A loop of wire enclosing an area `S` is placed in a region where the magnetic field is perpendicular to the plane. The magnetic field `B` varies with time according to the expression `B=B_0e^(-at)` where a is some constant. That is, at `t= 0`. The field is `B_0` and for `t gt 0`, the field decreases exponentially. Find the induced emf in the loop as a function of time.

Explanation

To find the induced emf in the loop as a function of time, we will follow these steps:

### Step 1: Understand the Magnetic Flux
The magnetic flux (Φ) through the loop is given by the formula:
\[ \Phi = B \cdot A \]
where \(B\) is the magnetic field and \(A\) is the area of the loop. Since the magnetic field is perpendicular to the plane of the loop, we can simplify this to:
\[ \Phi = B \cdot S \]
where \(S\) is the area of the loop.

### Step 2: Substitute the Expression for Magnetic Field
The magnetic field varies with time according to the expression:
\[ B = B_0 e^{-at} \]
Substituting this into the magnetic flux equation gives:
\[ \Phi = B_0 e^{-at} \cdot S \]

### Step 3: Apply Faraday's Law of Electromagnetic Induction
According to Faraday's law, the induced emf (ε) in the loop is given by the negative rate of change of magnetic flux:
\[ \varepsilon = -\frac{d\Phi}{dt} \]
Substituting the expression for magnetic flux:
\[ \varepsilon = -\frac{d}{dt}(B_0 S e^{-at}) \]

### Step 4: Differentiate the Magnetic Flux
Now, we differentiate the expression:
\[ \varepsilon = -B_0 S \frac{d}{dt}(e^{-at}) \]
Using the chain rule, the derivative of \(e^{-at}\) is:
\[ \frac{d}{dt}(e^{-at}) = -a e^{-at} \]
Thus, substituting this back into the equation gives:
\[ \varepsilon = -B_0 S (-a e^{-at}) = a B_0 S e^{-at} \]

### Step 5: Final Expression for Induced EMF
Therefore, the induced emf as a function of time is:
\[ \varepsilon(t) = a B_0 S e^{-at} \]

### Summary
The induced emf in the loop as a function of time is:
\[ \varepsilon(t) = a B_0 S e^{-at} \]

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