A tightly-wound, long solenoid has n turns per unit length, a radius r and carries a current i. A particle having charge q and mass m is projected from a point on the axis in a direction perpendicular to the axis. What can be the maximum speed for which the particle does not strike the solenoid?

Options

A. `(mu_0qrni)/(2m)`
B. `(mu_0qrni)/m`
C. `(2mu_0qrni)/(3m)`
D. none of these

Explanation

To solve the problem of finding the maximum speed \( v_{\text{max}} \) for which a charged particle does not strike a tightly-wound long solenoid, we can follow these steps:

### Step 1: Understand the setup
The solenoid has \( n \) turns per unit length, a radius \( r \), and carries a current \( i \). A particle with charge \( q \) and mass \( m \) is projected from the axis of the solenoid in a direction perpendicular to the axis.

### Step 2: Identify forces acting on the particle
When the charged particle moves in the magnetic field created by the solenoid, it experiences a magnetic force. This magnetic force will act as the centripetal force required to keep the particle in circular motion.

### Step 3: Determine the magnetic field inside the solenoid
The magnetic field \( B \) inside a long solenoid is given by the formula:
\[ B = \mu_0 n i \]
where \( \mu_0 \) is the permeability of free space, \( n \) is the number of turns per unit length, and \( i \) is the current flowing through the solenoid.

### Step 4: Write the expression for magnetic force
The magnetic force \( F_B \) acting on the particle is given by:
\[ F_B = q v B \]
where \( v \) is the speed of the particle and \( B \) is the magnetic field.

### Step 5: Set the magnetic force equal to the centripetal force
The centripetal force \( F_C \) required to keep the particle in circular motion is given by:
\[ F_C = \frac{m v^2}{r'} \]
where \( r' \) is the radius of the circular path of the particle. For the particle to not strike the solenoid, we can assume that the maximum radius of the circular path is \( \frac{r}{2} \) (half the radius of the solenoid).

Setting the magnetic force equal to the centripetal force gives:
\[ q v B = \frac{m v^2}{\frac{r}{2}} \]

### Step 6: Substitute the expression for magnetic field
Substituting \( B = \mu_0 n i \) into the equation gives:
\[ q v (\mu_0 n i) = \frac{m v^2}{\frac{r}{2}} \]

### Step 7: Simplify the equation
Rearranging the equation, we have:
\[ q v \mu_0 n i = \frac{2m v^2}{r} \]
Now, divide both sides by \( v \) (assuming \( v \neq 0 \)):
\[ q \mu_0 n i = \frac{2m v}{r} \]

### Step 8: Solve for \( v \)
Rearranging gives:
\[ v = \frac{q \mu_0 n i r}{2m} \]

### Conclusion
Thus, the maximum speed \( v_{\text{max}} \) for which the particle does not strike the solenoid is:
\[ v_{\text{max}} = \frac{q \mu_0 n i r}{2m} \]