1. **Problem statement:** A constant current flows through a long straight vertical wire. A Hall probe is placed at a distance $r$ from the wire's center. (i) Explain why rotating the Hall probe about the horizontal axis XY causes the Hall voltage to vary between a maximum positive and a maximum negative value. (ii) Given data of maximum Hall voltage $V_H$ at different distances $r$, analyze the relationship between $V_H$ and $r$. 2. **Relevant formula and concepts:** The magnetic field $B$ around a long straight current-carrying wire at distance $r$ is given by Ampère's law: $$B = \frac{\mu_0 I}{2 \pi r}$$ where $I$ is the current and $\mu_0$ is the permeability of free space. The Hall voltage $V_H$ is proportional to the magnetic field component perpendicular to the Hall probe surface: $$V_H \propto B \sin \theta$$ where $\theta$ is the angle between the magnetic field direction and the normal to the Hall probe. 3. **Step (i) explanation:** - The magnetic field lines around the wire form concentric circles perpendicular to the wire. - When the Hall probe is rotated about the horizontal axis XY, the angle $\theta$ between the magnetic field and the probe's sensitive axis changes. - At $\theta = 90^\circ$, $\sin \theta = 1$, so $V_H$ is maximum positive. - At $\theta = 270^\circ$, $\sin \theta = -1$, so $V_H$ is maximum negative. - Thus, the Hall voltage varies sinusoidally between positive and negative maxima as the probe rotates. 4. **Step (ii) analysis of data:** - The data shows $V_H$ decreases as $r$ increases. - Since $V_H \propto B \propto \frac{1}{r}$, we expect an inverse relationship. - To verify, plot $V_H$ versus $\frac{1}{r}$ or fit $V_H = \frac{k}{r}$ for some constant $k$. - For example, at $r=1.0$ cm, $V_H=0.290$ V, so $k = V_H \times r = 0.290 \times 1.0 = 0.290$. - Check at $r=2.0$ cm: predicted $V_H = \frac{0.290}{2.0} = 0.145$ V, close to measured 0.140 V. - This confirms the inverse proportionality. **Final conclusion:** (i) The Hall voltage varies between positive and negative maxima due to the sinusoidal dependence on the angle between the magnetic field and the Hall probe's sensitive axis during rotation. (ii) The maximum Hall voltage $V_H$ is inversely proportional to the distance $r$ from the wire, consistent with the magnetic field dependence $B \propto \frac{1}{r}$.