# 17.1 Current induced by a changing magnetic field

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## Current induced by a changing magnetic field

While Oersted's surprising discovery of electromagnetism paved the way for more practical applications of electricity, it was MichaelFaraday who gave us the key to the practical generation of electricity: electromagnetic induction .

Faraday discovered that a voltage was generated across a length of wire while moving a magnet nearby, such that the distance betweenthe two changed. This meant that the wire was exposed to a magnetic field flux of changing intensity. Furthermore, thevoltage also depended on the orientation of the magnet; this is easily understood again in terms of the magnetic flux. The fluxwill be at its maximum as the magnet is aligned perpendicular to the wire. The magnitude of the changingflux and the voltage are linked. In fact, if the lines of flux are parallel to the wire, there will be no induced voltage.

The emf, $ϵ$ , produced around a loop of conductor is proportional to the rate of change of the magnetic flux, $\phi$ , through the area, $A$ , of the loop. This can be stated mathematically as:

$ϵ=-N\frac{\Delta \phi }{\Delta t}$

where $\phi =B·A$ and $B$ is the strength of the magnetic field.

Faraday's Law relates induced emf to the rate of change of flux, which is the product of the magnetic field and the cross-sectionalarea the field lines pass through.

When the north pole of a magnet is pushed into a solenoid, the flux in the solenoid increases so the induced current will have anassociated magnetic field pointing out of the solenoid (opposite to the magnet's field). When the north pole is pulled out, theflux decreases, so the induced current will have an associated magnetic field pointing into the solenoid (same direction as themagnet's field) to try to oppose the change. The directions of currents and associated magnetic fields can all be found usingonly the Right Hand Rule. When the fingers of the right hand are pointed in the direction of the magnetic field, the thumb points in thedirection of the current. When the thumb is pointed in the direction of the magnetic field, the fingers point in thedirection of the current.

An easy way to create a magnetic field of changing intensity is to move a permanent magnet next to a wire or coil ofwire. The magnetic field must increase or decrease in intensity perpendicular to the wire (so that the magnetic field lines "cut across" the conductor), or else no voltage will be induced.
Finding the direction of the induced current

The induced current generates a magnetic field. The induced magnetic field isin a direction that tends to cancel out the change in the magnetic field in the loop of wire. So, you can use the Right Hand Rule to find the direction of the induced current by remembering that theinduced magnetic field is opposite in direction to the change in the magnetic field.

Electromagnetic induction is put into practical use in the construction of electrical generators, which use mechanical powerto move a magnetic field past coils of wire to generate voltage. However, this is by no means the only practical use for thisprinciple.

If we recall that the magnetic field produced by a current-carrying wire is always perpendicular to the wire, andthat the flux intensity of this magnetic field varies with the amount of current which passes through it, we can see that a wire is capable ofinducing a voltage along its own length if the current is changing. This effect is called self-induction . Self-induction is when a changing magnetic field is produced by changes in current through a wire, inducing a voltage along thelength of that same wire.

If the magnetic flux is enhanced by bending the wire into the shape of a coil, and/or wrapping that coil around a material of high permeability, this effect ofself-induced voltage will be more intense. A device constructed to take advantage of this effect is called an inductor , and will be discussed in greater detail in the next chapter.

## Lenz's law

The induced current will create a magnetic field that opposes the change in the magnetic flux.

Consider a flat square coil with 5 turns. The coil is 0,50 m on each side, and has a magnetic field of 0,5 Tpassing through it. The plane of the coil is perpendicular to the magnetic field: the field points out of the page. Use Faraday'sLaw to calculate the induced emf, if the magnetic field is increases uniformly from 0,5 T to 1 T in 10 s. Determine thedirection of the induced current.

1. We are required to use Faraday's Law to calculate the induced emf.

2. $ϵ=-N\frac{\Delta \phi }{\Delta t}$
3. $\begin{array}{ccc}\hfill ϵ& =& -N\frac{\Delta \phi }{\Delta t}\hfill \\ & =& -N\frac{{\phi }_{f}-{\phi }_{i}}{\Delta t}\hfill \\ & =& -N\frac{{B}_{f}·A-{B}_{i}·A}{\Delta t}\hfill \\ & =& -N\frac{A\left({B}_{f}-{B}_{i}\right)}{\Delta t}\hfill \\ & =& -\left(5\right)\frac{{\left(0,5\right)}^{2}\left(1-0,5\right)}{10}\hfill \\ & =& -0,0625\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}\hfill \end{array}$

The induced current is anti-clockwise as viewed from the direction of the increasing magnetic field.

## Real-life applications

The following devices use Faraday's Law in their operation.

• induction stoves
• tape players
• metal detectors
• transformers

## Research project : real-life applications of faraday's law

Choose one of the following devices and do some research on the internet or in a library how your device works. You will need to refer to Faraday's Law in your explanation.

• induction stoves
• tape players
• metal detectors
• transformers

1. State Faraday's Law in words and write down a mathematical relationship.
2. Describe what happens when a bar magnet is pushed into or pulled out of a solenoid connected to an ammeter. Draw pictures to support your description.
3. Use the Right Hand Rule to determine the direction of the induced current in the solenoid below.

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