A cylindrical region of radius R is filled with a uniform magnetic field B as shown in the figure. A metal wire (AB) of length L is placed inside the field such that its ends are symmetrically located with respect to the cnetre (O) of the circular cross section of the region. If the magnetic field is changed at a rate $\frac{d B}{d t}$ the emf induced in the metal wire is $ε$ . Find the percentage change in value of $ε$ if the wire is displaced by a small distance L/100 parallel to its own length. Assume that the wire remains inside the field region.

Question

A cylindrical region of radius R is filled with a uniform magnetic field B as shown in the figure. A metal wire (AB) of length L is placed inside the field such that its ends are symmetrically located with respect to the cnetre (O) of the circular cross section of the region. If the magnetic field is changed at a rate $\frac{d B}{d t}$ the emf induced in the metal wire is $ε$ . Find the percentage change in value of $ε$ if the wire is displaced by a small distance L/100 parallel to its own length. Assume that the wire remains inside the field region.

Accepted Answer

The electric field induced at a distance r from the centre is given by $\oint \vec{E} \vec{d} l = - \frac{d ϕ}{d t}$
$\Rightarrow E 2 π r = π r^{2} \frac{d B}{d t} \Rightarrow E = \frac{r}{2} \frac{d B}{d t}$
Component of E along AB is $E_{A B} = E \cos θ = \frac{1}{2} (r \cos θ) \frac{d B}{d t} = \frac{d}{2} (\frac{d B}{d t}) = a$ constant
The emf induced in wire AB is given by $E_{A B} L$ and it will not change even if the wire is displaced by a small amount parallel to itself.

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