Question

A charged particle with charge to mass ratio $(\frac{q}{m}=\frac{{10}^3}{19}{\text{C kg}}^{-1})$ enters a uniform magnetic field $\overrightarrow{B}=20\hat{i}+30\hat{j}$$+50\hat{k}\text{T}$ at time $t=0$ with velocity $\overrightarrow{v}=20\hat{i}+50\hat{j}+30\hat{k}\text{m/s}$. Assume that magnetic field exists in large space. The pitch of the helical path of the motion of the particle approximately will be

• A. $1\text{m}$
• B. $0.1\text{m}$
• C. $2.3\text{m}$
• D. $0.23\text{m}$

Answer 1

The correct option is B $0.1\text{m}$

Given,

$\frac{q}{m}=\frac{{10}^3}{19}{\text{Ckg}}^{-1}$

Magnetic field $\overrightarrow{B}=20\hat{i}+30\hat{j}+50\hat{k}\text{T}$

Velocity at $t=0$, $\overrightarrow{v}=20\hat{i}+50\hat{j}+30\hat{k}\text{m/s}$

We know that,

Time period, $T=\frac{2\pi m}{qB}=\frac{2\pi }{B}\times \frac{m}{q}$

Substituting the values,

$\Rightarrow T=\frac{2\pi }{\sqrt{{\left(20\right)}^2+{\left(30\right)}^2+{\left(50\right)}^2}}\times \frac{19}{{10}^3}$

$\Rightarrow T=\frac{38\pi }{1000\sqrt{3800}}$

Now, pitch

$P=v\cos \theta \times T.....\left(1\right)$

To calculate $\cos \theta$, using

$\overrightarrow{B}.\overrightarrow{v}=Bv\cos \theta .....\left(2\right)$

Using equations , $\left(1\right)$ and $\left(2\right)$

$P=\frac{\overrightarrow{B}.\overrightarrow{v}}{B}\times T$

Substituting the values,

$\Rightarrow P=\frac{20\times 20+30\times 50+50\times 30}{\sqrt{3800}}\times \frac{38\pi }{\sqrt{3800}\times 1000}$

$\Rightarrow P=\frac{3400\times 38\pi }{3800\times 1000}=\frac{34\pi }{1000}$

$\therefore P\approx 0.1\text{m}$

Hence, option (b) is the correct answer.