2 A solenoid is a device formed by winding a wire tightly around a hollow cylinder so that the wire forms (approximately) circular loops along the cylinder (see diagram).\\
\includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-2_161_691_1681_246}

When the wire carries an electrical current a magnetic field is created inside the solenoid which can cause a particle which is moving inside the solenoid to accelerate.

A student is carrying out experiments on particles moving inside solenoids. His professor suggests that, for a particle of mass $m$ moving with speed $v$ inside a solenoid of length $h$, the acceleration $a$ of the particle can be modelled by a relationship of the form $a = \mathrm { km } ^ { \alpha } \mathrm { v } ^ { \beta } \mathrm { h } ^ { \gamma }$, where $k$ is a constant. The professor tells the student that $[ k ] = \mathrm { MLT } ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Use dimensional analysis to find $\alpha , \beta$ and $\gamma$.
\item The mass of an electron is $9.11 \times 10 ^ { - 31 } \mathrm {~kg}$ and the mass of a proton is $1.67 \times 10 ^ { - 27 } \mathrm {~kg}$.

For an electron and a proton moving inside the same solenoid with the same speed, use the model to find the ratio of the acceleration of the electron to the acceleration of the proton. [3]
\item The professor tells the student that $a$ also depends on the number of turns or loops of wire, $N$, that the solenoid has.

Explain why dimensional analysis cannot be used to determine the dependence of $a$ on $N$. [1
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2019 Q2 [10]}}