A plane spiral with a great number `N` of turns wound tightly to one another is located in a uniform magnetic field perpendicular to the spiral's plane.The outside radius of the spiral's turns is equal to a and inner radius is zero.The magnetic induction varies with time as `B=B_(0) sin omega t`, where `B_(0)` and `omega` are constants.The amplitude of `emf` induced in the spiral is `epsilon_(im)=1/xpia^(2) NomegaB_(0)`.Find out value of `x`.

Explanation

To solve the problem, we need to find the value of \( x \) in the expression for the amplitude of the induced EMF in a plane spiral located in a varying magnetic field.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a plane spiral with \( N \) turns, an outer radius \( a \), and an inner radius of 0. The magnetic field \( B \) varies with time as \( B = B_0 \sin(\omega t) \).

2. **Induced EMF**:
The induced EMF \( \epsilon \) in the spiral can be expressed using Faraday's law of electromagnetic induction:
\[ \epsilon = -N \frac{d\Phi}{dt} \]
where \( \Phi \) is the magnetic flux through the spiral.

3. **Magnetic Flux Calculation**:
The magnetic flux \( \Phi \) through one turn of the spiral is given by:
\[ \Phi = B \cdot A \]
where \( A \) is the area of the spiral. The area \( A \) of the spiral can be approximated as:
\[ A = \pi r^2 \]
where \( r \) is the radius of the spiral.

4. **Differentiating the Magnetic Field**:
The magnetic field varies with time as \( B = B_0 \sin(\omega t) \). The rate of change of the magnetic field is:
\[ \frac{dB}{dt} = B_0 \omega \cos(\omega t) \]

5. **Substituting into the EMF Equation**:
Substituting the expression for \( \Phi \) into the EMF equation, we have:
\[ \epsilon = -N \frac{d}{dt} \left( B_0 \sin(\omega t) \cdot \pi r^2 \right) \]
This leads to:
\[ \epsilon = -N \cdot \pi r^2 \cdot \frac{dB}{dt} \]
Substituting \( \frac{dB}{dt} \):
\[ \epsilon = -N \cdot \pi r^2 \cdot B_0 \omega \cos(\omega t) \]

6. **Integrating to Find the Amplitude**:
To find the amplitude of the induced EMF, we consider the maximum value of \( \cos(\omega t) \), which is 1:
\[ \epsilon_{\text{max}} = N \cdot \pi a^2 \cdot B_0 \omega \]
Thus, the amplitude of the induced EMF is given by:
\[ \epsilon_{\text{im}} = \frac{N B_0 \omega}{x \pi a^2} \]

7. **Comparing with Given Expression**:
We are given that:
\[ \epsilon_{\text{im}} = \frac{1}{x \pi a^2} N \omega B_0 \]
By comparing both expressions, we find:
\[ x = 3 \]

### Final Answer:
The value of \( x \) is \( 3 \).