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# 02. Analysis Tools

### Abrupt Change in Flux

An N-turn loop of radius r is R from a very long straight wire carrying current i. If the current is reduced to zero in a time T, what is the average induced emf in the loop? What is the direction of the induced current? Assume r >> R.

pic 5

When the current is reduced, the magnetic flux through the small loop will change. By Faraday's Law, this change in flux will create an emf in the loop. The first step toward finding the emf involves finding the magnetic flux through the loop of interest. For this example, the direction of the magnetic field and the direction of the loop's area are parallel, so the dot product reduces to:

pic 6

Moreover, since the loop is far from the wire, and the loop's diameter is small, the magnetic field from the wire is approximately constant over the area of the loop, and the flux is given by:

pic 7

The N indicates that there are N loops of wire through which the flux passes.

This flux is reduced to zero over some finite time interval. Interpreting Faraday's Law over a finite time interval results in:

pic 8

This is the average emf induced during the time period under investigation.

Since initially the magnetic field through the loop was in the +z-direction, and was then removed, the induced emf will drive an induced current counterclockwise around the loop in an attempt to counter the reduction in flux. In other words, the induced current will try to maintain a constant magnetic flux through the loop.

### Continuous Change in Flux

A 1000-turn secondary coil of radius 2.0 cm is concentric with a 400-turn primary coil of radius 20 cm carrying AC current at 60 Hz with peak current 2.0 A. (AC current can be modeled as i(t) = i0 cos (2pft).) What is the induced emf in the secondary coil as a function of time?

pic 9

Again, the first step toward finding the emf involves finding the magnetic flux through the secondary loop. For this example, the direction of the magnetic field and the direction of the loop's area are parallel, flux reduces to:

pic 10

Moreover, since the magnetic field is approximately constant over the relatively small area of the secondary loop, the flux is given by:

pic 11

Plugging in values yields:

pic 12

Applying Faraday's Law:

pic 13

Thus, the secondary coil of wire has 60 Hz AC voltage 90° out of phase with the voltage in the primary coil. (Sine and cosine functions are 90° out of phase with each other.) The maximum emf in the secondary coil is 1.19 V.