Rod in magnetic field

C1  4.50 A weightless rod of a length $2R$ is placed perpendicular to a uniform magnetic field $\vec B$. Two identical small balls of mass $m$ and charge $q$ each are attached at the rod ends. Let us direct $z$-axis along the magnetic field and place the origin at the rod center. The balls are given the same initial velocity $v$ but in opposite directions so that one of the velocities is precisely in the $z$-direction. What are the maximum coordinates $z_\mathrm {max}$ of the balls? Express your answer in terms of $q, B, m, v$, and $R$. Find the magnitude of the ball accelerations at this moment and express your answer in terms of $q, B, m, v, R$, and $z_\mathrm{max}$.

The equations of motion of the centre of mass of balls is:

\[ 2m\dot{\vec{v}}_C = q \left[ \left( \vec{v}_1 + \vec{v}_2 \right) \times \vec{B} \right] = q \left[ \vec{v}_C \times \vec{B} \right],\]

where $\vec{v}_C$ is the velocity of the centre of mass, $\vec{v}_1$ and $\vec{v}_2$ are velocities of balls. Notice that $v_C=0$ at the start of motion then $v_C$ remains equal to $0$ always.
The equation for the rotation of the ball about the centre of mass is:
\[ \frac{d\vec{L}}{dt} = q \left[ \vec{R} \times \left[ \vec{v} \times \vec{B} \right] \right]. \]

Now let's use the formula of a triple vector product:
\[ \left[ \vec{R} \times \left[ \vec{v} \times \vec{B} \right] \right] =\vec{v} \left( \vec{R} \cdot \vec{B} \right) - \vec{B} \left( \vec{R} \cdot \vec{v} \right) . \]

So balls move around the sphere with the center in the middle of rod. It means that $\vec{R} \bot \vec{v}$ then $\vec{R} \cdot \vec{v} = 0$. Hence

\[ \frac{d\vec{L}}{dt} = q\vec{v} \left( \vec{R} \cdot \vec{B} \right) \]

Firstly, $\vec{B}$ directs along $z$-axis, then $\left( \vec{R} \cdot \vec{B} \right)=Bz$, where $z$ is a coordinat of the ball. Secondly, let look at projection of vector equation onto $z$-axis:

\[ \frac{dL_z}{dt} = qv_zBz = qBz\frac{dz}{dt} .\]

Multiplying to $dt$ and integrating gives us this equation:

\[ L_z = \frac{qBz^2}{2} + C,\]

where $C=0$ because $L_z=0$ at the start of the motion when $z=0$.
When $z$ is maximum $v_z = \dot{z}=0$. It means, that $L=L_z=mur$, where $r$ is a distance between the ball and $z$-axis, $u$ is the velocity of the ball at this moment. Recall that the ball is moving around the sphere then $r^2 + z^2 = R^2$.

The Lorenz force and the contact ball-rod force don't do the work. It means the kinetic energy of the system is constant. Together with $\vec{v}_C=0$, it gives that velocities of the balls don't change i.e. $u=v$.

To summarize, $L_z=mv\sqrt{R^2-z_\text{max}^2}$.
Let's combine the results of the two last blocks as a biquadratic equation for $z_\text{max}$:

\[ \left( \frac{qB}{2mv} \right)^2 z_\text{max}^4 + z_\text{max}^2 - R^2 = 0. \]

Solution is:

\[ z_\text{max} = \frac{\sqrt{2}mv}{qB} \sqrt{\sqrt{1+ \left(\frac{qBR}{mv}\right)^2 }-1 } \]
This result can be obtained in a faster way by those who are familiar with the generalized momentum of a charged particle in a magnetic field,
$$\vec{\mathcal{P}} =m\vec v+q\vec A,$$
where $\vec A$ is the vector potential which is defined by the condition $\mathrm{curl}\vec{A}=\vec{B}$. We can choose $\vec A = \frac 12\hat \tau Br$, where $r$ denotes the distance from the $z$-axis, and $\hat\tau$ stands for the unit vector perpendicular both to the radius vector and to the $z$-axis. This is a vector field obeying cylindrical symmetry around the $z$-axis and because of that, the corresponding generalized momentum is conserved, $$\mathcal L_z =\vec R\times\vec {\mathcal P}\cdot \hat z=mrv_\perp+\frac 12qBr^2\equiv \frac 12qBR^2;$$
here $v_\perp$ denotes the velocity component perpendicular to the $z$-axis. Noting that at the topmost position of the ball's trajectory, $v_\perp = v$ due to the fact that the speed is constant and horizontal (i.e. in $x-y$-plane), we obtain a quadratic equation for $\rho=\frac rR$:
$$\rho^2+2\kappa\rho-1=0,\;\;\;\kappa\equiv\frac {mv}{qBR}.$$
We select the positive root $\rho=\sqrt{\kappa^2+1}-\kappa$. Now we can write the final answer as $$z_{\max}=R\sqrt{1-\rho^2}=R\kappa\sqrt {2}\sqrt{\sqrt{\kappa^{-2}+1}-1}$$ which coincides with the previous result.
Consider three orthogonal unit vectors: 1-st is along with the velocity, 2-nd along with direction from ball to the middle of the rod, and 3-rd is perpendicular to the other two.

As we found earlier that $v=\text{const}$, so $a_1 = 0$. Further, $a_2$ coinsides with $v^2/R$ because ball is moving around sphere. Finally, the projection of 2-nd Newton law onto the 3-rd axis is: $ma_3 = qvB\dfrac{z_\text{max}}{R}$. Hence
\[ a = \sqrt{ \left(\frac{v^2}{R}\right)^2 + \left( \frac{qvBz_\text{max}}{mR} \right)^2 } \]