# Problem: A loop of wire of radius a = 25mm has an electrical resistance R = 0.039 Ω. The loop is initially inside a uniform magnetic field of magnitude B0 = 1.9T parallel to the loop's axis. The magnetic field is then reduced slowly at a constant rate, which induces a current I = 0.20A in the loop. How long does it take for the magnitude of the uniform magnetic field to drop from 1.9T to zero? Find the time Δt it takes the magnetic field to drop to zero.

###### FREE Expert Solution

Induced emf,

$\overline{){\mathbf{\epsilon }}{\mathbf{=}}{\mathbf{-}}{\mathbf{N}}{\mathbf{A}}\frac{\mathbf{∆}\mathbf{B}}{\mathbf{∆}\mathbf{t}}}$

Ohm's law,

$\overline{){\mathbf{\epsilon }}{\mathbf{=}}{\mathbf{I}}{\mathbf{R}}}$

A = πa2

ΔB = B1 - B0

$\begin{array}{rcl}\mathbf{\epsilon }& \mathbf{=}& \mathbf{-}\mathbf{N}\mathbf{\left(}{\mathbf{\pi a}}^{\mathbf{2}}\mathbf{\right)}\frac{\mathbf{\left(}{\mathbf{B}}_{\mathbf{1}}\mathbf{-}{\mathbf{B}}_{\mathbf{0}}\mathbf{\right)}}{\mathbf{∆}\mathbf{t}}\\ \mathbf{IR}& \mathbf{=}& \mathbf{-}\mathbf{N}\mathbf{\left(}{\mathbf{\pi a}}^{\mathbf{2}}\mathbf{\right)}\frac{\mathbf{\left(}{\mathbf{B}}_{\mathbf{1}}\mathbf{-}{\mathbf{B}}_{\mathbf{0}}\mathbf{\right)}}{\mathbf{∆}\mathbf{t}}\\ \mathbf{∆}\mathbf{t}& \mathbf{=}& \frac{\mathbf{-}\mathbf{N}\mathbf{\left(}{\mathbf{\pi a}}^{\mathbf{2}}\mathbf{\right)}\mathbf{\left(}{\mathbf{B}}_{\mathbf{1}}\mathbf{-}{\mathbf{B}}_{\mathbf{0}}\mathbf{\right)}}{\mathbf{IR}}\end{array}$

###### Problem Details

A loop of wire of radius a = 25mm has an electrical resistance R = 0.039 Ω. The loop is initially inside a uniform magnetic field of magnitude B0 = 1.9T parallel to the loop's axis. The magnetic field is then reduced slowly at a constant rate, which induces a current I = 0.20A in the loop. How long does it take for the magnitude of the uniform magnetic field to drop from 1.9T to zero? Find the time Δt it takes the magnetic field to drop to zero.