Question

A long, straight conductor passes through the center of a metal ring, perpendicular to its plane. If the current in the conductor increases, is a current induced in the ring? Explain.

Short answer

No, the current is not induced in the ring.

Step-by-step solution

The definition of magnetic flux

It can be defined as the number of magnetic field lines passing through the given closed surface.

Mathematically it can be given by

$\mathrm{\varphi }=\mathrm{BAcos}\left(\mathrm{\theta }\right)$

Where, $\mathrm{\varphi }$is the magnetic flux, B is applied magnetic field and A is surface area of closed object and $\mathrm{\theta }$is the angle between magnetic field lines and area vector of surface.

Faraday Law of EMF

Faraday Law of EMF states that the induced emf in a loop equals to the negative of the time rate of change of magnetic flux through the loop.

Mathematically it can be given by

$$\begin{gathered} \mathrm{e}=\frac{\mathrm{d\varphi }}{\mathrm{dt}} \\ =\frac{\mathrm{Bd}\left(\mathrm{A}\right)}{\mathrm{dt}} \end{gathered}$$

Where, e is the induced EMF.

Explanation

The magnetic field lines produced by long straight conductors are concentric circles and parallel to the plane of the ring. Therefore the magnetic field and area element are perpendicular to each other.

Hence, the magnitude of magnetic flux passing through the ring is zero.

Thus there is no current is induced in the ring.