Question

The figure shows a region of length$‘l’$ with a uniform magnetic field of $0.3T$ in it and a proton entering the region with velocity $4\times {10}^{5}m{s}^{-1}$ making an angle $60°$ with the field. If the proton completes $10revolution$ by the time it cross the region shown,$‘l’$ is close to (mass of proton =$1.67\times {10}^{–27}kg$, a charge of the proton = $1.6\times {10}^{–19}C$)

• A. $0.11m$
• B. $0.22m$
• C. $0.44m$
• D. $0.88m$

Answer 1

The correct option is C

$0.44m$

Step 1: Given data

Uniform magnetic field in region$\left(B\right)$= $0.3T$

The velocity of the proton is region$\left(v\right)$=$4\times {10}^{5}m{s}^{-1}$

The angle formed by a proton in region$\left(\theta \right)$= $60°$

Number of revolutions completed by proton$\left(n\right)$=$10revolution$

Mass of proton$\left(m\right)$ =$1.67\times {10}^{–27}kg$

Charge of the proton $\left(q\right)$= $1.6\times {10}^{–19}C$

Step 2: Formula used

The length can be computed using the formula $l=∆T\times \left(v_{\circ }\cos \theta \right)$……..(1)

where $∆T$ is the time period of revolution.

The period of revolution of $q$in a uniform magnetic field is given by-

$∆T=numberofrevolutions\times \frac{2\mathrm{\pi m}}{qB}$…….(2)

Step 3: Compute the value of $\text{'}l\text{'}$

We know that $∆T=numberofrevolutions\times \frac{2\mathrm{\pi m}}{qB}$……(3)

Substituting the known values in the above formula,(equation 3)

$∆T=10\times \frac{2\pi \times \left(1.67\times {10}^{-27}\right)}{\left(1.6\times {10}^{-19}\right)\times \left(0.3\right)}$

The length can be computed using the formula $l=∆T\times \left(v_{\circ }\cos \theta \right)$

Substitute the value of $∆T$ in the formula

Therefore,

the value of $\text{'}l\text{'}$ can be calculated as follows-

$$\begin{gathered} l=\left(10\times \frac{2\mathrm{\pi m}}{qB}\right)\left(v_{\circ }\cos \theta \right) \\ I=\left(10\times \frac{2\mathrm{\pi }\times 1.67\times {10}^{-27}}{1.6\times {10}^{-19}\times 0·3}\right)\left(4\times {10}^5\times \cos 60\right) \\ I=0.44m \end{gathered}$$

Hence, option C is the correct answer.