Magnetism — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Magnetism (Multi-step)

• Charged particle relations: $r=\dfrac{mv}{qB}=\dfrac{\sqrt{2mK}}{qB}=\dfrac{\sqrt{2mqV}}{qB}$. So for equal KE, $r\propto\dfrac{\sqrt m}{q}$; for equal accelerating voltage, $r\propto\dfrac{\sqrt{m/q}}{1}$.
• Helix: pitch $=v_\parallel T=(v\cos\theta)\dfrac{2\pi m}{qB}$. Cyclotron max energy: $E=\dfrac{q^2B^2R^2}{2m}$.
• Geometry: field at the centre of a square loop of side $a$ is $\dfrac{2\sqrt2\,\mu_0 I}{\pi a}$; a rotating charge $Q$ (radius $R$, angular speed $\omega$) has moment $\dfrac{Q\omega R^2}{2}$.
• Meter conversions: ammeter shunt $S=\dfrac{I_g G}{I-I_g}$; voltmeter series $R=\dfrac{V}{I_g}-G$.
• Earth: $B=\sqrt{B_H^2+B_V^2}=\dfrac{B_H}{\cos\delta}$.

Worked mini-example. A proton and an alpha particle accelerated through the same potential enter the same field. $r=\dfrac{\sqrt{2mqV}}{qB}\propto\sqrt{\dfrac{m}{q}}$; with $\dfrac{m_\alpha}{q_\alpha}=\dfrac{4}{2}=2$ times that of the proton, $r_\alpha=\sqrt2\,r_p$.