Averag induced emf

Question

A 50 loop circular coil has a radius of 0.03 m.  It is oriented so that the field lines of a magnetic field are normal to the area of the coil.  Suppose that the magnetic field is varied so that B increase from 0.10 T to 0.35 T in 0.002 seconds.  Find the average induced emf in the coil.

Answer 1

Hi Doug!

 

This turns out to be plug-and-play with Faraday's law:

 

EMF = N(ΔΦ_B/Δt)

 

where

 

Φ_B = magnetic flux

N = number of turns of coil

 

Since the magnetic flux is defined as:

 

Φ_B = BAcos(θ)

 

where

 

B = magnetic field strength

A = area of loop

θ = angle between the normal line to the loop and the magnetic field

 

then

 

ΔΦ_B = Δ(BAcosθ)

 

But θ (representing the orientation between the loop and field) is not changing, and neither is the area of the loop.  The only element that goes into the flux which is changing is the magnetic field.  So:

 

ΔΦ_B = Δ(BAcosθ) = Acosθ(ΔB)

 

So Faraday's law becomes:

 

 

EMF = N(Acosθ(ΔB/Δt)

 

where

 

N = 50 turns

A = area of a circle with radius 0.03 m (= π(0.03 m)²)

θ = 0^o (since the magnetic field is described as normal to the areas of the coil, and thus parallel to the normal to that area)

ΔB = B_f - B_i = 0.35 T - 0.10 T = 0.25 T

ΔT = 0.002 s

 

Hope that helps!  If you have further questions or problems, or would like to check an answer, just let me now.

                                                                                                                                                                                                                                                                  

Answer 2

Let E = the induced emf. According to Faraday's law of electromagnetic induction,

E = -dΦ/dt,

where Φ = magnetic flux = B(t)NA,
N = number of turns = 50
A = cross sectional area of coil = πr²
r = radius = 0.03 m.

If the magnetic field changes at a constant rate, then

dΦ/dt = NA (ΔB/Δt) = 50 π(0.03)² (0.35 - 0.10) / (0.002) volts = 17.67 volts

Then

E = -dΦ/dt = -17.67 volts

The minus sign indicates that the induced emf opposes the change in magnetic flux, per Lenz's law.