You want to design a generator that produces a peak EMF of 120 V and goes through 60 complete cycles every second. If you are able to produce a magnetic field of 0.15 T, how many coils of wire do you need on the armature? The cross sectional area of one of your loops is 0.71 m2.
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In this problem, you are asked to relate the EMF produced in a rotating loop of wire to the construction of that loop. Whenever you induce EMF or current in a loop by changing the magnetic flux through the loop, the physics you need to consider is induction.
So this is an induction problem. EMF in this loop is due to the changing magnetic flux through the loop as it rotates in the magnetic field. -
In this problem, you are not asked for direction of current and so will not need to use the right hand rule. For that reason, a picture is not really needed. However, it may help you to visualize the problem. The key to induction in a generator is that a coil of wire rotates in the presence of a magnetic field.
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Magnetic induction problems always begin with the definition of EMF:
EMF = -N ΔΦ/Δt = -N Δ(BA cosθ)/Δt (Calculus students use d rather than Δ.)
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No further mathematical solution is needed.
What is a generator?
You can look up generators in your book or on line. Generators produce an induced EMF due to the rotation of coils of wire through a magnetic field. Because this problem only asked about the EMF, you do not need to consider other components to their operation (such as what causes the coils to turn.)
My text gives the equation EMF = NBAω sin(ωt) for generators. Why didn’t you use that?
Algebra-based physics texts typically give the equation EMF = NBAω sin(ωt) for generators. This is just the general induction equation solved for the specific case of a generator. I don’ t like to start with the special case equation immediately because you lose the picture that all induction problems are approached in the same way.
In this case, you know that the magnetic flux through the loop continually changes because the loop rotates.
You can look up generators in your book or on line. Generators produce an induced EMF due to the rotation of coils of wire through a magnetic field. Because this problem only asked about the EMF, you do not need to consider other components to their operation (such as what causes the coils to turn.)
How do you know which right hand rule to use?
Any time you need to relate a magnetic field to the current that causes it, you use the two step right hand rule (there are two vectors: current and field.)
Any time you need to consider the force on a moving charge (or current) in an external magnetic field, you use the three step right hand rule (there are three vectors: force, motion and field.)How do you work induction problems in which the size of the loop or strength of the magnetic field changes?
You work them in exactly the same way. When you get to the point in the problem where you need to put in numbers and solve the equation, either A or B will remain inside the Δ__/Δt operator, and the other variables remain constant. The approach is identical!Generators produce an induced EMF due to the rotation of coils of wire through a magnetic field. The rotating part of a generator with the coil of wire is known as the armature.
Φ is magnetic flux through a loop. You only have an induced EMF when flux changes through the loop.
Magnetic flux through a loop depends on the strength of the magnetic field, the size of the loop, and the orientation of the loop with the magnetic field.
Algebra-based physics texts typically give the equation EMF = NBAω sin(ωt) for generators. This is just the general induction equation solved for the specific case of a generator. I don’ t like to start with the special case equation immediately because you lose the picture that all induction problems are approached in the same way.
The – sign in this equation is merely to remind you that the induced current is such that its magnetic field opposes the change in flux through the loop. It does not tell you the direction of either EMF or current.
“N” is the number of wraps in the loop of wire.
Both the size of the loop and the magnetic field remain the same—they can be factored out of BA cosθ and pulled outside the Δ__/Δt operator. If this doesn’t make sense to you, it is fine to leave them in. Then you will have (BA cosθ)final - (BA cosθ)initial, which is equivalent to BA(cosθfinal – cosθinitial) when A and B don’t change.
θ is the angle between the magnetic field and the orientation of the loop. In this case, that angle keeps changing because the loop continually rotates. Just as x = vt for linear motion in time, θ = ωt for rotational motion in time.
To go from "Δcos(ωt)/Δt"(technically, dcos(ωt)/dt) to "ω sin(ωt)" , you need to use calculus. The time derivative of cos(ωt) is ω(-sin(ωt)). Algebra-based physics books will give you this expression, but note that it merely restates the continually changing orientation of the loop.
You are told the peak EMF. This corresponds to the peak (or greatest) value for sinωt which is 1.
ω is given in radians (unitless) per second. Every rotation corresponds to 2π radians. So if the loops rotate 60 times per second, they go through (60)(2π) = 120π radians per second.
V/m2 T s-1 = V/[m2 (V∙s/m2) s-1] = V/V or no units
Note that you can find MKS units for Tesla on the internet.How do you know which right hand rule to use?
Any time you need to relate a magnetic field to the current that causes it, you use the two step right hand rule (there are two vectors: current and field.)
Any time you need to consider the force on a moving charge (or current) in an external magnetic field, you use the three step right hand rule (there are three vectors: force, motion and field.)
How do you work induction problems in which the size of the loop or strength of the magnetic field changes?
You work them in exactly the same way. When you get to the point in the problem where you need to put in numbers and solve the equation, either A or B will remain inside the Δ__/Δt operator, and the other variables remain constant. The approach is identical! -
In this problem, you are asked to design a generator that will have a peak induced EMF of 120 V. In general, if there is no source of voltage difference in a circuit, you should consider induction as the cause of current even though the word induction was not used in the problem. In this case, the magnetic flux through the loop continually changes because the loop. A changing magnetic flux results in an EMF, which has the same effect in a circuit as a voltage difference.