The magnetic flux `phi` linked with a conducting coil depends on time as `phi = 4t^(n) + 6`, where `n` is positive constant. The induced emf in the coil is `e`

Options

A. (a) if `0lt n lt1, e!= 0 and |e|` decreases with time.
B. (b) if `n = 1, e` is constant.
C. (c ) If `n gt 1, |e|` increases with time.
D. (d ) If `n gt 1, |e|` decreases with time.

Explanation

To solve the problem, we need to find the induced emf (E) in a conducting coil given the magnetic flux (Φ) as a function of time (t). The magnetic flux is given by:

\[ \Phi = 4t^n + 6 \]

where \( n \) is a positive constant.

### Step 1: Understand the relationship between magnetic flux and induced emf
The induced emf (E) in a coil is given by Faraday's law of electromagnetic induction, which states that:

\[ E = -\frac{d\Phi}{dt} \]

### Step 2: Differentiate the magnetic flux with respect to time
We need to differentiate the expression for magnetic flux with respect to time (t):

\[ \Phi = 4t^n + 6 \]

Differentiating this with respect to t:

\[ \frac{d\Phi}{dt} = \frac{d}{dt}(4t^n + 6) \]

Using the power rule of differentiation, we get:

\[ \frac{d\Phi}{dt} = 4n t^{n-1} \]

### Step 3: Substitute the derivative into the formula for induced emf
Now, substituting the derivative back into the equation for induced emf:

\[ E = -\frac{d\Phi}{dt} = -4n t^{n-1} \]

### Step 4: Analyze the behavior of E based on the value of n
Now we analyze the expression \( E = -4n t^{n-1} \) based on different ranges of n:

1. **If \( 0 < n < 1 \)**:
- Here, \( n - 1 < 0 \), which means \( t^{n-1} \) will decrease as time increases (since it is a negative exponent).
- Therefore, \( E \) will also decrease with time.

2. **If \( n = 1 \)**:
- Then \( E = -4n t^{0} = -4 \), which is a constant value.

3. **If \( n > 1 \)**:
- In this case, \( n - 1 > 0 \), so \( t^{n-1} \) will increase as time increases.
- Therefore, \( E \) will increase with time.

### Conclusion
Based on the analysis, we can summarize the behavior of the induced emf (E) as follows:

- **Option A**: If \( 0 < n < 1 \), then \( E \) is not equal to 0 and decreases with time. (Correct)
- **Option B**: If \( n = 1 \), then \( E \) is a constant. (Correct)
- **Option C**: If \( n > 1 \), then the magnitude of \( E \) increases with time. (Correct)
- **Option D**: If \( n > 1 \), then the magnitude of \( E \) decreases with time. (Incorrect)

Thus, the correct options are A, B, and C.