A magnetic field `vec(B)=-B_(0)hat(i)` exists within a sphere of radius `R=v_(0)Tsqrt(3)` where T is the time period of one revolution of a charged particle starting its motion form origin and moving with a velocity `vce(v)_(0) = (v_0)/(2) sqrt(3) hat(i) - v_(0)/(2) hat(j)`. Find the number of turns that the particle will take to come out of the magnetic field.

Explanation

To solve the problem, we need to find the number of turns a charged particle makes while moving out of a magnetic field within a sphere of radius \( R = v_0 T \sqrt{3} \). The particle starts from the origin and moves with a given velocity. Let's break down the solution step by step.

### Step 1: Understand the Given Information
- The magnetic field is given as \( \vec{B} = -B_0 \hat{i} \).
- The radius of the sphere is \( R = v_0 T \sqrt{3} \).
- The velocity of the particle is \( \vec{v}_0 = \frac{v_0}{2} \sqrt{3} \hat{i} - \frac{v_0}{2} \hat{j} \).

### Step 2: Calculate the Time Period of Revolution
The time period \( T \) of a charged particle in a magnetic field is given by the formula:
\[ T = \frac{2\pi m}{qB} \]
where \( m \) is the mass of the particle, \( q \) is the charge of the particle, and \( B \) is the magnitude of the magnetic field. In our case, the magnetic field is in the negative x-direction, so we can denote its magnitude as \( B = B_0 \).

### Step 3: Calculate the Distance Traveled in One Revolution
The distance traveled in one complete revolution can be calculated using the formula:
\[ \text{Distance in one revolution} = \text{Velocity} \times \text{Time Period} \]
The velocity of the particle is given as \( \vec{v}_0 \), and its magnitude can be calculated as:
\[ |\vec{v}_0| = \sqrt{\left(\frac{v_0}{2} \sqrt{3}\right)^2 + \left(-\frac{v_0}{2}\right)^2} = \sqrt{\frac{3v_0^2}{4} + \frac{v_0^2}{4}} = \sqrt{v_0^2} = v_0 \]
Thus, the distance traveled in one revolution is:
\[ \text{Distance in one revolution} = v_0 \times T \]

### Step 4: Calculate the Total Distance to Exit the Magnetic Field
The total distance the particle needs to travel to exit the magnetic field is twice the radius of the sphere:
\[ \text{Total distance} = 2R = 2(v_0 T \sqrt{3}) = 2v_0 T \sqrt{3} \]

### Step 5: Calculate the Number of Turns
The number of turns \( N \) can be calculated by dividing the total distance by the distance traveled in one revolution:
\[ N = \frac{\text{Total distance}}{\text{Distance in one revolution}} = \frac{2v_0 T \sqrt{3}}{v_0 T} = 2\sqrt{3} \]

### Final Answer
Thus, the number of turns that the particle will take to come out of the magnetic field is:
\[ N = 2\sqrt{3} \]