A proton is projected with a velocity `10^(7)ms^(-1)`, at right angles to a uniform magnetic field of induction `100mT`. The time `(` in second `)` taken by the proton to traverse `90^(@)` are is `: (` Mass of proton `=1.65xx10^(-27)kg` an charge of proton `=1.6 xx10^(-19)C)`

Options

A. `0.81xx10^(-7)`
B. `1.62xx10^(-7)`
C. `2.43xx10^(-7)`
D. `3.24xx10^(-7)`

Explanation

To solve the problem, we need to find the time taken by a proton to traverse a 90-degree arc in a magnetic field. Here's a step-by-step solution:

### Step 1: Understand the motion of the proton in the magnetic field
When a charged particle like a proton moves perpendicular to a magnetic field, it follows a circular path. The radius of this circular path can be calculated using the formula:

\[ r = \frac{mv}{qB} \]

where:
- \( m \) = mass of the proton = \( 1.65 \times 10^{-27} \, \text{kg} \)
- \( v \) = velocity of the proton = \( 10^7 \, \text{m/s} \)
- \( q \) = charge of the proton = \( 1.6 \times 10^{-19} \, \text{C} \)
- \( B \) = magnetic field induction = \( 100 \, \text{mT} = 100 \times 10^{-3} \, \text{T} \)

### Step 2: Calculate the radius \( r \)
Substituting the values into the formula for the radius:

\[ r = \frac{(1.65 \times 10^{-27} \, \text{kg})(10^7 \, \text{m/s})}{(1.6 \times 10^{-19} \, \text{C})(100 \times 10^{-3} \, \text{T})} \]

Calculating the denominator:

\[ (1.6 \times 10^{-19})(100 \times 10^{-3}) = 1.6 \times 10^{-19} \times 10^{-1} = 1.6 \times 10^{-20} \]

Now calculating the radius:

\[ r = \frac{(1.65 \times 10^{-27})(10^7)}{1.6 \times 10^{-20}} = \frac{1.65 \times 10^{-20}}{1.6 \times 10^{-20}} \approx 1.03125 \, \text{m} \]

### Step 3: Calculate the arc length for 90 degrees
The arc length \( s \) for a 90-degree angle (which is \( \frac{\pi}{2} \) radians) can be calculated as:

\[ s = \frac{\pi}{2} \times r \]

Substituting the radius we calculated:

\[ s = \frac{\pi}{2} \times 1.03125 \approx 1.62 \, \text{m} \]

### Step 4: Calculate the time taken to traverse the arc
The time \( t \) taken to traverse the arc can be calculated using the formula:

\[ t = \frac{s}{v} \]

Substituting the values:

\[ t = \frac{1.62}{10^7} = 1.62 \times 10^{-7} \, \text{s} \]

### Final Answer
The time taken by the proton to traverse 90 degrees is approximately:

\[ t \approx 1.62 \times 10^{-7} \, \text{s} \]