Question

A metal loop moves at constant velocity toward a long wire carrying a steady current I, as shown in the figure below. The current induced in the loop is directed

Answer 1

Concepts and reason

The concepts used to solve this problem are magnetic field due to a current carrying long straight wire, magnetic flux, Lenz’s law, and maxwell's right-hand thumb rule.

First, find the change in the magnetic field using the formula of the magnetic field due to a current carrying long straight wire then find the change in magnetic flux using its formula. Later, apply the Lenz's law and Maxwell’s right-hand thumb rule to find the direction of the induced current.

Fundamentals

Magnetic field due to current carrying long straight wire:

The magnetic field due to current carrying long straight wire can be defined by the relation:

$\vec B = \frac{{{\mu ^\circ }I}}{{2\pi r}}$

Here, $\overrightarrow B$ is the magnetic field, ${\mu ^\circ }$ is the magnetic permeability, $I$ is current and $r$ is the shortest distance of the point where the magnetic field needs to be found to the wire.

Magnetic flux:

Magnetic flux is the total number of magnetic field lines passing through some area per unit time.

The relation of magnetic flux can be written as:

$\phi = \int {\overrightarrow B } \cdot \overrightarrow {dS}$

Here, $\phi$ is the magnetic flux, $\overrightarrow B$ is the magnetic field, and $\overrightarrow {dS}$ is the area element.

Lenz’s law:

According to this law, if there is the change in magnetic flux through a closed circuit, there will be an induced electromotive force in the circuit. The induced electromotive force generates a current such a way that the magnetic field of the induced current opposes the change in the original magnetic flux.

It can be described as:

$\varepsilon = - \frac{{d\phi }}{{dt}}$

Here, $\phi$ is the magnetic flux, $t$ is the time, and $\varepsilon$ is the electromotive force.

Maxwell’s right-hand thumb rule:

Maxwell’s right-hand thumb explains the direction of the current and magnetic field. If a current carrying straight wire is held by the right hand such a way that the direction of thumb represents the direction of current then the fingers which warp fingers in a circular way represent the direction of the magnetic field that is generated by the current.

The same rule is applicable if the magnetic field and current are interchanged by each other. In the circuit like circular ring or coil if the warp fingers represent the direction of current then thumb will represent the direction of the magnetic field generated by the current.

A long wire is carrying steady current $I$ and a metal circular loop is moving towards the wire with a uniform velocity $\vec v$ . The situation is shown in the figure.

If the moment of time the ring is at distance $r$ from the wire, The magnetic field intensity can be written as,

$\vec B = \frac{{{\mu ^\circ }I}}{{2\pi r}}$

Here, the term $\frac{{{\mu ^\circ }I}}{{2\pi }}$ is constant; therefore the magnetic field only depends on the distance of the circular loop from the wire.

This dependency can be seen as:

$\vec B \propto \frac{1}{r}$

Therefore if distance will decrease the magnetic field intensity will increase and in the current situation, the loop is moving towards the wire. Hence, the intensity of the magnetic field at the loop will increase with time.

Now using the concept of magnetic flux:

$\phi = \int {\overrightarrow B } \cdot \overrightarrow {dS}$

Here, the area of the circular loop is constant, so flux through the circular ring will only depend upon the magnetic field.

The relation can also be written as:

$\phi = \overrightarrow B \cdot \overrightarrow A$

Here, $A$ is the constant area of the circular loop.

So it can be said that if the magnetic field will increase, the magnetic flux will also increase. Therefore, if the circular loop will move towards the wire, the flux through the wire will increase with time.

The concept of Lenz’s law,

$\varepsilon = - \frac{{d\phi }}{{dt}}$

According to formula if there is the change in the magnetic flux through the closed loop with the time, there will an induced electromotive force in the loop. According to Lenz’s law, the electromotive force will induce a current in the loop in such a way that the direction of the magnetic field due to induced current will oppose the original magnetic flux.

Therefore, to find the direction of the current, first, find the direction of the magnetic field due to current carrying long wire.

According to Maxwell’s right-hand thumb rule, if the direction of the right-hand thumb is towards the current in the long wire, then the direction of curled fingers will be in the outside direction through the loop. Hence the direction of magnetic flux due to current carrying log wire the magnetic flux will be outward to the loop.

The current is needed to be induced in such a way that its magnetic field should be opposing the initial flux. Therefore the direction of the magnetic field generated by the induced current will be inside the loop.

Now again applying Maxwell’s right-hand thumb rule, if the direction of thumb is inside the loop for defining the direction of the magnetic field, then the direction of curled fingers will tell about the direction of the induced current. The direction of the fingers will be clockwise. Therefore, the direction of induced current in the circular loop will be clockwise.

Ans:

The induced current in the circular loop will be directed clockwise.