Question Video: Solving for the Current Induced in a Coil in a Uniform Magnetic Field | Nagwa Question Video: Solving for the Current Induced in a Coil in a Uniform Magnetic Field | Nagwa

# Question Video: Solving for the Current Induced in a Coil in a Uniform Magnetic Field Physics • Third Year of Secondary School

A coil with 5 turns and a 12 cm radius rotates around an axis that is perpendicular to a uniform magnetic field, as shown in the diagram. The coil has a resistance of 25 Ω and completes 15 rotations per second. The strength of the magnetic field is 28 mT. What is the current induced in the coil?

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### Video Transcript

A coil with five turns and a 12-centimeter radius rotates around an axis that is perpendicular to a uniform magnetic field, as shown in the diagram. The coil has a resistance of 25 ohms and completes 15 rotations per second. The strength of the magnetic field is 28 milliteslas. What is the current induced in the coil?

In our diagram, we see this coil positioned between the north and south poles of a magnet. The magnetic field then between these poles points like this. Our coil rotates about an axis through its center. Based on the information given, we want to solve for the current induced in the coil. Current will be induced in the coil if there is an emf, or electromotive force, induced. This electromotive force, often represented using the symbol ε, is equal by Faraday’s law to negative the number of turns 𝑁 in the coil multiplied by the change in magnetic flux through the coil ΔΦ sub 𝐵 divided by the time Δ𝑡 over which that change occurs. Because an induced emf will lead to an induced current in the coil, we can begin by solving for this value ε.

When it comes to induced emf, the key is that there’s a change in magnetic flux through some object, in this case our coil. In general, the magnetic flux Φ sub 𝐵 through some surface is equal to the strength of the magnetic field involved, we’ll call that 𝐵, multiplied by the area of the surface exposed to that field, we’ll call it 𝐴. In order for there to be a change in magnetic flux, which is required for an emf to be induced, a given object needs to experience a change in 𝐵 or a change in 𝐴 or a change in both of these quantities. Considering our rotating coil, if we were to draw a top-down view of the coil, we would see that the magnetic field lines from the permanent magnet pass by it like this and that the coil rotates about an axis that runs right through its center. Moreover, this axis is perpendicular to the plane of the coil.

Now, because the permanent magnet that the coil is in doesn’t change as the coil rotates, that means that 𝐵 in our equation for magnetic flux is constant. And then, thinking further, even though our coil rotates in the magnetic field, this rotation doesn’t change anything about the area of the coil exposed to that field. In fact, the area of the coil exposed to the field is zero because none of the field lines as we draw them actually pass through any of the loops of the coil. We could think of them rather as passing through the sides of the coil. The important thing here though is not the particular value of the area exposed to the field but rather that that value doesn’t change over time.

Now, because neither 𝐵 nor 𝐴 changes in time, that means ΔΦ sub 𝐵 in our case is zero. There is no change in the magnetic flux through our coil as it rotates. From Faraday’s law, we can see that this implies that the emf induced in the coil is zero. That must be the case if ΔΦ sub 𝐵 is zero. And then if there’s no emf induced in this coil, then that means no current will be induced in it either. For our answer then, we can simply write that the current induced in the coil is zero amperes. This is because as we’ve seen, there is no emf induced in the coil. And that’s because there is no change in the magnetic flux through the coil. Our final answer then is zero amperes.

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