A rectangular coil rotates about an axis normal to the magnetic field, If `E_(m)` is the maximum value of the induced emf, then the instantaneous emf when plane of the coil makes an angle of `45^(@)` with the magnetic field is

Options

A. `(1)/(2)E_(m)`
B. `(1)/(4)E_(m)`
C. `(1)/(sqrt2)E_(m)`
D. `E_(m)`

Explanation

To solve the problem, we need to find the instantaneous induced electromotive force (emf) when the plane of a rectangular coil makes an angle of \(45^\circ\) with the magnetic field. Here are the steps to derive the solution:

### Step 1: Understanding the Magnetic Flux
The magnetic flux (\(\Phi\)) through the coil is given by the formula:
\[ \Phi = N \cdot B \cdot A \cdot \cos(\theta) \]
where:
- \(N\) = number of turns in the coil
- \(B\) = magnetic field strength
- \(A\) = area of the coil
- \(\theta\) = angle between the magnetic field and the normal to the coil's surface.

### Step 2: Expressing the Angle
Since the coil is rotating, we can express \(\theta\) as:
\[ \theta = \omega t \]
where \(\omega\) is the angular velocity and \(t\) is time.

### Step 3: Substituting into the Flux Formula
Substituting \(\theta = \omega t\) into the magnetic flux equation gives:
\[ \Phi = N \cdot B \cdot A \cdot \cos(\omega t) \]

### Step 4: Finding the Induced EMF
According to Faraday's law of electromagnetic induction, the induced emf (\(E\)) is given by the negative rate of change of magnetic flux:
\[ E = -\frac{d\Phi}{dt} \]
Differentiating the magnetic flux:
\[ E = -\frac{d}{dt}(N \cdot B \cdot A \cdot \cos(\omega t)) = N \cdot B \cdot A \cdot \sin(\omega t) \cdot \omega \]
Thus, we have:
\[ E = N \cdot B \cdot A \cdot \omega \cdot \sin(\omega t) \]

### Step 5: Maximum Induced EMF
The maximum induced emf (\(E_m\)) occurs when \(\sin(\omega t) = 1\):
\[ E_m = N \cdot B \cdot A \cdot \omega \]

### Step 6: Finding Instantaneous EMF at \(45^\circ\)
When the plane of the coil makes an angle of \(45^\circ\) with the magnetic field, we have:
\[ \omega t = 45^\circ = \frac{\pi}{4} \text{ radians} \]
Thus, we can substitute this into the equation for induced emf:
\[ E = E_m \cdot \sin\left(\frac{\pi}{4}\right) \]
Since \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), we get:
\[ E = E_m \cdot \frac{1}{\sqrt{2}} = \frac{E_m}{\sqrt{2}} \]

### Final Result
Therefore, the instantaneous emf when the plane of the coil makes an angle of \(45^\circ\) with the magnetic field is:
\[ E = \frac{E_m}{\sqrt{2}} \]