Solution

Given:

1. Dimensions of the coil:
   • Length ($l$) = $30 \, \text{cm} = 0.30 \, \text{m}$
   • Width ($w$) = $20 \, \text{cm} = 0.20 \, \text{m}$
   • Number of turns ($N$) = 40
2. Angular speed:
   • Speed in rotations per minute ($\text{r/min}$) = 1500
   • Angular velocity ($\omega$) in radians per second is calculated as: $\omega = \frac{1500 \times 2\pi}{60} = 50\pi \, \text{rad/s}$
3. Flux density ($B$) = $0.075 \, \text{T}$
4. Instantaneous emf evaluation time:
   • $t = 0.002 \, \text{s}$

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Expression for Induced EMF

Magnetic Flux through the Coil

The flux through one turn of the coil is: $\phi = B \cdot A \cdot \cos(\theta)$ where:

• $A = l \cdot w = 0.30 \cdot 0.20 = 0.06 \, \text{m}^2$
• $\theta = \omega t$ (the angle between the field and the normal to the coil's plane).

Thus, the flux through one turn is: $\phi = 0.075 \cdot 0.06 \cdot \cos(\omega t) = 0.0045 \cdot \cos(\omega t) \, \text{Wb}.$

The total flux through the $N = 40$ turns is: $\Phi = N \cdot \phi = 40 \cdot 0.0045 \cdot \cos(\omega t) = 0.18 \cdot \cos(\omega t) \, \text{Wb}.$

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Induced EMF ($e$)

The emf induced is the time derivative of the total flux: $e = -\frac{d\Phi}{dt}.$

Substituting $\Phi = 0.18 \cos(\omega t)$: $e = -\frac{d}{dt}[0.18 \cdot \cos(\omega t)] = -0.18 \cdot (-\omega \sin(\omega t)) = 0.18 \omega \sin(\omega t).$

Substituting $\omega = 50\pi \, \text{rad/s}$: $e = 0.18 \cdot 50\pi \cdot \sin(50\pi t) = 9\pi \sin(50\pi t) \, \text{V}.$

Thus, the instantaneous emf is: $\boxed{e = 9\pi \sin(50\pi t) \, \text{V}}.$

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Evaluate EMF at $t = 0.002 \, \text{s}$

Substitute $t = 0.002 \, \text{s}$ into $e = 9\pi \sin(50\pi t)$: $e = 9\pi \sin(50\pi \cdot 0.002) = 9\pi \sin(0.1\pi).$

The value of $\sin(0.1\pi)$ is: $\sin(0.1\pi) = \sin(18^\circ) \approx 0.309.$

Substitute this into the equation: $e = 9\pi \cdot 0.309 \approx 8.74 \, \text{V}.$

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Final Answer:

1. Expression for instantaneous emf: $e = 9\pi \sin(50\pi t) \, \text{V}.$
2. EMF at $t = 0.002 \, \text{s}$: $e \approx 8.74 \, \text{V}.$

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Would you like more details or clarification?

Here are 5 related questions:

1. What is the significance of the negative sign in Faraday’s law?
2. How would the emf change if the magnetic field strength were doubled?
3. What happens to the emf if the speed of rotation is halved?
4. How would the emf expression change for a non-uniform magnetic field?
5. Can we use this setup to measure the angular velocity of the coil?

Tip: Always ensure that angular velocity ($\omega$) is converted to radians per second for calculations involving rotations.