MAGNETISM AND ELECTROMAGNETISM: Electromagnetic Induction and the Electromotive Force ( emf )

Q: A metallic coil that consists of 200 ( N ) turns encloses an area ( A ) of 100 cm². The coil is placed within a magnetic field ( B ) that is perpendicular to its area, and it has a magnetic flux density of 0.50 T. Next, the field is shut off, and its magnitude decreases to zero in 200 milliseconds ( ms ). What is the average emf induced in the conducting material? If the coil has a resistance ( R ) of 25 Ω , what magnitude of current will be induced within it?

A: Let’s begin by taking a look at the equation for the average emf ( Ɛ ) that will be induced in a coil that contains a given quantity of loops:

Ɛ = – NΔɸ/Δt

Please recall that the negative sign is indicative of an induced force that always acts in opposition to the accelerating magnetic flux ( ɸ ) passing through the conducting loop at hand. Furthermore, the voltage that is induced across the conductor at hand is proportional to the number of loops it contains. Unfortunately, we are not given the value of the flux that is initially present within the system, so we must derive it using the following equation:

ɸ = BA cos θ

A common source of confusion with this equation regards the use of the cosine θ function. The area within each loop is indeed perpendicular to the flux when the maximum amount of flux penetrates it. Thus, since the sin 90^o function is equal to 1, it seems as if the sin θ function would be used in the flux expression; however, we must consider how the magnetic field associated with the induced current affects the region of space immediately outside of the loop’s area. If we extend the circular fields far enough outward, they will intersect at the midpoint of a normal that is perpendicular to the plane of interest:

When incoming lines of flux directly overlap with the imaginary normal in question, a 0^o angle exists between them. Since cos 0^o is 1, the expression above will model this and other circumstances accordingly:

ɸ = BA cos 0^o

cos 0^o = 1

ɸ = BA

ɸ = ( 0.50 T )( 0.0100 m² )

ɸ = 0.0050 Tm²

Hold up!!!

This value gets us close to where we need to be, but we must keep in mind that the change in flux is of interest to us. The final flux value is zero. In order to arrive at this value, a decrease of 0.0050 Tm² had to occur within the system:

Δɸ = – 0.0050 Tm²

We now have all of the terms needed to determine the average emf induced within the coil:

Ɛ = – NΔɸ/Δt

Ɛ = – ( 200 )( – 0.0050 Tm² / 0.200 s )

Ɛ = 5.0 V

If the magnetic flux reversed direction and entered the loop over the same time interval, this increase in flux would induce a negative voltage value. Fortunately, we determine the resulting current in the coil via usage of Ohm’s Law:

( V / R ) = I

( 5.0 V / 25 Ω ) = 0.20 A