• When a particle of mass $m$ and charge $q$ is accelerated from rest through potential difference $V$:
  $$q V = \frac{1}{2} m u^2 \implies \frac{q}{m} = \frac{u^2}{2V}$$
• In a perpendicular uniform magnetic field $B$, the magnetic Lorentz force provides the necessary centripetal force:
  $$q u B = \frac{m u^2}{r} \implies r = \frac{m u}{q B}$$
• Substituting $\frac{m}{q} = \frac{2V}{u^2}$:
  $$r = \frac{u}{B} \left(\frac{2V}{u^2}\right) = \frac{2V}{B u}$$
• Since $B$ is constant, the radius is directly proportional to $V/u$:
  $$r \propto \frac{V}{u}$$

The magnetic field produced by an infinitely long straight wire carrying current $I_1$ at a distance $d$ is:

$$B_1 = \frac{\mu_0 I_1}{2\pi d}$$

The magnetic force exerted on a segment of length $L$ of the second parallel conductor carrying current $I_2$ is:

$$F = I_2 L B_1 = \frac{\mu_0 I_1 I_2 L}{2\pi d}$$

The condition for a balanced Wheatstone bridge is:

$$\frac{P}{Q} = \frac{R}{S_{\text{eff}}}$$

Given $P = 3\,\Omega, Q = 3\,\Omega, R = 3\,\Omega$:

$$\frac{3}{3} = \frac{3}{S_{\text{eff}}} \implies S_{\text{eff}} = 3\,\Omega$$

The arm has resistance $S = 4\,\Omega$. To reduce its equivalent resistance to $S_{\text{eff}} = 3\,\Omega$, a shunt resistance $S'$ is connected in parallel with $S$:

$$\frac{1}{S_{\text{eff}}} = \frac{1}{S} + \frac{1}{S'}\implies \frac{1}{3} = \frac{1}{4} + \frac{1}{S'}$$ $$\frac{1}{S'} = \frac{1}{3} - \frac{1}{4} = \frac{4 - 3}{12} = \frac{1}{12} \implies S' = 12\,\Omega$$

The radius of the circular path of a charged particle of mass $m$, charge $q$, and kinetic energy $K$ moving perpendicular to a magnetic field $B$ is:

$$r = \frac{m v}{q B} = \frac{\sqrt{2 m K}}{q B}$$

For a proton ($p$) and an $\alpha$-particle ($\alpha$):

• $m_p = m$, $q_p = e$
• $m_\alpha = 4m$, $q_\alpha = 2e$

Taking the ratio of their radii for identical kinetic energy $K$ and field $B$:

$$\frac{r_p}{r_\alpha} = \frac{\sqrt{m_p}}{q_p} \times \frac{q_\alpha}{\sqrt{m_\alpha}} = \frac{\sqrt{m}}{e} \times \frac{2e}{\sqrt{4m}} = 1 \times \frac{2}{2} = 1$$