A 30.0 cm ×60.0 cm rec- tangular circuit containing a a 15Ωresistor is perpendicular to a unifor

step 1 For this problem, we have a 30 centimeter by 60 centimeter circuit. In this circuit, there's a 1

A 30.0 cm ×60.0 cm rec- tangular circuit containing a a...

A $30.0 \mathrm{cm} \times 60.0 \mathrm{cm} \mathrm{rec}-$ tangular circuit containing a a 15$\Omega$ resistor is perpendicular to a uniform magnetic field that starts out at 2.65 $\mathrm{T}$ and steadily decreases at 0.25 $\mathrm{T} / \mathrm{s}$ . (See Figure $21.48 . )$ While this field is changing, what does the ammeter read?

Key Concepts

Ohm's Law

Ohm's Law relates the current flowing through a circuit to the voltage across it and the resistance of the circuit, typically expressed as I = V/R. When an electromotive force is induced in a circuit, Ohm's Law allows us to determine the magnitude of the resulting current by accounting for the total resistance encountered in the circuit. This concept is key in analyzing how induced voltages lead to observable currents.

Magnetic Flux

Magnetic flux is a measure of the amount of magnetic field passing through a given area. It is calculated as the product of the magnetic field strength and the area through which the field lines pass, typically taking into account the angle between the field direction and the normal (perpendicular) to the surface. This concept is fundamental in understanding how changes in magnetic environments induce electrical effects.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a changing magnetic flux through a closed loop induces an electromotive force (emf) in the loop. The magnitude of the induced emf is directly proportional to the rate at which the magnetic flux changes. This principle is the basis for understanding how electric generators, transformers, and inductors operate, by converting mechanical energy or a changing magnetic environment into electrical energy.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from a changing magnetic flux. It states that the induced current will flow in such a direction that its own magnetic field opposes the change in the original magnetic flux. This opposition is a manifestation of the conservation of energy, ensuring that the induced electromotive force does not exaggerate the change causing it.

Transcript

00:03 For this problem, we have a 30 centimeter by 60 centimeter circuit.

00:08 In this circuit, there's a 15 -on resistor and an ammeter.

00:13 The initial b -field that this is in, and the b -field is perpendicular to the circuit, it starts out at 2 .65 tesla, and it is decreasing in value at 0 .25 teslas per second.

00:28 Our goal is to find the induced current and this would be what our am meter would read in this setting.

00:37 So let's start off by writing what our induced current would be.

00:46 So we know that our emf is going to be equal to our induced current multiplied by our resistance.

00:55 So our induced current is going to be our emf divided by that resistance.

01:01 We know the resistance, so now we have to find what our emf would be.

01:07 So in this setting, our emf is going to be the change in the flux through our circuit over the change in time.

01:18 We can rewrite our flux as the b field multiplied by the area, and these two are perpendicular, so we just have b times a, no cosine, the angle 0, so the cosine is 1.

01:34 Divided by our time.

01:37 The area is not changing, so we can pull that out and that's going to be our change in b over change in time.

01:45 And that's the information we're given.

01:46 We're told that this is decreasing at 0 .25 tesla's per second...

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