Question

A particle of mass $m$ and charge $q$ moving with a velocity $\vec{v} = v_0(\hat{i} + \hat{j} - \hat{k})$ is placed in a uniform magnetic field $\vec{B} = B_0(\hat{i} + \hat{j} + \hat{k})$. It executes a helical trajectory of radius $r$ and pitch $p$. Which of the following options is correct?

• A. $r = \frac{2\sqrt{2}mv_0}{3qB_0}$ and $p = \frac{2\pi mv_0}{3qB_0}$
• B. $r = \frac{mv_0}{3qB_0}$ and $p = \frac{2\pi mv_0}{3qB_0}$
• C. $r = \frac{2\sqrt{2}mv_0}{3qB_0}$ and $p = \frac{4\sqrt{2}\pi mv_0}{3qB_0}$
• D. $r = \frac{2\pi mv_0}{3qB_0}$ and $p = \frac{2\sqrt{2}mv_0}{3qB_0}$

Hint:

Notice that the denominator for both $r$ and $p$ contains $3qB_0$.
Evaluating $v_{\parallel} = v_0/\sqrt{3}$ quickly shows that the pitch $p \propto v_{\parallel}/B \propto v_0/3B_0$, which immediately narrows down the choices.
This vector projection trick is highly effective for competitive exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This problem asks for the parameters (radius and pitch) of the helical trajectory of a charged particle moving obliquely relative to a uniform magnetic field.
Step 2: Key Formulas and Approach:
1. Resolve velocity $\vec{v}$ into parallel ($v_{\parallel}$) and perpendicular ($v_{\perp}$) components relative to the magnetic field direction.
2. Magnitude of magnetic field:
\[ B = |\vec{B}| = \sqrt{3} B_0 \]
3. Helical radius $r$:
\[ r = \frac{m v_{\perp}}{q B} \]
4. Pitch of the helix $p$:
\[ p = v_{\parallel} T = v_{\parallel} \left( \frac{2\pi m}{q B} \right) \]
Step 3: Detailed Explanation:

Let us first find the magnitude of the magnetic field:
\[ B = |\vec{B}| = B_0 \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} B_0 \]

Let $\hat{b}$ be the unit vector in the direction of the magnetic field:
\[ \hat{b} = \frac{\vec{B}}{B} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} \]

The parallel component of the velocity, $v_{\parallel}$, is:
\[ v_{\parallel} = \vec{v} \cdot \hat{b} = v_0 (\hat{i} + \hat{j} - \hat{k}) \cdot \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}} \]
\[ v_{\parallel} = \frac{v_0}{\sqrt{3}} \left( 1 + 1 - 1 \right) = \frac{v_0}{\sqrt{3}} \]

The square of the magnitude of the total velocity is:
\[ v^2 = v_0^2 \left( 1^2 + 1^2 + (-1)^2 \right) = 3 v_0^2 \]

The perpendicular velocity component $v_{\perp}$ satisfies:
\[ v_{\perp}^2 = v^2 - v_{\parallel}^2 = 3 v_0^2 - \frac{v_0^2}{3} = \frac{8}{3} v_0^2 \]
\[ v_{\perp} = \sqrt{\frac{8}{3}} v_0 = \frac{2\sqrt{2}}{\sqrt{3}} v_0 \]

Using the formula for the radius $r$ of helical motion:
\[ r = \frac{m v_{\perp}}{q B} = \frac{m \left( \frac{2\sqrt{2}}{\sqrt{3}} v_0 \right)}{q \left( \sqrt{3} B_0 \right)} = \frac{2\sqrt{2} m v_0}{3 q B_0} \]

The time period $T$ for one complete revolution is:
\[ T = \frac{2\pi m}{q B} = \frac{2\pi m}{q \sqrt{3} B_0} \]

The pitch $p$ of the helix is the distance traveled along the magnetic field during one time period:
\[ p = v_{\parallel} T = \left( \frac{v_0}{\sqrt{3}} \right) \left( \frac{2\pi m}{q \sqrt{3} B_0} \right) = \frac{2\pi m v_0}{3 q B_0} \]

Step 4: Final Answer:
The radius is $r = \frac{2\sqrt{2}mv_0}{3qB_0}$ and the pitch is $p = \frac{2\pi mv_0}{3qB_0}$, corresponding to Option (A).