"You build a coil of radius r(m) and place it in a uniform B⃗ field oriented perpendicular to th

"You build a coil of radius r(m) and place it in a uniform...

Key Concepts

Integration of Current Over Time

Integrating the induced current over time gives the total charge that flows through the circuit. This concept is key in determining how the transient behavior of induced currents results in net charge transfer, underlying analyses in various time-dependent electromagnetic processes.

Ohm's Law

Ohm's Law establishes the relationship between voltage, current, and resistance in an electrical circuit. It is used to connect the induced emf in a circuit to the resulting induced current, by stating that the current is equal to the emf divided by the resistance of the circuit.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in the magnetic flux through a loop induces an electromotive force (emf) in the loop. This principle is fundamental in understanding how varying magnetic fields can generate currents in conductors, forming the basis of many electrical generation and transformation devices.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetic field passing through a given area. It is calculated as the product of the magnetic field strength and the area perpendicular to the field, playing a central role in the analysis of electromagnetic phenomena, particularly in systems involving coils and loops.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from a change in magnetic flux. It asserts that the induced current will flow in such a way as to oppose the change in flux that produced it, ensuring the conservation of energy in electromagnetic processes.

Transcript

00:01 This question tells us that we have a coil of radius r, and it is in a uniform magnetic field oriented perpendicular to the coil's surface.

00:14 So this tells me that i don't have to worry about having cosine or sign because the field is already perpendicular to the surface, which means it will be parallel to the surface area vector.

00:28 And that simplifies our calculation for flux later in case that that's something we need.

00:34 And then it's asking us, what is the total electric charge that passes through the coil's wire loop if the magnetic field decreases at a constant rate to zero? so a lot of information there.

00:47 They're looking for the total charge that passes through, so essentially the change in charge.

00:54 And they tell us that b field will decrease.

00:59 At a constant rate to zero.

01:04 So b field is changing.

01:06 And then finally, they say that the resistance of the coils wire is r.

01:11 So how do we approach this question? well, if ultimately changing charge is what we're looking for, hopefully you'll remember that changing charge can be found by the current times the changing time.

01:25 Because by definition, current is how much charge that passes through, a cross -sectional area of a wire over time.

01:34 So rearranging that equation, you have delta q is equal to i times delta t.

01:39 Well, how do i find delta i then? hopefully we remember that once we have the resistance, we can use oms law to find current being emf divided by resistance because v equals i are.

01:57 And how then do i find emf? well, that's where we have to use faraday's law of induction.

02:04 And from there, we can find the induced emf.

02:10 So let's go ahead and solve for that.

02:13 Induced emf is equal to the change in magnetic flux over the changing time.

02:19 Again, i care about the magnitude of emf in this case because it seems like they want us to express a everything algebraically.

02:28 And of course, we're only looking at one coil, so there's no need for n here because it's just going to be one.

02:35 Now, the problem tells us it is b that is changing, so i am going to write that out.

02:45 Basically, i expanded phi into b times a, and i note that because b is already perpendicular to a...

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