# Magnetic flux and induced emf

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Question

# A uniform magnetic field B = 0.5 T exists in a circular region of radius R = 5 m. A loop of radius R = 5 m encloses the magnetic field at t = 0 and then pulled at uniform speed v = 2 ms-1 in the plane of the paper. Find the induced emf (in V) in the loop at time t = 3 s.

Moderate
Solution

## ${\mathrm{O}}_{1}\mathrm{M}=\left(\mathrm{vt}-\mathrm{R}\right)+\left(\frac{2\mathrm{R}-\mathrm{vt}}{2}\right)=\frac{\mathrm{v}}{2}$$\mathcal{l}=\mathrm{PN}=2\sqrt{{\mathrm{R}}^{2}-{\left({\mathrm{O}}_{1}\mathrm{M}\right)}^{2}}2\sqrt{{\mathrm{R}}^{2}-{\left(\frac{\mathrm{vt}}{2}\right)}^{2}}=\sqrt{4{\mathrm{R}}^{2}-{\mathrm{v}}^{2}{\mathrm{t}}^{2}}$so $\mathrm{E}=\mathrm{Bvl}=\mathrm{Bv}\sqrt{4{\mathrm{R}}^{2}-{\mathrm{v}}^{2}{\mathrm{t}}^{2}}$given $\mathrm{E}=0.5×2\sqrt{4×25-4×9}=8\mathrm{V}$

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