The coil of a generator has a radius of 0.14 m . When this coil is unwound, the wire from which

The coil of a generator has a radius of 0.14 m . When this...

Key Concepts

Angular Velocity

Angular velocity is the rate at which an object rotates about an axis. In the context of a rotating coil in a magnetic field, the angular velocity determines how quickly the magnetic flux through the coil changes, thereby affecting the magnitude of the induced emf.

Rotating Coil

A rotating coil in a magnetic field experiences a periodic change in magnetic flux as its orientation relative to the field changes with time. This results in a sinusoidally varying induced emf, which forms the basis of alternating current generation in electrical generators.

Number of Turns in a Coil

The total induced emf in a coil is directly proportional to the number of turns the coil has. Each turn contributes to the net emf, so a coil with more turns will produce a higher voltage for the same rate of change of magnetic flux.

Faraday's Law of Induction

Faraday's Law states that the magnitude of the induced emf in a circuit is proportional to the rate of change of magnetic flux through the circuit. This principle quantitatively describes how a time-varying magnetic field can generate an emf, and it is central to analyzing systems like rotating coils in generators.

Electromagnetic Induction

This concept refers to the process by which a changing magnetic field creates (induces) an electromotive force (emf) in a circuit. It is the fundamental principle that underlies the operation of generators and transformers, where mechanical energy is converted into electrical energy.

Magnetic Flux

Magnetic flux is the measure of the quantity of magnetic field lines passing through a given area. It is defined as the product of the magnetic field strength, the area it penetrates, and the cosine of the angle between the field and the normal to the area. Variations in magnetic flux are essential in inducing an emf according to Faraday's Law.

Transcript

00:02 So in this example, we have a coil inside a generator that is really just a conductive loop of wire that is getting spun around at a certain angular velocity, in our case, which is 25 radiance per second.

00:19 It's getting spun around in a uniform magnetic field, and because of that, it is generating a induced emf.

00:27 So what we want to know is what the maximum emf is that gets produced because of this rotation.

00:37 So we're given the radius of our circular coil, the total length of wire used, the strength of the magnetic field, and the rate at which it's spinning.

00:48 So first, we're going to have to remember that the induced emf, or a loop spinning in a magnetic field like this, it's going to be n, big n, where n is the number of turns in our loop times the cross -sectional area of a single turn times b times omega times the sine of omega t and this is going to be maximized whenever the sign of omega t here is equal to one so we have an ac voltage here and it would look something like this in time so whenever the sign is maximized we get these peaks here so the maximum mf is just given by nab w or nab omega so we're almost where we want to be we don't have a and we don't have n well lucky for us a is just the area of a circle which is just going to be pi r squared and we have r so that's good to go now n is going to be a little more complicated because n is the number of turns number of turns number of turns but all we have is the length of the wire we used.

02:23 So the thing to realize here is that the number of turns we're going to have is going to be equal to our total length of wire divided by the length of one loop of wire.

02:39 And what is the length of one loop of our wire? well it's just going to be the circumference of a circle with radius r.

02:49 And the circumference of a circle is just 2x r.

02:54 So it's total length to the circle...

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