induced emf. The expanding radius is what drives the change in flux, so pay attention to how R(t)R(t) appears in your integration limits.

Intuitive explanation

This problem combines Faraday's law with a spatially varying magnetic field—the key is that the field strength itself depends on position, not just time. You'll need to find the magnetic flux through the expanding loop by integrating the non-uniform field over the area, then differentiate that flux with respect to time to get the induced emf. The expanding radius is what drives the change in flux, so pay attention to how R(t)R(t) appears in your integration limits.

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