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).
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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 \([ k ] = \mathrm { MLT } ^ { - 1 }\).
- Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
- 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]
- 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