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A metallic wheel with 8 metallic spokes, each of length r, is rotating at an angular frequency ω in a plane perpendicular to a magnetic field B. The magnitude of the induced emf between the axle and the rim of the wheel is
- a)1/2ωr2B
- b)2 ωr2B
- c)4 ωr2B
- d)8 ωr2B
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A metallic wheel with 8 metallic spokes, each of length r, is rotatin...
Explanation:
To understand the induced emf in the metallic wheel, we need to consider Faraday's law of electromagnetic induction.
1. Faraday's Law:
- Faraday's law states that the induced emf in a wire loop is equal to the rate of change of magnetic flux passing through the loop.
- Mathematically, it can be represented as:
emf = -dΦ/dt
where emf is the induced electromotive force, Φ is the magnetic flux, and t is time.
2. Magnetic Flux:
- Magnetic flux (Φ) is the product of the magnetic field (B) and the area (A) through which the magnetic field passes perpendicular to the area.
- Mathematically, it can be represented as:
Φ = B * A
3. Induced emf in the metallic wheel:
- In the given scenario, the metallic wheel with 8 metallic spokes is rotating at an angular frequency (ω) in a plane perpendicular to the magnetic field (B).
- Each metallic spoke of length (r) can be considered as a small wire loop.
- As the wheel rotates, the area (A) enclosed by each spoke changes, resulting in a change in magnetic flux.
- The rate of change of magnetic flux is given by the product of angular frequency (ω) and the change in area (∆A).
- Mathematically, it can be represented as:
dΦ/dt = B * dA/dt = B * ω * ∆A
- Since there are 8 spokes in the metallic wheel, the total induced emf between the axle and the rim of the wheel is the sum of the emf induced in each spoke.
- Therefore, the magnitude of the induced emf is:
Emf = 8 * (dΦ/dt) = 8 * B * ω * ∆A
4. Calculating the change in area (∆A):
- The change in area (∆A) can be calculated by considering the length of each spoke (r) and the displacement (∆x) of the spoke from its initial position.
- Since the wheel is rotating, the displacement (∆x) can be calculated using the formula: ∆x = r * θ
- Here, θ is the angular displacement of the spoke.
- As the wheel rotates with an angular frequency (ω), the angular displacement (θ) can be represented as: θ = ω * t
- Substituting the value of θ in ∆x, we get: ∆x = r * ω * t
- The change in area (∆A) can be calculated using the formula: ∆A = 2 * ∆x * r
- Substituting the value of ∆x, we get: ∆A = 2 * r * r * ω * t = 2 * r^2 * ω * t
5. Final expression for the induced emf:
- Substituting the value of
To understand the induced emf in the metallic wheel, we need to consider Faraday's law of electromagnetic induction.
1. Faraday's Law:
- Faraday's law states that the induced emf in a wire loop is equal to the rate of change of magnetic flux passing through the loop.
- Mathematically, it can be represented as:
emf = -dΦ/dt
where emf is the induced electromotive force, Φ is the magnetic flux, and t is time.
2. Magnetic Flux:
- Magnetic flux (Φ) is the product of the magnetic field (B) and the area (A) through which the magnetic field passes perpendicular to the area.
- Mathematically, it can be represented as:
Φ = B * A
3. Induced emf in the metallic wheel:
- In the given scenario, the metallic wheel with 8 metallic spokes is rotating at an angular frequency (ω) in a plane perpendicular to the magnetic field (B).
- Each metallic spoke of length (r) can be considered as a small wire loop.
- As the wheel rotates, the area (A) enclosed by each spoke changes, resulting in a change in magnetic flux.
- The rate of change of magnetic flux is given by the product of angular frequency (ω) and the change in area (∆A).
- Mathematically, it can be represented as:
dΦ/dt = B * dA/dt = B * ω * ∆A
- Since there are 8 spokes in the metallic wheel, the total induced emf between the axle and the rim of the wheel is the sum of the emf induced in each spoke.
- Therefore, the magnitude of the induced emf is:
Emf = 8 * (dΦ/dt) = 8 * B * ω * ∆A
4. Calculating the change in area (∆A):
- The change in area (∆A) can be calculated by considering the length of each spoke (r) and the displacement (∆x) of the spoke from its initial position.
- Since the wheel is rotating, the displacement (∆x) can be calculated using the formula: ∆x = r * θ
- Here, θ is the angular displacement of the spoke.
- As the wheel rotates with an angular frequency (ω), the angular displacement (θ) can be represented as: θ = ω * t
- Substituting the value of θ in ∆x, we get: ∆x = r * ω * t
- The change in area (∆A) can be calculated using the formula: ∆A = 2 * ∆x * r
- Substituting the value of ∆x, we get: ∆A = 2 * r * r * ω * t = 2 * r^2 * ω * t
5. Final expression for the induced emf:
- Substituting the value of
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Community Answer
A metallic wheel with 8 metallic spokes, each of length r, is rotatin...
Let v be the frequency of rotation.
The time taken for 1 full rotation is T = 1/v.
Therefore, rate of change of area is
Now, the emf induced between the axle and rim is e = B × rate of change of area
= 
Since the same emf is produced between the ends of each spoke, and these emfs are in parallel as is evident from Fig. the net emf between the axle and the rim of the wheel will be the same as that across each spoke. We notice that all the eight spokes are connected with one end at the rim and the other at the axle. Hence the magnitude of the net emf between the axle and the rim is independent of the number of spokes.
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A metallic wheel with 8 metallic spokes, each of length r, is rotating at an angular frequency ω in a plane perpendicular to a magnetic field B. The magnitude of the induced emf between the axle and the rim of the wheel isa)1/2ωr2Bb)2 ωr2Bc)4 ωr2Bd)8 ωr2BCorrect answer is option 'A'. Can you explain this answer?
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