A toy electric generator has a 20 -turn circular coil with...

A toy electric generator has a 20 -turn circular coil with each turn of radius 1.8 $\mathrm{cm} .$ The coil resides in a $1.0-\mathrm{T}$ magnitude $\vec{B}$ field. It also has a lightbulb that lights if the potential difference across it is about 1 $\mathrm{V}$ . You start rotating the coil, which is initially perpendicular to the $\vec{B}$ field. ( a ) Determine the time interval needed for a $90^{\circ}$ rotation that will produce an average induced emf of 1.0 $\mathrm{V}$ . (b) Use a proportion technique to show that the same emf can be produced if the time interval for one rotation is reduced by one-fourth while the radius of the coil is reduced by one-half.

A toy electric generator has a 20 -turn circular coil with each turn of radius 1.8 $\mathrm{cm} .$ The coil resides in a $1.0-\mathrm{T}$ magnitude $\vec{B}$ field. It also has a lightbulb that lights if the potential difference across it is about 1 $\mathrm{V}$ . You start rotating the coil, which is initially perpendicular to the $\vec{B}$ field. ( a ) Determine the time interval needed for a $90^{\circ}$ rotation that will produce an average induced emf of 1.0 $\mathrm{V}$ . (b) Use a proportion technique to show
that the same emf can be produced if the time interval for one rotation is reduced by one-fourth while the radius of the coil is reduced by one-half.

Key Concepts

Angular Velocity

Angular velocity is the rate at which an object rotates and is crucial in determining the time rate of change of the magnetic flux through a rotating coil. It directly influences the magnitude of the induced EMF as per Faraday's law.

Coil Geometry

The shape and size of a coil, specifically its area, directly affect the amount of magnetic flux it intercepts. For a circular coil, the radius determines its area, and thus alterations in the coil's dimensions will change the magnetic flux passing through it.

Scaling and Proportional Reasoning

This involves understanding how modifying parameters such as coil radius and rotation time proportionally affects the induced EMF. It is important for analyzing and predicting how adjustments in the system can maintain or change the performance of electromagnetic devices.

Faraday's Law

Faraday's law quantitatively relates the induced EMF in a circuit to the rate of change of magnetic flux through the circuit. It is a fundamental principle that explains how varying magnetic fields, such as those created by a rotating coil, produce an electromotive force.

Electromagnetic Induction

This is the process through which a changing magnetic flux induces an electromotive force (EMF) in a circuit. It forms the basis for the operation of generators and transformers by converting mechanical energy into electrical energy, and vice versa.

Magnetic Flux

Magnetic flux measures the quantity of magnetic field passing through a given area and is dependent on both the strength of the magnetic field and the orientation of the surface with respect to the field. Changes in flux as the coil rotates are what cause the induced EMF.

Transcript

00:01 In this question, we have a toy electric generator that has a 20 -turn coil, so n is 20, each with radius 1 .8 centimeters.

00:13 So r is equal to 1 .8 centimeters.

00:16 I'm just going to go ahead and convert it to meters, so i don't have to worry about it later.

00:21 And it says the coil resides in a 1 .0 tesla magnitude of magnetic field.

00:29 So b is equal to 0 .1 tesla.

00:33 It has a light bulb that lights if the potential difference across it is about 1 volts.

00:39 So since i have a coil, i have magnetic field, i bet this is about induction again.

00:44 So this is going to be induced emf of 1 volts.

00:49 This is necessary to light the light bulb.

00:53 And you start rotating the coil, which is initially perpendicular to the.

00:59 The magnetic field.

01:01 So when the coil, i'm just going to go draw it really quickly, when the coil is initially perpendicular to the field, that's my coil.

01:13 And if that's my field, then the angle between the field, magnetic field vector and the area vector, which is normal to the surface of the coil, notice that the angle between them initially is zero degrees.

01:34 Okay, so that's what that information tells us.

01:38 And letter a says, determine the time interval needed for a 90 degree rotation that will produce an average induced emf of one volts.

01:51 So that's letter a.

01:53 In order to have a 90 degree rotation that means my change in angle will just be 90 degrees and initially my angle is 0 and at the end my angle will be 90 degrees.

02:12 So to set this question up i have to use faraday's law of induction.

02:18 So emf is equal to n times change in magnetic flux over the changing time.

02:25 And i'm going to keep expanding this equation.

02:29 We know n.

02:30 We know b.

02:31 It's a constant.

02:33 Because we have r, we can figure out a.

02:36 The main thing is you have to recognize it is the angle that is changing.

02:41 So my changing angle will go from a cosine of 90 to cosine of zero...