Question

An electron follows a helical path in a uniform magnetic field given by $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{=}\left(20\hat{i}-50\hat{j}-30\hat{k}\right)\mathit{m}\mathit{T}$ . At time t = 0, the electron’s velocity is given by $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}\mathbf{=}\left(20\hat{i}-30\hat{j}+50\hat{k}\right)\mathit{m}\mathbf{/}\mathit{s}$.

(a)What is the angle$\mathit{\varphi }$between$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$ and$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}$ The electron’s velocity changes with time.

(b) Do its speed change with time?

(c) Do the angle$\mathit{\varphi }$change with time?

(d) What is the radius of the helical path?

Short answer

1. The angle between $\stackrel{\rightharpoonup }{v}$and $\stackrel{\rightharpoonup }{B}$is 84^o .
2. The electron’s velocity does not change with time.
3. The angle does not change with time.
4. The radius of the helical path is $5.7\times {10}^{-9}m$.

Step-by-step solution

Listing the given quantities

$$\begin{gathered} \stackrel{\rightharpoonup }{B}=20\hat{i}-50\hat{j}-30\hat{k}mT=\left(20\hat{i}-50\hat{j}-30\hat{k}\right)\times {10}^{-3}T \\ \stackrel{\rightharpoonup }{v}=20\hat{i}-30\hat{j}+50\hat{k}m/s \end{gathered}$$

Understanding the concept of the radius of the helical path

We are given the vectors for$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$and$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}$. We can find the cross product for them. From this, we can find the angle between them. We have the formula for the radius of the helical path so we can find the required answer.

Formula:

localid="1663951121578" $\begin{array}{rcl}\stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{B}& =& \left|v\right|\left|B\right|\sin \varphi \\ qvB& =& \frac{m{v}^2}{r}\end{array}$

Step 3:(a) Calculation of the engle between v⇀ and B⇀

$\begin{array}{rcl}\stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{B}& =& \left(20\hat{i}-30\hat{j}+50\hat{k}\right)\times \left(20\hat{i}-50\hat{j}-30\hat{k}\right)\times {10}^{-3}\\ & =& 3.4\hat{i}+1.6\hat{j}-0.4\hat{k}\\ \left|\stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{B}\right|& =& \sqrt{3.4^2+1.6^2-0.4^2}\\ & =& 3.78\frac{m}{s}.T\\ \left|\stackrel{\rightharpoonup }{v}\right|& =& \sqrt{{20}^2+{30}^2+{50}^2}\\ & =& 62\frac{m}{s}\\ \left|\stackrel{\rightharpoonup }{B}\right|& =& \left(\sqrt{{20}^2+{50}^2+{30}^2}\right)\times {10}^{-3}\\ & =& 61.6\times {10}^{-3}T\\ \stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{B}& =& \left|\stackrel{\rightharpoonup }{v}\right|\left|\stackrel{\rightharpoonup }{B}\right|\sin \varphi \\ \sin \varphi & =& \frac{\left|\stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{B}\right|}{\left|\stackrel{\rightharpoonup }{v}\right|\left|\stackrel{\rightharpoonup }{B}\right|}\end{array}$

The angle can be calculated as:

$\begin{array}{rcl}\varphi & =& {\sin }^{-1}\left(\frac{\left|\stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{B}\right|}{\left|\stackrel{\rightharpoonup }{v}\right|\left|\stackrel{\rightharpoonup }{B}\right|}\right)\\ & =& {\sin }^{-1}\left(\frac{3.78}{61.6\times 61.6\times {10}^{-3}}\right)\\ & =& {84}^o\end{array}$

The angle between $\stackrel{\rightharpoonup }{v}$ and$\stackrel{\rightharpoonup }{B}$ is 84^o .

Step 4:(b) Change of electron’s speed with its time

The magnetic field changes the direction of motion of the particle, but it can’t change its speed, so the speed does not change with time.

Step 5:(c) Change of angle with its time

From the above diagram, we can see that the angle between velocity and magnetic field does not change with time. As the particle changes its path according to the magnetic field, the angle between them remains constant.

Step 6:(d) Calculation of the radius of the helical path

If v is the speed of the electron, then the speed of the electron along the plane perpendicular to the magnetic field will be $v\sin \varphi$, as the angle between velocity and field is given by $\varphi$.

We know that:

role="math" localid="1662723909566" $\begin{array}{rcl}qvB& =& \frac{m{v}^2}{r}\\ qB& =& \frac{mv}{r}\\ r& =& \frac{mv}{qB}\\ r& =& \frac{m\times v\sin \varphi }{qB}\\ & =& \frac{9.1\times {10}^{-31}\times 62\times \sin {84}^o}{1.6\times {10}^{-19}\times 61.6\times {10}^{-3}}\\ & =& 5.7\times {10}^{-9}m\end{array}$

The radius of the helical path is $\begin{array}{rcl}& & 5.7\times {10}^{-9}m\end{array}$.