Faraday’s Law states that the electromotive force, E, induced by N
turns of a coil with a flux, φ, passing through it, is given by E = −N*dφ/dt .
If φ = K sin(2πf t), where K and f are constants, determine E.

To find the electromotive force \( E \) induced by the coil, we need to calculate the time derivative of the magnetic flux \( \phi \). The expression for the flux is given as:

\[
\phi = K \sin(2 \pi f t)
\]

where:
- \( K \) is a constant,
- \( f \) is the frequency,
- \( t \) is time.

### Step 1: Compute the Derivative of \( \phi \)

The electromotive force \( E \) is given by Faraday's Law:

\[
E = -N \frac{d\phi}{dt}
\]

To find \( \frac{d\phi}{dt} \), let's differentiate \( \phi = K \sin(2 \pi f t) \) with respect to \( t \):

\[
\frac{d\phi}{dt} = \frac{d}{dt} \left( K \sin(2 \pi f t) \right)
\]

Applying the chain rule:

\[
\frac{d\phi}{dt} = K \cdot \frac{d}{dt} \left( \sin(2 \pi f t) \right)
\]

Since the derivative of \( \sin(x) \) is \( \cos(x) \), we have:

\[
\frac{d}{dt} \left( \sin(2 \pi f t) \right) = (2 \pi f) \cos(2 \pi f t)
\]

Therefore:

\[
\frac{d\phi}{dt} = K \cdot (2 \pi f) \cos(2 \pi f t)
\]

\[
\frac{d\phi}{dt} = 2 \pi f K \cos(2 \pi f t)
\]

### Step 2: Determine the Induced EMF, \( E \)

Now substitute \( \frac{d\phi}{dt} \) into Faraday's Law:

\[
E = -N \frac{d\phi}{dt}
\]

\[
E = -N (2 \pi f K \cos(2 \pi f t))
\]

Simplifying:

\[
E = -2 \pi N f K \cos(2 \pi f t)
\]

### Final Answer

The induced electromotive force is:

\[
\boxed{E = -2 \pi N f K \cos(2 \pi f t)}
\]

This is the expression for the induced EMF in terms of the constants \( N \), \( f \), \( K \), and time \( t \).

Would you like more details, or do you have any questions?

### Related Questions to Explore:
1. What happens to the EMF if the frequency \( f \) is doubled?
2. How does the negative sign in Faraday's Law relate to Lenz's Law?
3. What is the physical significance of the constant \( K \) in this context?
4. How would the expression for \( E \) change if \( \phi = K \cos(2 \pi f t) \) instead?
5. What are some practical applications of Faraday's Law in electrical engineering?

**Tip:** When dealing with sinusoidal functions, always pay attention to their derivatives, as they often involve phase shifts.

Faraday’s Law states that the electromotive force, E, induced by N turns of a coil with a flux, φ, passing through it, is given by E = −N*dφ/dt . If φ = K sin(2πf t), where K and f are constants, determine E.

Learn how to calculate the induced electromotive force (EMF) using Faraday's Law, given a sinusoidal magnetic flux. This detailed solution covers differentiation techniques and the application of trigonometric identities. Suitable for advanced physics and engineering students.

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